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In [[combinatorics]], a branch of [[mathematics]], a '''matroid''' ({{IPAc-en|icon|ˈ|m|eɪ|t|r|ɔɪ|d}}) or '''independence structure''' is a structure that captures and generalizes the notion of [[linear independence]] in [[vector space]]s.
{{Millennium Problems}}
The '''Hodge conjecture''' is a major unsolved problem in [[algebraic geometry]] that relates the [[algebraic topology]] of a [[non-singular]] [[complex number|complex]] [[algebraic variety]] and the subvarieties of that variety. More specifically, the conjecture says that certain [[de Rham cohomology]] classes are algebraic, that is, they are sums of [[Poincaré duality|Poincaré duals]] of the [[homology class]]es of subvarieties. It was formulated by the Scottish mathematician [[William Vallance Douglas Hodge]] as a result of a work in between 1930 and 1940 to enrich the description of de Rham cohomology to include extra structure that is present in the case of complex algebraic varieties. It received little attention before Hodge presented it in an address during the 1950 [[International Congress of Mathematicians]], held in Cambridge, [[Massachusetts]], U.S. The Hodge conjecture is one of the [[Clay Mathematics Institute]]'s [[Millennium Prize Problems]], with a prize of $1,000,000 for whoever can prove or disprove the Hodge conjecture using "some argument".


There are many equivalent ways to define a matroid, a phenomenon sometimes called [[cryptomorphism]].  Significant definitions of matroid include those in terms of independent sets, bases, circuits, closed sets or flats, closure operators, and rank functions.
== Motivation ==
Let ''X'' be a [[compact space|compact]] [[complex manifold]] of complex dimension ''n''. Then ''X'' is an [[orientable]] [[smooth manifold]] of real dimension 2''n'', so its [[cohomology]] groups lie in degrees zero through 2''n''.  Assume ''X'' is a [[Kähler manifold]], so that there is a decomposition on its cohomology with complex coefficients:


Matroid theory borrows extensively from the terminology of [[linear algebra]] and [[graph theory]], largely because it is the abstraction of various notions of central importance in these fields.
:<math>H^k(X, \mathbf{C}) = \bigoplus_{p+q=k} H^{p,q}(X),\,</math>


==Definition==<!-- [[Hereditary property (matroid)]] redirects to this section title-->
where ''H<sup>p, q</sup>''(''X'') is the subgroup of cohomology classes which are represented by [[harmonic form]]s of type (''p'', ''q''). That is, these are the cohomology classes represented by [[differential form]]s which, in some choice of local coordinates ''z''<sub>1</sub>, ..., ''z<sub>n</sub>'', can be written as a [[harmonic function]] times
There are many equivalent ([[Cryptomorphism|cryptomorphic]]) ways to define a (finite) matroid.
:<math>dz_{i_1} \wedge \cdots \wedge dz_{i_p} \wedge d\bar z_{j_1} \wedge \cdots \wedge d\bar z_{j_q}.</math>
(See [[Hodge theory]] for more details.) Taking wedge products of these harmonic representatives corresponds to the [[cup product]] in cohomology, so the cup product is compatible with the Hodge decomposition:


===Independent sets===
:<math>\cup : H^{p,q}(X) \times H^{p',q'}(X) \rightarrow H^{p+p',q+q'}(X).\,</math>
In terms of independence, a finite matroid <math>M</math> is a pair <math>(E,\mathcal{I})</math>, where <math>E</math> is a [[finite set]] (called the '''ground set''') and <math>\mathcal{I}</math> is a [[family of sets|family]] of [[subset]]s of <math>E</math> (called the '''independent sets''') with the following properties:
# The [[empty set]] is independent, i.e., <math>\emptyset\in\mathcal{I}</math>. Alternatively, at least one subset of <math>E</math> is independent, i.e., <math>\mathcal{I}\neq\emptyset</math>.
# Every subset of an independent set is independent, i.e., for each <math>A'\subset A\subset E</math>, if <math>A\in\mathcal{I}</math> then <math>A'\in\mathcal{I}</math>. This is sometimes called the '''hereditary property'''.
# If <math>A</math> and <math>B</math> are two independent sets of <math>\mathcal{I}</math> and <math>A</math> has more elements than<math>B</math>, then there exists an element in <math>A</math> that when added to <math>B</math> gives a larger independent set.  This is sometimes called the '''augmentation property''' or the '''independent set exchange property'''.


The first two properties define a combinatorial structure known as an [[independence system]].
Since ''X'' is a compact oriented manifold, ''X'' has a [[fundamental class]].


===Bases and circuits===
Let ''Z'' be a complex submanifold of ''X'' of dimension ''k'', and let ''i'' : ''Z'' → ''X'' be the inclusion map.  Choose a differential form α of type (''p'', ''q'').  We can integrate α over ''Z'':
A subset of the ground set <math>E</math> that is not independent is called '''dependent'''.  A maximal independent set—that is, an independent set which becomes dependent on adding any element of <math>E</math>—is called a '''basis''' for the matroid. A '''circuit''' in a matroid <math>M</math> is a minimal dependent subset of <math>E</math>—that is, a dependent set whose proper subsets are all independent.  The terminology arises because the circuits of [[graphic matroid]]s are cycles in the corresponding graphs.


The dependent sets, the bases, or the circuits of a matroid characterize the matroid completely: a set is independent if and only if it is not dependent, if and only if it is a subset of a basis, and if and only if it does not contain a circuit. The collection of dependent sets, or of bases, or of circuits each has simple properties that may be taken as axioms for a matroid.  For instance, one may define a matroid <math>M</math> to be a pair <math>(E,\mathcal{B})</math>, where <math>E</math> is a finite set as before and <math>\mathcal{B}</math> is a collection of subsets of <math>E</math>, called "bases", with the following properties:
:<math>\int_Z i^*\alpha.\!\,</math>
# <math>\mathcal{B}</math> is nonempty.
# If <math>A</math> and <math>B</math> are distinct members of <math>\mathcal{B}</math> and <math>a\in A\setminus B</math>, then there exists an element <math>b\in B\setminus A</math> such that <math>A\setminus\{a\}\cup\{b\}\in\mathcal{B}</math>.  (Here the backslash symbol stands for the [[Complement (set theory)|difference of sets]]. This property is called the '''basis exchange property'''.)
It follows from the basis exchange property that no member of <math>\mathcal{B}</math> can be a proper subset of another.


===Rank functions===
To evaluate this integral, choose a point of ''Z'' and call it 0.  Around 0, we can choose local coordinates ''z''<sub>1</sub>, ..., ''z<sub>n</sub>'' on ''X'' such that ''Z'' is just ''z''<sub>''k'' + 1</sub> = ... = ''z<sub>n</sub>'' = 0.  If ''p'' > ''k'', then α must contain some ''dz<sub>i</sub>'' where ''z<sub>i</sub>'' pulls back to zero on ''Z''.  The same is true if ''q'' > ''k''. Consequently, this integral is zero if (''p'', ''q'') ≠ (''k'', ''k'').
It is a basic result of matroid theory, directly analogous to a similar [[basis (linear algebra)|theorem of linear algebra]], that any two bases of a matroid <math>M</math> have the same number of elements.  This number is called the '''rank''' of&nbsp;<math>M</math>. If <math>M</math> is a matroid on <math>E</math>, and <math>A</math> is a subset of <math>E</math>, then a matroid on <math>A</math> can be defined by considering a subset of <math>A</math> to be independent if and only if it is independent in <math>M</math>. This allows us to talk about '''submatroids''' and about the rank of any subset of <math>E</math>.


The '''rank function''' <math>r</math> assigns a non-negative integer to every subset of <math>E</math> and has the following properties:
More abstractly, the integral can be written as the [[cap product]] of the homology class of ''Z'' and the cohomology class represented by α.  By Poincaré duality, the homology class of ''Z'' is dual to a cohomology class which we will call [''Z''], and the cap product can be computed by taking the cup product of [''Z''] and α and capping with the fundamental class of ''X''.  Because [''Z''] is a cohomology class, it has a Hodge decomposition.  By the computation we did above, if we cup this class with any class of type (''p'', ''q'') ≠ (''k'', ''k''), then we get zero. Because ''H''<sup>2''n''</sup>(''X'', '''C''') = ''H<sup>n, n</sup>''(''X''), we conclude that [''Z''] must lie in ''H<sup>n-k, n-k</sup>''(''X'', '''C'''). Loosely speaking, the Hodge conjecture asks:
# If <math>A\subset B\subset E</math>, then <math>r(A)\leq r(B)\leq r(E)</math>. That is, the rank is a [[monotonic function]].
# For any two subsets <math>A</math> and <math>B</math> of <math>E</math>, <math>r(A\cup B)+r(A\cap B)\le r(A)+r(B)</math>. That is, the rank is a [[submodular function]].


These properties can be used as one of the alternative definitions of a finite matroid: if <math>(E,r)</math> satisfies these properties, then the independent sets of a matroid over <math>E</math> can be defined as those subsets <math>A</math> of <math>E</math> with <math>r(A)=|A|</math>.
:''Which cohomology classes in ''H<sup>k, k</sup>''(''X'') come from complex subvarieties ''Z''?''


The difference <math>|A|-r(A)</math> is called the '''nullity''' or '''corank''' of the subset <math>A</math>. It is the minimum number of elements that must be removed from <math>A</math> to obtain an independent set. The nullity of <math>E</math> in <math>M</math> is called the nullity or corank of <math>M</math>.
== Statement of the Hodge conjecture ==
Let:


===Closure operators===
:<math>\operatorname{Hdg}^k(X) = H^{2k}(X, \mathbf{Q}) \cap H^{k,k}(X).</math>
Let <math>M</math> be a matroid on a finite set <math>E</math>, with rank function <math>r</math> as above.  The '''closure'''  <math>\operatorname{cl}(A)</math>  of a subset <math>A</math> of <math>E</math> is the set
:<math>\operatorname{cl}(A) = \Bigl\{x\in E\mid r(A)=r\bigl(A\cup\{x\}\bigr)\Bigr\}</math>.
This defines a [[closure operator]] <math>\operatorname{cl}: \mathcal{P}(E)\mapsto \mathcal{P}(E)</math> where <math>\mathcal{P}</math> denotes the [[power set]], with the following property:
* For all elements <math>a</math>, and <math>b</math> of <math>E</math>  and all subsets <math>Y</math> of <math>E</math>, if <math>a\in\operatorname{cl}(Y\cup b) \setminus Y</math> then <math>b\in\operatorname{cl}(Y\cup a) \setminus Y</math>.
This is sometimes called the '''Mac Lane–Steinitz exchange property''' and may be taken as another definition of matroid: any closure operator on <math>E</math> with this property determines a matroid.


===Flats===
We call this the group of ''Hodge classes'' of degree 2''k'' on ''X''.
A set whose closure equals itself is said to be '''closed''', or a '''flat''' or '''subspace''' of the matroid.  A set is closed if it is [[maximal element|maximal]] for its rank, meaning that the addition of any other element to the set would increase the rank. The closed sets of a matroid are characterized by a covering partition property:
* The whole point set <math>E</math> is closed.
* If <math>S</math> and <math>T</math> are flats, then <math>S\cap T</math> is a flat.
* If <math>S</math> is a flat, then the flats <math>T</math> that [[Covering relation|cover]] <math>S</math> (meaning that <math>T</math> properly contains <math>S</math> but there is no flat <math>U</math> between <math>S</math> and <math>T</math>), partition the elements of&nbsp;<math>E\setminus S</math>.


The class <math>\mathcal{L}(M)</math> of all flats, [[partially ordered set|partially ordered]] by set inclusion, forms a [[matroid lattice]].
The modern statement of the Hodge conjecture is:
Conversely, every matroid lattice <math>L</math> forms a matroid over its set <math>E</math> of [[Atom_(order_theory)|atoms]] under the following closure operator:  for a set <math>S</math> of atoms with join <math>\vee S</math>,
:<math>\operatorname{cl}(S) = \{ x\in E\mid x\le\vee S \}</math>.
The flats of this matroid correspond one-for-one with the elements of the lattice; the flat corresponding to lattice element <math>y</math> is the set
:<math>\{ x\in E\mid x\le y\}</math>.
Thus, the lattice of flats of this matroid is naturally isomorphic to&nbsp;<math>L</math>.


== Examples ==
::'''Hodge conjecture.'''  Let ''X'' be a non-singular complex projective manifold.  Then every Hodge class on ''X'' is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of ''X''.
===Uniform matroids===


Let ''E'' be a finite set and ''k'' a [[natural number]].  One may define a matroid on ''E'' by taking every ''k''-element subset of ''E'' to be a basis.  This is known as the [[uniform matroid]] of rank ''k''.  All uniform matroids of rank at least 2 are simple. The uniform matroid of rank 2 on ''n'' points is called the ''n''-'''point line'''. A matroid is uniform if and only if it has no circuits of size less than the one plus the rank of the matroid. The direct sums of uniform matroids are called [[partition matroid]]s.
A projective complex manifold is a complex manifold which can be embedded in [[complex projective space]].  Because projective space carries a Kähler metric, the [[Fubini–Study metric]], such a manifold is always a Kähler manifold.  By [[Algebraic geometry and analytic geometry#Chow.27s theorem|Chow's theorem]], a projective complex manifold is also a smooth projective algebraic variety, that is, it is the zero set of a collection of homogeneous polynomials.


===Discrete matroids===
=== Reformulation in terms of algebraic cycles ===
Another way of phrasing the Hodge conjecture involves the idea of an algebraic cycle.  An ''[[algebraic cycle]]'' on ''X'' is a formal combination of subvarieties of ''X'', that is, it is something of the form:


A matroid is called '''discrete''' if every element is a loop or a coloop. Equivalently, every proper, non-empty subset of the ground set ''E'' is a separator (loop, coloop, and separator are defined in Additional Terminology below).
: <math>\sum_i c_iZ_i.\,</math>


===Matroids from linear algebra===
The coefficients are usually taken to be integral or rational.  We define the cohomology class of an algebraic cycle to be the sum of the cohomology classes of its components. This is an example of the cycle class map of de Rham cohomology, see [[Weil cohomology]].  For example, the cohomology class of the above cycle would be:


Matroid theory developed mainly out of a deep examination of the properties of independence and dimension in vector spaces.  Matroids from vector spaces are still the main examples.  There are two ways to present them.
:<math>\sum_i c_i[Z_i].\,</math>


* If ''E'' is any finite subset of a [[vector space]] ''V'', then we can define a matroid ''M'' on ''E'' by taking the independent sets of ''M'' to be the [[linear independence|linearly independent]] subsets of ''E''.  We say the set ''E'' '''represents''' ''M''.  Matroids of this kind are called '''vector matroids'''.
Such a cohomology class is called ''algebraic''.  With this notation, the Hodge conjecture becomes:
* A [[matrix (mathematics)|matrix]] ''A'' with entries in a [[field (mathematics)|field]] gives rise to a matroid ''M'' on its set of columns.  The dependent sets of columns in the matroid are those that are linearly dependent as vectors.  This matroid is called the '''column matroid''' of ''A'', and ''A'' is said to '''represent''' ''M''.  Column matroids are just vector matroids under another name, but there are often reasons to favor the matrix representation.  (There is one technical difference: a column matroid can have distinct elements that are the same vector, but a vector matroid as defined above cannot.  Usually this difference is insignificant and can be ignored, but by letting ''E'' be a [[multiset]] of vectors one brings the two definitions into complete agreement.)


A matroid that is equivalent to a vector matroid, although it may be presented differently, is called '''representable'''.  If ''M'' is equivalent to a vector matroid over a [[field (mathematics)|field]] ''F'', then we say ''M'' is '''representable over''' ''F''&nbsp;; in particular, ''M'' is '''real-representable''' if it is representable over the real numbers.  For instance, although a graphic matroid (see below) is presented in terms of a graph, it is also representable by vectors over any field.  A basic problem in matroid theory is to determine whether a given matroid ''M'' is representable over a given field ''F''.  The main results so far are characterizations of binary matroids due to Tutte (1950s), of ternary matroids (representable over the 3-element field) due to Reid and Bixby, and separately to Seymour (1970s), and of quaternary matroids (representable over the 4-element field) due to Geelen, Gerards, and Kapoor (2000).  This is very much an open area.
::Let ''X'' be a projective complex manifold.  Then every Hodge class on ''X'' is algebraic.


A [[regular matroid]] is a matroid that is representable over all possible fields. [[Rota's conjecture]] states that, for every [[finite field]] ''F'', the ''F''-representable matroids have a finite set of [[matroid minor|forbidden minors]].
The assumption in the Hodge conjecture that ''X'' be algebraic (projective complex manifold) cannot be weakened. In 1977 Zucker showed that it is possible to construct a counterexample to the Hodge conjecture as complex tori with analytic rational cohomology of type (p,p), which is not projective algebraic. (see the appendix B: in {{Harvtxt|Zucker|1977}})


===Matroids from graph theory===
== Known cases of the Hodge conjecture ==


A second original source for the theory of matroids is [[graph theory]].
=== Low dimension and codimension ===
The first result on the Hodge conjecture is due to {{Harvtxt|Lefschetz|1924}}.  In fact, it predates the conjecture and provided some of Hodge's motivation.


Every finite graph (or [[multigraph]]) ''G'' gives rise to a matroid as follows: take as ''E'' the set of all edges in ''G'' and consider a set of edges independent if and only if it is a [[tree (graph theory)|forest]]; that is, if it does not contain a [[simple cycle]].  This is called the [[graphic matroid]] of ''G'' and is usually written ''M''(''G''). Every graphic matroid is regular.
::'''Theorem''' ([[Lefschetz theorem on (1,1)-classes]]) Any element of ''H''<sup>2</sup>(''X'', '''Z''') ∩ ''H''<sup>1,1</sup>(''X'') is the cohomology class of a [[divisor (algebraic geometry)|divisor]] on ''X''.  In particular, the Hodge conjecture is true for ''H''<sup>2</sup>.


Other matroids on graphs were discovered subsequently:
A very quick proof can be given using [[sheaf cohomology]] and the [[exponential exact sequence]]. (The cohomology class of a divisor turns out to equal to its first [[Chern class]].)  Lefschetz's original proof proceeded by [[normal function (geometry)|normal function]]s, which were introduced by [[Henri Poincaré]].  However, [[Griffiths transversality theorem]] shows that this approach cannot prove the Hodge conjecture for higher codimensional subvarieties.
*The [[bicircular matroid]] of a graph is defined by calling a set of edges independent if every connected subset contains at most one cycle.
*In any directed or undirected graph ''G'' let ''E'' and ''F'' be two distinguished sets of vertices.  In the set ''E'', define a subset ''U'' to be independent if there are |''U''| vertex-disjoint paths from ''F'' onto ''U''.  This defines a matroid on ''E'' called a [[gammoid]].
*Graphic matroids have been generalized to matroids from [[signed graph]]s, [[gain graph]]s, and [[biased graph]]s.  A graph ''G'' with a distinguished linear class '''''B''''' of cycles, known as a "biased graph" (''G'','''''B'''''), has two matroids, known as the '''frame matroid''' and the '''lift matroid''' of the biased graph.  If every cycle belongs to the distinguished class, these matroids coincide with the cycle matroid of ''G''.  If no cycle is distinguished, the frame matroid is the bicircular matroid of ''G''. A signed graph, whose edges are labeled by signs, and a gain graph, which is a graph whose edges are labeled orientably from a group, each give rise to a biased graph and therefore have frame and lift matroids.
*The [[Laman graph]]s form the bases of the two-dimensional [[rigidity matroid]], a matroid defined in the theory of [[structural rigidity]].


===Transversal matroids===
By the [[Hard Lefschetz theorem]], one can prove:
Given a set of "points", ''E'', and a class ''A'' of subsets of ''E'', a '''[[Transversal (combinatorics)|transversal]]''' of ''A'' is a subset ''S'' of ''E'' such that there is a [[one-to-one function]] ''f'' from ''S'' to ''A'' by which ''x'' belongs to ''f'' (''x'') for each ''x'' in ''S''.  The set of transversals forms the class of independent sets of a matroid, called the '''transversal matroid''' of (''E'', ''A'').  Transversal matroids are a special case of gammoids, and are dual to strict gammoids (gammoids in which the set of elements consists of all vertices in the defining graph).


===Matroids from field extensions===
::'''Theorem.'''  If the Hodge conjecture holds for Hodge classes of degree ''p'', ''p'' &lt; ''n'', then the Hodge conjecture holds for Hodge classes of degree 2''n''&nbsp;&minus;&nbsp;''p''.
A third original source of matroid theory is [[field theory (mathematics)|field theory]].


An [[extension field|extension]] of a field gives rise to a matroid.  Suppose ''F'' and ''K'' are fields with ''K'' containing ''F''.  Let ''E'' be any finite subset of ''K''.  Define a subset ''S'' of ''E'' to be independent if the extension field ''F''[''S''] has [[transcendence degree]] equal to |''S''|.
Combining the above two theorems implies that Hodge conjecture is true for Hodge classes of degree 2''n''&nbsp;&minus;&nbsp;2.  This proves the Hodge conjecture when ''X'' has dimension at most three.


A matroid that is equivalent to a matroid of this kind is called an '''algebraic matroid'''.  The problem of characterizing algebraic matroids is extremely difficult; little is known about it.
The Lefschetz theorem on (1,1)-classes also implies that if all Hodge classes are generated by the Hodge classes of divisors, then the Hodge conjecture is true:


===The Fano matroid===
::'''Corollary.'''  If the algebra
[[Image:fano plane.svg|thumb|right|Fano matroid]]
:: <math>\operatorname{Hdg}^*(X) = \sum_k \operatorname{Hdg}^k(X)\,</math>
: is generated by Hdg<sup>1</sup>(''X''), then the Hodge conjecture holds for ''X''.


Matroids with a small number of elements are often portrayed with a diagram. The dots are the elements of the underlying set, and a curve has been drawn through every rank-2 flat. The diagram shows a rank 3 matroid called the '''Fano matroid''', an example that appeared in the original 1935 paper of [[Hassler Whitney|Whitney]].
===Hypersurfaces===
By the strong and weak [[Lefschetz theorem]], the only non-trivial part of the Hodge conjecture for [[hypersurface]]s is the degree ''m'' part (i.e., the middle cohomology) of a 2''m''-dimensional hypersurface <math>X \subset \mathbf P^{2m+1}</math>. If the degree ''d'' is 2, i.e., ''X'' is a [[quadric]]s, the Hodge conjecture holds for all ''m''. For ''m''=2, i.e., [[fourfold]]s, the Hodge conjecture is known for <math>d \le 5</math>.<ref>James Lewis: ''A Survey of the Hodge Conjecture'', 1991, Example 7.21</ref>


The name arises from the fact that the Fano matroid is the [[projective plane]] of order 2, known as the [[Fano plane]], whose coordinate field is the 2-element field.  This means the Fano matroid is the vector matroid associated to the seven nonzero vectors in a three-dimensional vector space over a [[finite field|field with two elements]].
=== Abelian varieties ===
For most [[abelian variety|abelian varieties]], the algebra Hdg*(''X'') is generated in degree one, so the Hodge conjecture holds.  In particular, the Hodge conjecture holds for sufficiently general abelian varieties, for products of elliptic curves, and for simple abelian varieties {{Citation needed|date=April 2012}}.  However, {{Harvtxt|Mumford|1969}} constructed an example of an abelian variety where Hdg<sup>2</sup>(''X'') is not generated by products of divisor classes.  {{Harvtxt|Weil|1977}} generalized this example by showing that whenever the variety has [[complex multiplication]] by an [[imaginary quadratic field]], then <Hdg<sup>2</sup>(''X'') is not generated by products of divisor classes.  {{Harvtxt|Moonen|Zarhin|1999}} proved that in dimension less than 5, either Hdg*(''X'') is generated in degree one, or the variety has complex multiplication by an imaginary quadratic field.  In the latter case, the Hodge conjecture is only known in special cases.


It is known from [[projective geometry]] that the Fano matroid is not representable by any set of vectors in a real or complex vector space (or in any vector space over a field whose [[characteristic (algebra)|characteristic]] differs from 2).
== Generalizations ==


A less famous example is the '''anti-Fano matroid''', defined in the same way as the Fano matroid with the exception that the circle in the above diagram is missing. The anti-Fano matroid is representable over a field if and only if its characteristic differs from 2.
=== The integral Hodge conjecture ===
Hodge's original conjecture was:


The direct sum of a Fano matroid and an anti-Fano matroid is the simplest example for a matroid which is not representable over any field.
::'''Integral Hodge conjecture.'''  Let ''X'' be a projective complex manifold.  Then every cohomology class in ''H''<sup>2''k''</sup>(''X'', '''Z''') ∩ ''H<sup>k, k</sup>''(''X'') is the cohomology class of an algebraic cycle with integral coefficients on ''X''.


===Non-examples===
This is now known to be false.  The first counterexample was constructed by {{Harvtxt|Atiyah|Hirzebruch|1961}}.  Using [[K-theory]], they constructed an example of a torsion Hodge class, that is, a Hodge class α such that for some positive integer ''n'', ''n'' α = 0.  Such a cohomology class cannot be the class of a cycle.  {{Harvtxt|Totaro|1997}} reinterpreted their result in the framework of [[cobordism]] and found many examples of torsion classes.


[[Image:Maximal three-colorable.png|right|frame|Two maximal three-colorings of different sizes. The one on the left cannot be enlarged because the only remaining vertex is already adjacent to all three colors.]]
The simplest adjustment of the integral Hodge conjecture is:
On the other hand, consider this '''non-example''': let ''E'' be a set of pairs (''v'',''c'') where ''v'' ranges over the vertices of a graph and ''c'' ranges over the set {red, blue, yellow}. Let the independent sets be the sets of pairs that associate only one color with each vertex and do not associate the same color with two adjacent vertices; that is, they represent valid [[graph coloring]]s. The empty set is a valid three-coloring, and any subset of a valid three-coloring is a valid three-coloring, but the exchange property does not hold, because it's possible to have two maximal three-colored subgraphs of different sizes, as shown to the right. It's no surprise that this is not a matroid, since if it were, it would give us a greedy algorithm for the NP-complete 3-coloring problem, showing P = NP.


== Basic constructions ==
::'''Integral Hodge conjecture modulo torsion.'''  Let ''X'' be a projective complex manifold.  Then every cohomology class in ''H''<sup>2''k''</sup>(''X'', '''Z''') ∩ ''H<sup>k,k</sup>''(''X'') is the sum of a torsion class and the cohomology class of an algebraic cycle with integral coefficients on ''X''.
There are some standard ways to make new matroids out of old ones.


===Duality===
Equivalently, after dividing ''H''<sup>2''k''</sup>(''X'', '''Z''') ∩ ''H<sup>k,k</sup>''(''X'') by torsion classes, every class is the image of the cohomology class of an integral algebraic cycle.  This is also false.  {{Harvtxt|Kollár|1992}} found an example of a Hodge class α which is not algebraic, but which has an integral multiple which is algebraic.
If ''M'' is a finite matroid, we can define the [[dual matroid]] ''M''* by taking the same underlying set and calling a set a ''basis'' in ''M''* if and only if its complement is a basis in ''M''.  It is not difficult to verify that ''M''* is a matroid and that the dual of ''M''* is ''M''.


The dual can be described equally well in terms of other ways to define a matroid.  For instance:
=== The Hodge conjecture for Kähler varieties ===
A natural generalization of the Hodge conjecture would ask:


* A set is independent in ''M''* if and only if its complement spans ''M''.
::'''Hodge conjecture for Kähler varieties, naive version.''' Let ''X'' be a complex Kähler manifold. Then every Hodge class on ''X'' is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of ''X''.
* A set is a circuit of ''M''* if and only if its complement is a coatom in ''M''.
* The rank function of the dual is r*(S) = |S|- r(E) + r(E\S).


According to a matroid version of [[Kuratowski's theorem]], the dual of a graphic matroid ''M'' is a graphic matroid if and only if ''M'' is the matroid of a [[planar graph]]. In this case, the dual of ''M'' is the matroid of the [[dual graph]] of ''G''. A the dual of a vector matroid representable over a particular field ''F'' is also representable over ''F''. The dual of a transversal matroid is a strict gammoid and vice versa.
This is too optimistic, because there are not enough subvarieties to make this work. A possible substitute is to ask instead one of the two following questions:


===Minors===
::'''Hodge conjecture for Kähler varieties, vector bundle version.''' Let ''X'' be a complex Kähler manifold.  Then every Hodge class on ''X'' is a linear combination with rational coefficients of Chern classes of vector bundles on ''X''.
If ''M'' is a matroid with element set ''E'', and ''S'' is a subset of ''E'', the '''restriction''' of ''M'' to ''S'', written ''M''&nbsp;|''S'', is the matroid on the set ''S'' whose independent sets are the independent sets of ''M'' that are contained in ''S''.  Its circuits are the circuits of ''M'' that are contained in ''S'' and its rank function is that of ''M'' restricted to subsets of ''S''. In linear algebra, this corresponds to restricting to the subspace generated by the vectors in ''S''.
::'''Hodge conjecture for Kähler varieties, coherent sheaf version.''' Let ''X'' be a complex Kähler manifold.  Then every Hodge class on ''X'' is a linear combination with rational coefficients of Chern classes of coherent sheaves on ''X''.
The dual operation of restriction is contraction. If ''T'' is a subset of ''E'', the '''contraction''' of ''M'' by ''T'', written ''M''/''T'', is the matroid on the underlying set ''E &minus; T'' whose rank function is <math>r'(A) = r(A \cup T) - r(T).</math> In linear algebra, this corresponds to looking at the quotient space by the linear space generated by the vectors in ''T'', together with the images of the vectors in ''E - T''.


A matroid ''N'' that is obtained from ''M'' by a sequence of restriction and contraction operations is called a [[matroid minor|minor]] of ''M''.  We say ''M'' '''contains''' ''N'' '''as a minor'''. Many important families of matroids may be characterized by the [[minimal element|minor-minimal]] matroids that do not belong to the family; these are called '''forbidden minors'''.
{{Harvtxt|Voisin|2002}} proved that the Chern classes of coherent sheaves give strictly more Hodge classes than the Chern classes of vector bundles and that the Chern classes of coherent sheaves are insufficient to generate all the Hodge classes.  Consequently, the only known formulations of the Hodge conjecture for Kähler varieties are false.


===Sums and unions===
=== The generalized Hodge conjecture ===
Let ''M'' be a matroid with an underlying set of elements ''E'', and let ''N'' be another matroid on an underlying set ''F''.
Hodge made an additional, stronger conjecture than the integral Hodge conjecture.  Say that a cohomology class on ''X'' is of ''level c'' if it is the pushforward of a cohomology class on a ''c''-codimensional subvariety of ''X''. The cohomology classes of level at least ''c'' filter the cohomology of ''X'', and it is easy to see that the ''c''th step of the filtration ''N<sup>c</sup>'' ''H<sup>k</sup>''(''X'', '''Z''') satisfies
The '''direct sum''' of matroids ''M'' and ''N'' is the matroid whose underlying set is the [[disjoint union]] of ''E'' and ''F'', and whose independent sets are the disjoint unions of an independent set of ''M'' with an independent set of ''N''.


The '''union''' of ''M'' and ''N'' is the matroid whose underlying set is the union (not the disjoint union) of ''E'' and ''F'', and whose independent sets are those subsets which are the union of an independent set in ''M'' and one in ''N''.  Usually the term "union" is applied when ''E'' = ''F'', but that assumption is not essential.  If ''E'' and ''F'' are disjoint, the union is the direct sum.
:<math>N^cH^k(X, \mathbf{Z}) \subseteq H^k(X, \mathbf{Z}) \cap (H^{k-c,c}(X) \oplus\cdots\oplus H^{c,k-c}(X)).</math>


== Additional terminology ==
Hodge's original statement was:
Let ''M'' be a matroid with an underlying set of elements ''E''.
::'''Generalized Hodge conjecture, Hodge's version.'''  <math>N^cH^k(X, \mathbf{Z}) = H^k(X, \mathbf{Z}) \cap (H^{k-c,c}(X) \oplus\cdots\oplus H^{c,k-c}(X)).</math>
* ''E'' may be called the '''ground set''' of ''M''.  Its elements may be called the '''points''' of ''M''.
{{harvtxt|Grothendieck|1969}} observed that this cannot be true, even with rational coefficients, because the right-hand side is not always a Hodge structure. His corrected form of the Hodge conjecture is:
* A subset of ''E'' '''spans''' ''M'' if its closure is ''E''. A set is said to '''span''' a closed set ''K'' if its closure is ''K''.
::'''Generalized Hodge conjecture.'''  ''N<sup>c</sup>'' ''H<sup>k</sup>''(''X'', '''Q''') is the largest sub-Hodge structure of ''H<sup>k</sup>''(''X'', '''Z''') contained in <math>H^{k-c,c}(X) \oplus\cdots\oplus H^{c,k-c}(X).</math>
* A maximal closed proper subset of ''E'' is called a '''coatom''' or '''copoint''' or '''hyperplane''' of ''M''.  An equivalent definition:  A coatom is a subset of ''E'' that does not span ''M'', but such that adding any other element to it does make a spanning set.
This version is open.
* An element that forms a single-element circuit of ''M'' is called a '''loop'''. Equivalently, an element is a loop if it belongs to no basis.
* An element that belongs to no circuit is called a '''coloop'''. Equivalently, an element is a coloop if it belongs to every basis.
* If a two-element set {''f, g''} is a circuit of ''M'', then ''f'' and ''g'' are '''parallel''' in ''M''.
* A matroid is called '''simple''' if it has no circuits consisting of 1 or 2 elements. That is, it has no loops and no parallel elements. A simple matroid obtained from another matroid ''M'' by deleting all loops and deleting one element from each 2-element circuit until no 2-element circuits remain is called a '''simplification''' of ''M''.
* A union of circuits is sometimes called a '''cycle''' of ''M''.  A cycle is therefore the complement of a flat of the dual matroid.  (This usage conflicts with the common meaning of "cycle" in graph theory.)
* A '''separator''' of ''M'' is a subset ''S'' of ''E'' such that <math>r(S) + r(E-S) = r(M).</math> A '''proper separator''' is a separator that is neither ''E'' nor the empty set.  An '''irreducible separator''' is a separator that contains no other non-empty separator.  The irreducible separators partition the ground set ''E''.
* A matroid which cannot be written as the direct sum of two nonempty matroids, or equivalently which has no proper separators, is called '''connected''' or '''irreducible'''.
* A maximal irreducible submatroid of ''M'' is called a '''component''' of ''M''.  A component is the restriction of ''M'' to an irreducible separator, and contrariwise, the restriction of ''M'' to an irreducible separator is a component.
* A matroid ''M'' is called a '''frame matroid''' if it, or a matroid that contains it, has a basis such that all the points of ''M'' are contained in the lines that join pairs of basis elements.


== Further topics ==
== Algebraicity of Hodge loci ==
 
The strongest evidence in favor of the Hodge conjecture is the algebraicity result of {{Harvtxt|Cattani|Deligne|Kaplan|1995}}. Suppose that we vary the complex structure of ''X'' over a simply connected base. Then the topological cohomology of ''X'' does not change, but the Hodge decomposition does change. It is known that if the Hodge conjecture is true, then the locus of all points on the base where the cohomology of a fiber is a Hodge class is in fact an algebraic subset, that is, it is cut out by polynomial equations.  Cattani, Deligne & Kaplan (1995) proved that this is always true, without assuming the Hodge conjecture.
===Greedy algorithms===
A [[weighted matroid]] is a matroid together with a function from its elements to the nonnegative [[real number]]s. The weight of a subset of elements is defined to be the sum of the weights of the elements in the subset. A [[greedy algorithm]] can be used to find the maximum weight independent set of the matroid, by starting from the empty set and repeatedly adding one element at a time, at each step choosing the maximum weight element among the elements whose addition would preserve the independence of the augmented set.
 
This optimization algorithm may be used to characterize matroids: if a family ''F'' of sets has the property that, no matter how the sets are weighted, the greedy algorithm finds the maximum-weight set in the family, then ''F'' must be the family of independent sets of a matroid.
 
The notion of matroid has been generalized to allow for other types of sets on which greedy algorithms give optimal solutions; see [[greedoid]] for more information.
 
===Matroid intersection===
The [[matroid intersection|intersection]] of two or more matroids is the family of sets that are simultaneously independent in each of the matroids. The problem of finding the largest set, or the maximum weighted set, in the intersection of two matroids can be found in [[polynomial time]], and provides a solution to many other important combinatorial optimization problems. For instance, [[maximum matching]] in [[bipartite graph]]s can be expressed as a problem of intersecting two [[partition matroid]]s. However, finding the largest set in an intersection of three or more matroids is [[NP-complete]].
 
=== Infinite matroids ===
<!-- [[Infinite matroid]] redirects here. -->
The theory of infinite matroids is much more complicated than that of finite matroids and forms a subject of its own.  For a long time, one of the difficulties has been that there were many reasonable and useful definitions, none of which appeared to capture all the important aspects of finite matroid theory.  For instance, it seemed to be hard to have bases, circuits, and duality together in one notion of infinite matroids.
 
The simplest definition of an infinite matroid is to require ''finite rank''; that is, the rank of ''E'' is finite.  This theory is similar to that of finite matroids except for the failure of duality due to the fact that the dual of an infinite matroid of finite rank does not have finite rank.  Finite-rank matroids include any subsets of finite-dimensional vector spaces and of [[Field (mathematics)|field extensions]] of finite [[transcendence degree]].
 
The next simplest infinite generalization is finitary matroids.  A matroid is '''finitary''' if it has the property that
:<math>x \in cl(Y) \Leftrightarrow (\exists Y' \subseteq Y) Y' \text{ is finite and } x \in cl(Y').</math>
Equivalently, every dependent set contains a finite dependent set.
Examples are linear dependence of arbitrary subsets of infinite-dimensional [[vector spaces]] (but not infinite dependencies as in [[Hilbert space|Hilbert]] and [[Banach space]]s), and algebraic dependence in arbitrary subsets of field extensions of possibly infinite transcendence degree.  Again, the class of finitary matroid is not self-dual, because the dual of a finitary matroid is not finitary.
Finitary infinite matroids are studied in [[model theory]], a branch of [[mathematical logic]] with strong ties to [[algebra]].
 
In the late 1960s matroid theorists asked for a more general notion that shares the different aspects of finite matroids and generalizes their duality. Many notions of infinite matroids were defined in response to this challenge, but the question remained open. One of the approaches examined by D.A. Higgs became known as ''B-matroids'' and was studied by Higgs, Oxley and others in the 1960s and 1970s. According to a recent result by Bruhn, Diestel, Kriesell, Pendavingh and Wollan ([[#CITEREFBruhnDiestelKriesellWollan2010|2010]]), it solves the problem: Arriving at the same notion independently, they provided four different systems of axioms – in terms of independence, bases, circuits, closure and rank. The duality of B-matroids generalizes dualities that can be observed in infinite graphs.
 
==Polynomial invariants==
 
There are two especially significant polynomials associated to a finite matroid ''M'' on the ground set ''E''.  Each is a '''matroid invariant''', which means that isomorphic matroids have the same polynomial.
 
===Characteristic polynomial===
 
The '''characteristic polynomial''' of ''M'' (which is sometimes called the '''chromatic polynomial''', although it does not count colorings), is defined to be
:<math>p_M(\lambda) := \sum_{S \subseteq E} (-1)^{|S|}\lambda^{r(M)-r(S)},</math>
or equivalently (as long as the empty set is closed in ''M'') as
:<math>p_M(\lambda) := \sum_A \mu(\emptyset,A) \lambda^{r(M)-r(S)}.</math>
 
When ''M'' is the cycle matroid ''M''(''G'') of a graph ''G'', the characteristic polynomial is a slight transformation of the [[chromatic polynomial]], which is given by χ<sub>''G''</sub>&nbsp;(λ) = λ<sup>c</sup>''p''<sub>''M''(''G'')</sub>&nbsp;(λ), where ''c'' is the number of connected components of ''G''.
 
When ''M'' is the bond matroid ''M''*(''G'') of a graph ''G'', the characteristic polynomial equals the [[Tutte polynomial#Flow polynomial|flow polynomial]] of ''G''.
 
When ''M'' is the matroid of an [[Arrangement of hyperplanes|arrangement]] ''A'' of linear hyperplanes in '''R'''<sup>''n''</sup>, the characteristic polynomial of the arrangement is given by ''p''<sub>''A''</sub>&nbsp;(λ) = λ<sup>n&minus;r(M)</sup>''p''<sub>''M''(''A'')</sub>&nbsp;(λ).
 
===Tutte polynomial===
 
The '''[[Tutte polynomial]]''' of a matroid, ''T''<sub>''M''</sub>&nbsp;(''x'',''y''), generalizes the characteristic polynomial to two variables.  This gives it more combinatorial interpretations, and also gives it the duality property
:<math>T_{M^*}(x,y) = T_M(y,x),</math>
which implies a number of dualities between properties of ''M'' and properties of ''M''&nbsp;*.  One definition of the Tutte polynomial is
:<math>T_M(x,y) = \sum_{S\subseteq E} (x-1)^{r(M)-r(S)}(y-1)^{|S|-r(S)}.</math>
This expresses the Tutte polynomial as an evaluation of the '''corank-nullity''' or '''rank generating polynomial''',
:<math>R_M(x,y) = \sum_{S\subseteq E} v^{r(M)-r(S)}u^{|S|-r(S)}.</math>
 
Another definition is in terms of internal and external activities and a sum over bases.  This, which sums over fewer subsets but has more complicated terms, was Tutte's original definition.
 
The [[Tutte polynomial]] ''T''<sub>''G''</sub>&nbsp; of a graph is the Tutte polynomial ''T''<sub>''M''(''G'')</sub> of its cycle matroid.
 
==Matroid software==
 
Two systems for calculations with matroids are Kingan's [http://userhome.brooklyn.cuny.edu/skingan/software.html Oid] and Hlineny's [http://www.fi.muni.cz/~hlineny/MACEK/ Macek].
 
"Oid" is an open source, interactive, extensible software system for experimenting with matroids.
 
"Macek" is a specialized software system with tools and routines for reasonably efficient combinatorial computations with representable matroids.
 
==History==
 
Matroid theory was introduced by {{harvs|last=Whitney|first=Hassler|authorlink=Hassler Whitney|year=1935|txt}}. It was also independently discovered by [[Takeo Nakasawa]], whose work was forgotten for many years {{harv|Nishimura|Kuroda|2009}}.
 
In his seminal paper, Whitney provided two axioms for independence, and defined any structure adhering to these axioms to be "matroids".
(Although it was perhaps implied, he did not include an axiom requiring at least one subset to be independent.)
His key observation was that these axioms provide an abstraction of "independence" that is common to both graphs and matrices.
Because of this, many of the terms used in matroid theory resemble the terms for their analogous concepts in [[linear algebra]] or [[graph theory]].
 
Almost immediately after Whitney first wrote about matroids, an important article was written by {{harvs|last=Mac Lane|first=Saunders|authorlink=Saunders Mac Lane|year=1936|txt}} on the relation of matroids to projective geometry. A year later, {{harvs|last=van der Waerden|first=Bartel|authorlink=Bartel Leendert van der Waerden|year=1937|txt}} noted similarities between algebraic and linear dependence in his classic textbook on Modern Algebra.
 
In the 1940s [[Richard Rado]] developed further theory under the name "independence systems" with an eye towards [[transversal theory]], where his name for the subject is still sometimes used.
 
In the 1950s [[W. T. Tutte]] became the foremost figure in matroid theory, a position he retained for many years.  His contributions were plentiful, including the characterization of [[binary matroid|binary]], [[regular matroid|regular]], and [[graphic matroid|graphic]] matroids by [[Matroid minor|excluded minors]]; the regular-matroid representability theorem; the theory of chain groups and their matroids; and the tools he used to prove many of his results, the "Path Theorem" and "Homotopy Theorem" (see, e.g., {{harvnb|Tutte|1965}}), which are so complex that later theorists have gone to great trouble to eliminate the necessity of using them in proofs.  (A fine example is [[A. M. H. Gerards]]' short proof ([[#CITEREFGerard1989|1989]]) of Tutte's characterization of regular matroids.)
 
{{harvtxt|Crapo|1969}} and {{harvtxt|Brylawski|1972}} generalized to matroids Tutte's "dichromate", a graphic polynomial now known as the [[Tutte polynomial]] (named by Crapo).  Their work has recently been followed by a flood of papers&mdash;though not as many as on the Tutte polynomial of a graph.
 
In 1976 Dominic Welsh published the first comprehensive book on matroid theory.
 
[[Paul Seymour (mathematician)|Paul Seymour]]'s decomposition theorem for regular matroids ([[#CITEREFSeymour1980|1980]]) was the most significant and influential work of the late 1970s and the 1980s.
Another fundamental contribution, by {{harvtxt|Kahn|Kung|1982}}, showed why projective geometries and Dowling geometries play such an important role in matroid theory.
 
By this time there were many other important contributors, but one should not omit to mention Geoff Whittle's extension to ternary matroids of Tutte's characterization of binary matroids that are representable over the rationals {{harv|Whittle|1995}}, perhaps the biggest single contribution of the 1990s.  In the current decade{{When|date=August 2011}} the Matroid Minors Project of [[Jim Geelen|Geelen]], Gerards, Whittle, and others, which attempts to duplicate for matroids that are representable over a finite field the success of the Robertson&ndash;Seymour Graph Minors Project (see [[Robertson–Seymour theorem]]), has produced substantial advances in the structure theory of matroids.  Many others have also contributed to that part of matroid theory, which is presently{{When|date=August 2011}} flourishing.


==See also==
==See also==
* [[Algebraic independence]]
*[[Tate conjecture]]
* [[Antimatroid]]
*[[Hodge theory]]
* [[Oriented matroid]]
*[[Hodge structure]]
* [[Pregeometry (model theory)]]
*[[period mapping]]
 
==Researchers==
 
Mathematicians who pioneered the study of matroids include the following:
* [[Saunders Mac Lane]]
* [[Richard Rado]]
* [[W. T. Tutte]]
* [[Bartel Leendert van der Waerden|B. L. van der Waerden]]
* [[Hassler Whitney]]
 
Other major contributors include:
* [[Henry Crapo (mathematician)|Henry Crapo]]
* [[Jack Edmonds]]
* [[Jim Geelen]]
* [[Gian-Carlo Rota]]
* [[Paul Seymour (mathematician)|P. D. Seymour]]
* [[Geoff Whittle]]


There is an on-line list of [http://userhome.brooklyn.cuny.edu/skingan/matroids/people.html current researchers].
==References==
 
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==Notes==
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*{{citation|last=Tutte|first=W. T.|authorlink=W. T. Tutte|year=1959|title=Matroids and graphs|journal=Transactions of the American Mathematical Society|volume=90|pages=527–552|doi=10.2307/1993185|issue=3|publisher=American Mathematical Society|mr=0101527|jstor=1993185}}.
*{{citation|last=Tutte|first=W. T.|authorlink=W. T. Tutte|year=1965|title=Lectures on matroids|journal=Journal of Research of the National Bureau of Standards (U.S.A.), Sect. B|volume=69|pages=1&ndash;47}}.
*{{citation|last=Tutte|first=W. T.|authorlink=W. T. Tutte|year=1971|title=Introduction to the Theory of Matroids|location=New York|publisher=American Elsevier}}.
*{{citation|last=Vámos|first=Peter|year=1978|title=The missing axiom of matroid theory is lost forever|journal=Journal of the London Mathematical Society, II. Ser.|volume=18|pages=403–408|doi=10.1112/jlms/s2-18.3.403|issue=3}}.
*{{citation|last=van der Waerden|first=B. L.|authorlink=Bartel Leendert van der Waerden|year=1937|title=Moderne Algebra}}.
*{{citation|last=Welsh|first=D. J. A.|year=1976|title=Matroid Theory|publisher=Academic Press|isbn=0-12-744050-X}}.
*{{citation|editor-last=White|editor-first=Neil|year=1986|title=Theory of Matroids|series=Encyclopedia of Mathematics and its Applications|volume=26|publisher=Cambridge University Press|location=Cambridge|isbn=978-0-521-30937-0}}.
*{{citation|editor-last=White|editor-first=Neil|year=1992|title=Matroid Applications|series=Encyclopedia of Mathematics and its Applications|volume=40|publisher=Cambridge University Press|location=Cambridge|isbn=978-0-521-38165-9}}.
*{{citation|last=Whitney|first=Hassler|authorlink=Hassler Whitney|year=1935|title=On the abstract properties of linear dependence|journal=American Journal of Mathematics|volume=57|pages=509–533|doi=10.2307/2371182|issue=3|publisher=The Johns Hopkins University Press|mr=1507091|jstor=2371182}}. Reprinted in {{harvtxt|Kung|1986}}, pp.&nbsp;55–79.
*{{citation|last=Whittle|first=Geoff|year=1995|title=A characterization of the matroids representable over ''GF''(3) and the rationals|journal=Journal of Combinatorial Theory Series B|volume=65|issue=2|pages=222&ndash;261|url=http://eprints.kfupm.edu.sa/39296/1/39296.pdf|doi=10.1006/jctb.1995.1052}}.


== External links ==
== External links ==
* [http://www.claymath.org/sites/default/files/hodge.pdf The Clay Math Institute Official Problem Description by P. Deligne (pdf)]
* Popular lecture on Hodge Conjecture by Dan Freed (University of Texas) [http://claymath.msri.org/hodgeconjecture.mov (Real Video)]  [http://www.ma.utexas.edu/users/dafr/HodgeConjecture/netscape_noframes.html (Slides)]
* [[Indranil Biswas]], Kapil Paranjape. [http://arxiv.org/abs/math/0007192v1 The Hodge Conjecture for general Prym varieties]
* [[Burt Totaro]], [http://burttotaro.wordpress.com/2012/03/18/why-believe-the-hodge-conjecture/ Why believe the Hodge Conjecture?]
* [[Claire Voisin]], [http://www.math.polytechnique.fr/~voisin/Articlesweb/hodgeloci.pdf Hodge loci]


* Kingan, Sandra : [http://userhome.brooklyn.cuny.edu/skingan/matroids/ Matroid theory]. A large bibliography of matroid papers, matroid software, and links.
[[Category:Homology theory]]
* Locke, S. C. : [http://www.math.fau.edu/locke/Greedy.htm Greedy Algorithms].
[[Category:Hodge theory]]
* Pagano, Steven R. : [http://www.math.binghamton.edu/zaslav/Pagano/Matridx.htm Matroids and Signed Graphs].
[[Category:Conjectures]]
* Mark Hubenthal: [http://www.math.washington.edu/~hubenjm/matroid2.pdf A Brief Look At Matroids] ([[pdf]]) (contain proofs for staments of this article)
[[Category:Millennium Prize Problems]]
* James Oxley : [http://www.cs.cornell.edu/courses/CS6820/2012sp/Handouts/oxley-matroids.pdf What is a matroid?]
[[Category:Algebraic geometry]]
 
[[Category:Matroid theory]]
[[Category:Dimension]]
[[Category:Closure operators]]
[[Category:Set families]]
 
[[ca:Matroide]]
[[cs:Matroid]]
[[de:Matroid]]
[[es:Matroide]]
[[fr:Matroïde]]
[[it:Matroide]]
[[he:מטרואיד]]
[[hu:Matroid]]
[[nl:Matroïde]]
[[ja:マトロイド]]
[[pl:Matroid]]
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Revision as of 00:40, 11 August 2014

Template:Millennium Problems The Hodge conjecture is a major unsolved problem in algebraic geometry that relates the algebraic topology of a non-singular complex algebraic variety and the subvarieties of that variety. More specifically, the conjecture says that certain de Rham cohomology classes are algebraic, that is, they are sums of Poincaré duals of the homology classes of subvarieties. It was formulated by the Scottish mathematician William Vallance Douglas Hodge as a result of a work in between 1930 and 1940 to enrich the description of de Rham cohomology to include extra structure that is present in the case of complex algebraic varieties. It received little attention before Hodge presented it in an address during the 1950 International Congress of Mathematicians, held in Cambridge, Massachusetts, U.S. The Hodge conjecture is one of the Clay Mathematics Institute's Millennium Prize Problems, with a prize of $1,000,000 for whoever can prove or disprove the Hodge conjecture using "some argument".

Motivation

Let X be a compact complex manifold of complex dimension n. Then X is an orientable smooth manifold of real dimension 2n, so its cohomology groups lie in degrees zero through 2n. Assume X is a Kähler manifold, so that there is a decomposition on its cohomology with complex coefficients:

where Hp, q(X) is the subgroup of cohomology classes which are represented by harmonic forms of type (p, q). That is, these are the cohomology classes represented by differential forms which, in some choice of local coordinates z1, ..., zn, can be written as a harmonic function times

(See Hodge theory for more details.) Taking wedge products of these harmonic representatives corresponds to the cup product in cohomology, so the cup product is compatible with the Hodge decomposition:

Since X is a compact oriented manifold, X has a fundamental class.

Let Z be a complex submanifold of X of dimension k, and let i : Z → X be the inclusion map. Choose a differential form α of type (p, q). We can integrate α over Z:

To evaluate this integral, choose a point of Z and call it 0. Around 0, we can choose local coordinates z1, ..., zn on X such that Z is just zk + 1 = ... = zn = 0. If p > k, then α must contain some dzi where zi pulls back to zero on Z. The same is true if q > k. Consequently, this integral is zero if (p, q) ≠ (k, k).

More abstractly, the integral can be written as the cap product of the homology class of Z and the cohomology class represented by α. By Poincaré duality, the homology class of Z is dual to a cohomology class which we will call [Z], and the cap product can be computed by taking the cup product of [Z] and α and capping with the fundamental class of X. Because [Z] is a cohomology class, it has a Hodge decomposition. By the computation we did above, if we cup this class with any class of type (p, q) ≠ (k, k), then we get zero. Because H2n(X, C) = Hn, n(X), we conclude that [Z] must lie in Hn-k, n-k(X, C). Loosely speaking, the Hodge conjecture asks:

Which cohomology classes in Hk, k(X) come from complex subvarieties Z?

Statement of the Hodge conjecture

Let:

We call this the group of Hodge classes of degree 2k on X.

The modern statement of the Hodge conjecture is:

Hodge conjecture. Let X be a non-singular complex projective manifold. Then every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X.

A projective complex manifold is a complex manifold which can be embedded in complex projective space. Because projective space carries a Kähler metric, the Fubini–Study metric, such a manifold is always a Kähler manifold. By Chow's theorem, a projective complex manifold is also a smooth projective algebraic variety, that is, it is the zero set of a collection of homogeneous polynomials.

Reformulation in terms of algebraic cycles

Another way of phrasing the Hodge conjecture involves the idea of an algebraic cycle. An algebraic cycle on X is a formal combination of subvarieties of X, that is, it is something of the form:

The coefficients are usually taken to be integral or rational. We define the cohomology class of an algebraic cycle to be the sum of the cohomology classes of its components. This is an example of the cycle class map of de Rham cohomology, see Weil cohomology. For example, the cohomology class of the above cycle would be:

Such a cohomology class is called algebraic. With this notation, the Hodge conjecture becomes:

Let X be a projective complex manifold. Then every Hodge class on X is algebraic.

The assumption in the Hodge conjecture that X be algebraic (projective complex manifold) cannot be weakened. In 1977 Zucker showed that it is possible to construct a counterexample to the Hodge conjecture as complex tori with analytic rational cohomology of type (p,p), which is not projective algebraic. (see the appendix B: in Template:Harvtxt)

Known cases of the Hodge conjecture

Low dimension and codimension

The first result on the Hodge conjecture is due to Template:Harvtxt. In fact, it predates the conjecture and provided some of Hodge's motivation.

Theorem (Lefschetz theorem on (1,1)-classes) Any element of H2(X, Z) ∩ H1,1(X) is the cohomology class of a divisor on X. In particular, the Hodge conjecture is true for H2.

A very quick proof can be given using sheaf cohomology and the exponential exact sequence. (The cohomology class of a divisor turns out to equal to its first Chern class.) Lefschetz's original proof proceeded by normal functions, which were introduced by Henri Poincaré. However, Griffiths transversality theorem shows that this approach cannot prove the Hodge conjecture for higher codimensional subvarieties.

By the Hard Lefschetz theorem, one can prove:

Theorem. If the Hodge conjecture holds for Hodge classes of degree p, p < n, then the Hodge conjecture holds for Hodge classes of degree 2n − p.

Combining the above two theorems implies that Hodge conjecture is true for Hodge classes of degree 2n − 2. This proves the Hodge conjecture when X has dimension at most three.

The Lefschetz theorem on (1,1)-classes also implies that if all Hodge classes are generated by the Hodge classes of divisors, then the Hodge conjecture is true:

Corollary. If the algebra
is generated by Hdg1(X), then the Hodge conjecture holds for X.

Hypersurfaces

By the strong and weak Lefschetz theorem, the only non-trivial part of the Hodge conjecture for hypersurfaces is the degree m part (i.e., the middle cohomology) of a 2m-dimensional hypersurface . If the degree d is 2, i.e., X is a quadrics, the Hodge conjecture holds for all m. For m=2, i.e., fourfolds, the Hodge conjecture is known for .[1]

Abelian varieties

For most abelian varieties, the algebra Hdg*(X) is generated in degree one, so the Hodge conjecture holds. In particular, the Hodge conjecture holds for sufficiently general abelian varieties, for products of elliptic curves, and for simple abelian varieties Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.. However, Template:Harvtxt constructed an example of an abelian variety where Hdg2(X) is not generated by products of divisor classes. Template:Harvtxt generalized this example by showing that whenever the variety has complex multiplication by an imaginary quadratic field, then <Hdg2(X) is not generated by products of divisor classes. Template:Harvtxt proved that in dimension less than 5, either Hdg*(X) is generated in degree one, or the variety has complex multiplication by an imaginary quadratic field. In the latter case, the Hodge conjecture is only known in special cases.

Generalizations

The integral Hodge conjecture

Hodge's original conjecture was:

Integral Hodge conjecture. Let X be a projective complex manifold. Then every cohomology class in H2k(X, Z) ∩ Hk, k(X) is the cohomology class of an algebraic cycle with integral coefficients on X.

This is now known to be false. The first counterexample was constructed by Template:Harvtxt. Using K-theory, they constructed an example of a torsion Hodge class, that is, a Hodge class α such that for some positive integer n, n α = 0. Such a cohomology class cannot be the class of a cycle. Template:Harvtxt reinterpreted their result in the framework of cobordism and found many examples of torsion classes.

The simplest adjustment of the integral Hodge conjecture is:

Integral Hodge conjecture modulo torsion. Let X be a projective complex manifold. Then every cohomology class in H2k(X, Z) ∩ Hk,k(X) is the sum of a torsion class and the cohomology class of an algebraic cycle with integral coefficients on X.

Equivalently, after dividing H2k(X, Z) ∩ Hk,k(X) by torsion classes, every class is the image of the cohomology class of an integral algebraic cycle. This is also false. Template:Harvtxt found an example of a Hodge class α which is not algebraic, but which has an integral multiple which is algebraic.

The Hodge conjecture for Kähler varieties

A natural generalization of the Hodge conjecture would ask:

Hodge conjecture for Kähler varieties, naive version. Let X be a complex Kähler manifold. Then every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X.

This is too optimistic, because there are not enough subvarieties to make this work. A possible substitute is to ask instead one of the two following questions:

Hodge conjecture for Kähler varieties, vector bundle version. Let X be a complex Kähler manifold. Then every Hodge class on X is a linear combination with rational coefficients of Chern classes of vector bundles on X.
Hodge conjecture for Kähler varieties, coherent sheaf version. Let X be a complex Kähler manifold. Then every Hodge class on X is a linear combination with rational coefficients of Chern classes of coherent sheaves on X.

Template:Harvtxt proved that the Chern classes of coherent sheaves give strictly more Hodge classes than the Chern classes of vector bundles and that the Chern classes of coherent sheaves are insufficient to generate all the Hodge classes. Consequently, the only known formulations of the Hodge conjecture for Kähler varieties are false.

The generalized Hodge conjecture

Hodge made an additional, stronger conjecture than the integral Hodge conjecture. Say that a cohomology class on X is of level c if it is the pushforward of a cohomology class on a c-codimensional subvariety of X. The cohomology classes of level at least c filter the cohomology of X, and it is easy to see that the cth step of the filtration Nc Hk(X, Z) satisfies

Hodge's original statement was:

Generalized Hodge conjecture, Hodge's version.

Template:Harvtxt observed that this cannot be true, even with rational coefficients, because the right-hand side is not always a Hodge structure. His corrected form of the Hodge conjecture is:

Generalized Hodge conjecture. Nc Hk(X, Q) is the largest sub-Hodge structure of Hk(X, Z) contained in

This version is open.

Algebraicity of Hodge loci

The strongest evidence in favor of the Hodge conjecture is the algebraicity result of Template:Harvtxt. Suppose that we vary the complex structure of X over a simply connected base. Then the topological cohomology of X does not change, but the Hodge decomposition does change. It is known that if the Hodge conjecture is true, then the locus of all points on the base where the cohomology of a fiber is a Hodge class is in fact an algebraic subset, that is, it is cut out by polynomial equations. Cattani, Deligne & Kaplan (1995) proved that this is always true, without assuming the Hodge conjecture.

See also

References

  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

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    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

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    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 Reprinted in Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 http://archive.numdam.org/ARCHIVE/CM/CM_1977__34_2/CM_1977__34_2_199_0/CM_1977__34_2_199_0.pdf

External links

  1. ↑ James Lewis: A Survey of the Hodge Conjecture, 1991, Example 7.21
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