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| In [[mathematics]], in particular in [[algebraic topology]], [[differential geometry and topology|differential geometry]] and [[algebraic geometry]], the '''Chern classes''' are [[characteristic class]]es associated to complex [[vector bundle]]s.
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| Chern classes were introduced by {{harvs|txt|authorlink=Shiing-Shen Chern|first=Shiing-Shen|last=Chern|year=1946}}.
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| == Geometric approach ==
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| === Basic idea and motivation ===
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| Chern classes are [[characteristic class]]es. They are [[topological invariant]]s associated to vector bundles on a smooth manifold. The question of whether two ostensibly different vector bundles are the same can be quite hard to answer. The Chern classes provide a simple test: if the Chern classes of a pair of vector bundles do not agree, then the vector bundles are different. The converse, however, is not true.
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| In topology, differential geometry, and algebraic geometry, it is often important to count how many [[linearly independent]] sections a vector bundle has. The Chern classes offer some information about this through, for instance, the [[Riemann-Roch theorem]] and the [[Atiyah-Singer index theorem]].
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| Chern classes are also feasible to calculate in practice. In differential geometry (and some types of algebraic geometry), the Chern classes can be expressed as polynomials in the coefficients of the [[curvature form]].
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| === Construction of Chern classes ===
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| There are various ways of approaching the subject, each of which focuses on a slightly different flavor of Chern class.
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| The original approach to Chern classes was via algebraic topology: the Chern classes arise via [[homotopy theory]] which provides a mapping associated to ''V'' to a [[classifying space]] (an infinite [[Grassmannian]] in this case). Any vector bundle ''V'' over a manifold may be realized as the pullback of a universal bundle over the classifying space, and the Chern classes of ''V'' can therefore be defined as the pullback of the Chern classes of the universal bundle; these universal Chern classes in turn can be explicitly written down in terms of [[Schubert cycle]]s.
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| Chern's approach used differential geometry, via the curvature approach described predominantly in this article. He showed that the earlier definition was in fact equivalent to his.
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| There is also an approach of [[Alexander Grothendieck]] showing that axiomatically one need only define the line bundle case.
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| Chern classes arise naturally in [[algebraic geometry]]. The generalized Chern classes in algebraic geometry can be defined for vector bundles (or more precisely, [[locally free sheaves]]) over any nonsingular variety. Algebro-geometric Chern classes do not require the underlying field to have any special properties. In particular, the vector bundles need not necessarily be complex.
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| Regardless of the particular paradigm, the intuitive meaning of the Chern class concerns 'required zeroes' of a [[Section (category theory)|section]] of a vector bundle: for example the theorem saying one can't comb a hairy ball flat ([[hairy ball theorem]]). Although that is strictly speaking a question about a ''real'' vector bundle (the "hairs" on a ball are actually copies of the real line), there are generalizations in which the hairs are complex (see the example of the complex hairy ball theorem below), or for 1-dimensional projective spaces over many other fields.
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| See [[Chern-Simons]] for more discussion.
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| == The Chern class of line bundles ==
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| {{for|a sheaf theoretic description|Exponential sheaf sequence}}
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| An important special case occurs when ''V'' is a [[line bundle]]. Then the only nontrivial Chern class is the first Chern class, which is an element of the second cohomology group of ''X''. As it is the top Chern class, it equals the [[Euler class]] of the bundle.
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| The first Chern class turns out to be a [[complete invariant]] with which to classify complex line bundles, topologically speaking. That is, there is a [[bijection]] between the isomorphism classes of line bundles over ''X'' and the elements of ''H''<sup>2</sup>(''X'';''Z''), which associates to a line bundle its first Chern class. Addition in the second dimensional cohomology group coincides with [[tensor product]] of complex line bundles. {{Citation needed|date=April 2011}}
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| In algebraic geometry, this classification of (isomorphism classes of) complex line bundles by the first Chern class is a crude approximation to the classification of (isomorphism classes of) holomorphic line bundles by [[linear equivalence]] classes of [[divisor]]s.
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| For complex vector bundles of dimension greater than one, the Chern classes are not a complete invariant.
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| ==The Chern class in [[Chern-Weil theory]]==
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| ===The Chern class of a Hermitian vector bundle on a smooth manifold===
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| Given a complex [[Hermitian metric|hermitian]] [[vector bundle]] ''V'' of [[vector bundle|complex rank]] ''n'' over a [[smooth manifold]] ''M'', a representative of each Chern class (also called a '''Chern form''') ''c<sub>k</sub>''(''V'') of ''V'' are given as the coefficients of the [[characteristic polynomial]] of the [[curvature form]] Ω of ''V''.
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| :<math>\det \left(\frac {it\Omega}{2\pi} +I\right) = \sum_k c_k(V) t^k</math>
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| The determinant is over the ring of ''n'' × ''n'' matrices whose entries are polynomials in ''t'' with coefficients in the commutative algebra of even complex differential forms on ''M''. The [[curvature form]] Ω of ''V'' is defined as
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| : <math>\Omega=d\omega+\tfrac{1}{2}[\omega,\omega]</math>
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| with ω the [[connection form]] and ''d'' the [[exterior derivative]], or via the same expression in which ω is a [[gauge form]] for the [[gauge group]] of ''V''. The scalar ''t'' is used here only as an [[indeterminate (variable)|indeterminate]] to [[generating function|generate]] the sum from the determinant, and ''I'' denotes the ''n'' × ''n'' [[identity matrix]]. | |
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| To say that the expression given is a ''representative'' of the Chern class indicates that 'class' here means [[up to]] addition of an [[exact differential form]]. That is, Chern classes are [[cohomology class]]es in the sense of [[de Rham cohomology]]. It can be shown that the cohomology class of the Chern forms do not depend on the choice of connection in ''V''.
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| Using the matrix identity tr(ln(''X''))=ln(det(''X'')) and the Maclaurin series for ln(''X''+''I''), this expression for the Chern form expands as
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| :<math>\sum_k c_k(V) t^k = \left[ I
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| + i \frac{\mathrm{tr}(\Omega)}{2\pi} t
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| + \frac{\mathrm{tr}(\Omega^2)-\mathrm{tr}(\Omega)^2}{8\pi^2} t^2
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| + i \frac{-2\mathrm{tr}(\Omega^3)+3\mathrm{tr}(\Omega^2)\mathrm{tr}(\Omega)-\mathrm{tr}(\Omega)^3}{48\pi^3} t^3
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| + \cdots
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| \right].</math>
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| ===Example: the complex tangent bundle of the Riemann sphere===
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| Let '''CP'''<sup>1</sup> be the [[Riemann sphere]]: 1-dimensional [[complex projective space]]. Suppose that ''z'' is a [[holomorphic]] [[manifold|local coordinate]] for the Riemann sphere. Let ''V'' = ''T'''''CP'''<sup>1</sup> be the bundle of complex tangent vectors having the form ''a''∂/∂''z'' at each point, where ''a'' is a complex number. We prove the complex version of the ''[[hairy ball theorem]]'': ''V'' has no section which is everywhere nonzero.
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| For this, we need the following fact: the first Chern class of a trivial bundle is zero, i.e.,
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| : <math>c_1({\mathbf C\mathbf P}^1\times {\mathbf C})=0.</math>
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| This is evinced by the fact that a trivial bundle always admits a flat connection.
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| So, we shall show that
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| :<math>c_1(V) \not= 0.</math>
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| Consider the [[Kähler metric]]
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| :<math>h = \frac{dzd\bar{z}}{(1+|z|^2)^2}.</math>
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| One readily shows that the curvature 2-form is given by
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| :<math>\Omega=\frac{2dz\wedge d\bar{z}}{(1+|z|^2)^2}.</math>
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| Furthermore, by the definition of the first Chern class
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| :<math>c_1= \left[\frac{i}{2\pi} \mathrm{tr} \ \Omega\right] .</math>
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| We must show that this cohomology class is non-zero. It suffices to compute its integral over the Riemann sphere:
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| :<math>\int c_1 dz\wedge d\bar{z} =\frac{i}{\pi}\int \frac{dz\wedge d\bar{z}}{(1+|z|^2)^2}=2</math>
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| after switching to [[polar coordinates]]. By [[Stokes' theorem]], an [[exact form]] would integrate to 0, so the cohomology class is nonzero.
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| This proves that ''T'''''CP'''<sup>1</sup> is not a trivial vector bundle.
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| == Properties of Chern classes ==
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| Given a complex [[vector bundle]] ''V'' over a [[topological space]] ''X'', the Chern classes of ''V'' are a sequence of elements of the [[cohomology]] of ''X''. The '''''k''-th Chern class''' of ''V'', which is usually denoted ''c<sub>k</sub>''(''V''), is an element of
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| :''H''<sup>2''k''</sup>(''X'';'''Z'''),
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| the cohomology of ''X'' with [[integer]] coefficients. One can also define the '''total Chern class''' | |
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| :<math>c(V) = c_0(V) + c_1(V) + c_2(V) + \cdots .</math>
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| Since the values are in integral cohomology groups, rather than cohomology with real coefficients, these Chern classes are slightly more refined than those in the Riemannian example.
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| ===Classical axiomatic definition===
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| The Chern classes satisfy the following four axioms:
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| '''Axiom 1.''' <math>c_0(V) = 1</math> for all ''V''.
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| '''Axiom 2.''' Functoriality: If <math>f : Y \to X</math> is [[continuous function (topology)|continuous]] and ''f*V'' is the [[pullback bundle|vector bundle pullback]] of ''V'', then <math>c_k(f^* V) = f^* c_k(V)</math>.
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| '''Axiom 3.''' [[Hassler Whitney|Whitney]] sum formula: If <math>W \to X</math> is another complex vector bundle, then the Chern classes of the [[direct sum of vector bundles|direct sum]] <math>V \oplus W</math> are given by
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| :<math>c(V \oplus W) = c(V) \smile c(W);</math>
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| that is,
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| :<math>c_k(V \oplus W) = \sum_{i = 0}^k c_i(V) \smile c_{k - i}(W).</math>
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| '''Axiom 4.''' Normalization: The total Chern class of the [[tautological line bundle]] over '''CP'''<sup>''k''</sup> is 1−''H'', where ''H'' is [[Poincaré duality|Poincaré-dual]] to the [[hyperplane]] <math>\mathbf{CP}^{k - 1} \subseteq \mathbf{CP}^k</math>.
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| ===Alexander Grothendieck axiomatic approach=== | |
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| Alternatively, {{harvs|txt|authorlink=Alexander Grothendieck|first=Alexander|last=Grothendieck|year=1958}} replaced these with a slightly smaller set of axioms:
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| * Functoriality: (Same as above)
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| * Additivity: If <math>\ 0\to E'\to E\to E''\to 0</math> is an [[exact sequence]] of vector bundles, then <math>c(E)=c(E')\smile c(E'')</math>.
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| * Normalization: If ''E'' is a [[line bundle]], then <math>c(E)=1+e(E_{\mathbf R})</math> where <math>e(E_{\mathbf R})</math> is the [[Euler class]] of the underlying real vector bundle.
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| He shows using the [[Leray-Hirsch theorem]] that the total Chern class of an arbitrary finite rank complex vector bundle can be defined in terms of the first Chern class of a tautologically-defined line bundle.
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| Namely, introducing the projectivization '''P'''(''E'') of the rank ''n'' complex vector bundle ''E'' → ''B'' as the fiber bundle on ''B'' whose fiber at any point <math>b\in B</math> is the projective space of the fiber ''E<sub>b</sub>''. The total space of this bundle '''P'''(''E'') is equipped with its tautological complex line bundle, that we denote τ, and the first Chern class
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| :<math>c_1(\tau)=: -a</math>
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| restricts on each fiber '''P'''(''E<sub>b</sub>'') to minus the (Poincaré-dual) class of the hyperplane, that spans the cohomology of the fiber, in view of the cohomology of [[complex projective space]]s.
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| The classes
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| :<math>1, a, a^2, \ldots , a^{n-1}\in H^*(\mathbf{P}(E))</math>
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| therefore form a family of ambient cohomology classes restricting to a basis of the cohomology of the fiber. The [[Leray-Hirsch theorem]] then states that
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| any class in ''H*''('''P'''(''E'')) can be written uniquely as a linear combination of the 1, ''a'', ''a''<sup>2</sup>, ..., ''a''<sup>''n''−1</sup> with classes on the basis as coefficients.
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| In particular, one may define the Chern classes of ''E'' in the sense of Grothendieck, denoted <math>c_1(E), \ldots c_{n}(E)</math> by expanding this way the class <math>-a^n</math>, with the relation:
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| :<math> - a^n = c_1(E). a^{n-1}+ \ldots c_{n-1}(E) .a + c_{n}(E) .</math>
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| One then may check that this alternative definition coincides with whatever other definition one may favor, or use the previous axiomatic characterization.
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| ===The top Chern class===
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| In fact, these properties uniquely characterize the Chern classes. They imply, among other things:
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| * If ''n'' is the complex rank of ''V'', then <math>c_k(V) = 0</math> for all ''k'' > ''n''. Thus the total Chern class terminates.
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| * The top Chern class of ''V'' (meaning <math>c_n(V)</math>, where ''n'' is the rank of ''V'') is always equal to the [[Euler class]] of the underlying real vector bundle.
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| ==Proximate notions==
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| ===The Chern character===
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| Chern classes can be used to construct a homomorphism of rings from the [[topological K-theory]] of a space to (the completion of) its rational cohomology. For a line bundle ''L'', the Chern character ch is defined by
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| :<math>\operatorname{ch}(L) = \exp(c_{1}(L)) := \sum_{m=0}^\infty \frac{c_1(L)^m}{m!}.</math>
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| More generally, if <math>V = L_1 \oplus ... \oplus L_n</math> is a direct sum of line bundles, with first Chern classes <math>x_i = c_1(L_i),</math> the Chern character is defined additively
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| :<math> \operatorname{ch}(V) = e^{x_1} + \dots + e^{x_n} :=\sum_{m=0}^\infty \frac{1}{m!}(x_1^m + ... + x_n^m). </math>
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| Observe that when ''V'' is a sum of line bundles, the Chern classes of ''V'' can be expressed as [[elementary symmetric polynomials]] in the <math>x_i</math>, <math>c_i(V) = e_i(x_1,...,x_n).</math>
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| In particular, on the one hand
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| :<math>c(V) := \sum_{i=0}^n c_i(V),</math>
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| while on the other hand
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| :<math>c(V) = c(L_1 \oplus \dots \oplus L_n) = \prod_{i=1}^n c(L_i) = \prod_{i=1}^n (1+x_i) = \sum_{i=0}^n e_i(x_1,\dots,x_n). </math>
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| Consequently, [[Newton's_identities#Expressing_power_sums_in_terms_of_elementary_symmetric_polynomials|Newton's identities]] may be used to reexpress the power sums in ''ch(V)'' above solely in terms of the Chern classes of ''V'', giving
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| :<math> \operatorname{ch}(V) = \operatorname{dim}(V) + c_1(V) + \frac{1}{2}(c_1(V)^2 - 2c_2(V)) + \frac{1}{6} (c_1(V)^3 - 3c_1(V)c_2(V) + 3c_3(V)) + ..., </math>
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| This last expression, justified by invoking the [[splitting principle]], is taken as the definition ''ch(V)'' for arbitrary vector bundles ''V''.
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| If a connection is used to define the Chern classes, then the explicit form of the Chern character is
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| : <math>\hbox{ch}(V)=\hbox{tr}\left(\exp\left(\frac{i\Omega}{2\pi}\right)\right)</math>
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| where Ω is the curvature of the connection.
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| The Chern character is useful in part because it facilitates the computation of the Chern class of a tensor product. Specifically, it obeys the following identities:
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| :<math>\hbox{ch}(V\oplus W)=\hbox{ch}(V)+\hbox{ch}(W)</math>
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| :<math>\hbox{ch}(V\otimes W)=\hbox{ch}(V)\hbox{ch}(W).</math>
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| As stated above, using the Grothendieck additivity axiom for Chern classes, the first of these identities can be generalized to state that ''ch'' is a [[homomorphism]] of [[abelian groups]] from the [[K-theory]] ''K''(''X'') into the rational cohomology of ''X''. The second identity establishes the fact that this homomorphism also respects products in ''K''(''X''), and so ''ch'' is a homomorphism of rings.
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| The Chern character is used in the [[Hirzebruch-Riemann-Roch theorem]].
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| ===Chern numbers===
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| If we work on an [[orientable manifold|oriented manifold]] of dimension ''2n'', then any product of Chern classes of total degree ''2n'' can be paired with the [[orientation homology class]] (or "integrated over the manifold") to give an integer, a '''Chern number''' of the vector bundle. For example, if the manifold has dimension 6, there are three linearly independent Chern numbers, given by ''c''<sub>1</sub><sup>3</sup>, ''c''<sub>1</sub>''c''<sub>2</sub>, and ''c''<sub>3</sub>. In general, if the manifold has dimension 2''n'', the number of possible independent Chern numbers is the number of [[integer partition|partition]]s of ''n''.
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| The Chern numbers of the tangent bundle of a complex (or almost complex) manifold are called the Chern numbers of the manifold, and are important invariants.
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| ===The Chern class in generalized cohomology theories===
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| There is a generalization of the theory of Chern classes, where ordinary cohomology is replaced with a [[generalized cohomology theory]]. The theories for which such generalization is possible are called ''[[Complex_cobordism#Formal group laws|complex orientable]]''. The formal properties of the Chern classes remain the same, with one crucial difference: the rule which computes the first Chern class of a tensor product of line bundles in terms of first Chern classes of the factors is not (ordinary) addition, but rather a [[formal group law]].
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| === Chern classes of manifolds with structure ===
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| The theory of Chern classes gives rise to [[cobordism]] invariants for [[almost complex manifold]]s.
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| If ''M'' is an almost complex manifold, then its [[tangent bundle]] is a complex vector bundle. The '''Chern classes''' of ''M'' are thus defined to be the Chern classes of its tangent bundle. If ''M'' is also [[Compact space|compact]] and of dimension 2''d'', then each [[monomial]] of total degree 2''d'' in the Chern classes can be paired with the [[fundamental class]] of ''M'', giving an integer, a '''Chern number''' of ''M''. If ''M''′ is another almost complex manifold of the same dimension, then it is cobordant to ''M'' if and only if the Chern numbers of ''M''′ coincide with those of ''M''.
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| The theory also extends to real [[Symplectic geometry|symplectic]] vector bundles, by the intermediation of compatible almost complex structures. In particular, [[symplectic manifold]]s have a well-defined Chern class.
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| === Chern classes on arithmetic schemes and Diophantine equations ===
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| (See [[Arakelov geometry]])
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| == See also ==
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| * [[Pontryagin class]]
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| * [[Stiefel-Whitney class]]
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| * [[Euler class]]
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| * [[Segre class]]
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| ==References==
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| * {{Citation | last1=Chern | first1=S. S. | title=Characteristic classes of Hermitian Manifolds | year=1946 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=47 | issue=1 | pages=85–121 | doi=10.2307/1969037 | publisher=The Annals of Mathematics, Vol. 47, No. 1 |jstor=1969037}}
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| * {{Citation | last1=Grothendieck | first1=Alexander | author1-link=Alexander Grothendieck | title=La théorie des classes de Chern | mr=0116023 | year=1958 | journal=Bulletin de la Société Mathématique de France | issn=0037-9484 | volume=86 | pages=137–154
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| |url= http://www.numdam.org/item?id=BSMF_1958__86__137_0}}
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| * {{Citation | last1=Jost | first1=Jürgen | title=Riemannian Geometry and Geometric Analysis | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=4th | isbn=978-3-540-25907-7 | year=2005}} (Provides a very short, introductory review of Chern classes).
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| * {{Citation | last1=Milnor | first1=John Willard | author1-link=John Milnor | last2=Stasheff | first2=James D. |author2-link=Jim Stasheff| title=Characteristic classes | publisher=Princeton University Press; University of Tokyo Press | series=Annals of Mathematics Studies | isbn=978-0-691-08122-9 | year=1974 | volume=76}}
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| ==External links==
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| * [http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html Vector Bundles & K-Theory] - A downloadable book-in-progress by [[Allen Hatcher]]. Contains a chapter about characteristic classes.
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| *[[Dieter Kotschick]], [http://www.physorg.com/news163858041.html Chern numbers of algebraic varieties]
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| [[Category:Characteristic classes]]
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