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| In [[mathematics]], the '''Hirzebruch–Riemann–Roch theorem''', named after [[Friedrich Hirzebruch]], [[Bernhard Riemann]], and [[Gustav Roch]], is Hirzebruch's 1954 result contributing to the [[Riemann–Roch theorem|Riemann–Roch problem]] for complex [[algebraic varieties]] of all dimensions. It was the first successful generalisation of the classical [[Riemann–Roch theorem]] on [[Riemann surface]]s to all higher dimensions, and paved the way to the [[Grothendieck–Hirzebruch–Riemann–Roch theorem]] proved about three years later.
| | Myrtle Benny is how I'm known as and I feel comfortable when people use the complete title. Bookkeeping is her day job now. Doing ceramics is what adore doing. For a while she's been in South Dakota.<br><br>Here is my webpage; [http://www.pornextras.info/user/G49Z at home std test] |
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| ==Statement of Hirzebruch–Riemann–Roch theorem==
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| The Hirzebruch–Riemann–Roch theorem applies to any holomorphic [[vector bundle]] ''E'' on a [[compact space|compact]] [[complex manifold]] ''X'', to calculate the [[holomorphic Euler characteristic]] of ''E'' in [[sheaf cohomology]], namely the alternating sum
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| :<math> \chi(X,E) = \sum_{i=0}^{\dim_{\mathbb{C}} X} (-1)^{i} \dim_{\mathbb{C}} H^{i}(X,E) </math>
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| of the dimensions as complex vector spaces.
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| Hirzebruch's theorem states that χ(''X'', ''E'') is computable in terms of the [[Chern class]]es ''C''<sub>''j''</sub>(''E'') of ''E'', and the [[Todd polynomial]]s ''T''<sub>''j''</sub> in the Chern classes of the holomorphic [[tangent bundle]] of ''X''. These all lie in the [[cohomology ring]] of ''X''; by use of the [[fundamental class]] (or, in other words, integration over ''X'') we can obtain numbers from classes in ''H''<sup>2''n''</sup>(''X''). The Hirzebruch formula asserts that
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| :<math> \chi(X,E) = \sum \operatorname{ch}_{n-j}(E) \frac{T_{j}}{j!} </math>
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| taken over all relevant ''j'' (so 0 ≤ ''j'' ≤ ''n''), using the [[Chern character]] ch(''E'') in cohomology. In other words, the cross products are formed in cohomology ring of all the 'matching' degrees that add up to 2''n'', where to 'massage' the ''C''<sub>''j''</sub>(''E'') a formal manipulation is done, setting
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| :<math>\operatorname{ch}(E) = \sum \exp(x_{i}) </math>
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| and the total Chern class
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| :<math> C(E) = \sum C_{j}(E) = \prod (1 + x_{i}). </math>
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| Formulated differently the theorem gives the equality
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| :<math> \chi(X,E) = \int_X \operatorname{ch}(E) \operatorname{td}(X)</math>
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| where ''td(X)'' is the [[Todd class]] of the tangent bundle of ''X''.
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| Significant special cases are when ''E'' is a complex [[line bundle]], and when ''X'' is an [[algebraic surface]] ('''Noether's formula'''). Weil's Riemann–Roch theorem for vector bundles on curves, and the Riemann–Roch theorem for algebraic surfaces (see below), are included in its scope. The formula also expresses in a precise way the vague notion that the [[Todd class]]es are in some sense reciprocals of [[characteristic class]]es.
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| ==Riemann Roch theorem for curves==
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| For curves, the Hirzebruch–Riemann–Roch theorem is essentially the classical [[Riemann–Roch theorem]]. To see this,
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| recall that for each [[divisor (algebraic geometry)|divisor]] ''D'' on a curve there is an [[invertible sheaf]] O(''D'') (which corresponds to a line bundle) such that
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| the [[linear systems of divisors|linear system]] of ''D'' is more or less the space of sections of O(''D'').
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| For curves the Todd class is 1 + ''c''<sub>1</sub>(''T(X)'')/2, and the Chern character of a sheaf O(''D'') is just
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| 1+''c''<sub>1</sub>(O(''D'')), so the Hirzebruch–Riemann–Roch theorem
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| states that
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| : <math>h^0(\mathcal{O}(D)) - h^1(\mathcal{O}(D)) = c_1(\mathcal{O}(D)) +c_1(T(X))/2\ \ \ </math> (integrated over ''X'').
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| But ''h''<sup>0</sup>(O(''D'')) is just ''l''(''D''), the dimension of the linear system of ''D'', and by [[Serre duality]] ''h''<sup>1</sup>(O(''D'')) = ''h''<sup>0</sup>(O(''K'' − ''D'')) = ''l''(''K'' − ''D'') where ''K'' is the [[canonical divisor]]. Moreover ''c''<sub>1</sub>(O(''D'')) integrated over ''X'' is the degree of ''D'', and ''c''<sub>1</sub>(''T''(''X'')) integrated over ''X'' is the Euler class 2 − 2''g''
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| of the curve ''X'', where ''g'' is the genus. So we get the classical Riemann Roch theorem
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| : <math>\ell(D)-\ell(K-D) = \text{deg}(D)+1-g.</math> | |
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| For vector bundles ''V'', the Chern character is rank(''V'') + ''c''<sub>1</sub>(''V''), so we get Weil's Riemann Roch theorem for vector bundles over curves:
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| : <math>h^0(V) - h^1(V) = c_1(V) + \text{rank}(V)(1-g).</math>
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| ==Riemann Roch theorem for surfaces==
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| {{main|Riemann-Roch theorem for surfaces}}
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| For surfaces, the Hirzebruch–Riemann–Roch theorem is essentially the [[Riemann–Roch theorem for surfaces]]
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| : <math>\chi(D) = \chi(\mathcal{O}) + ((D.D)-(D.K))/2.</math>
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| combined with the Noether formula.
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| If we want, we can use Serre duality to express ''h''<sup>2</sup>(O(''D'')) as ''h''<sup>0</sup>(O(''K'' − ''D'')),
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| but unlike the case of curves there is in general no easy way to write the ''h''<sup>1</sup>(O(''D'')) term in a form not involving sheaf cohomology (although in practice it often vanishes).
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| ==References==
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| * ''Topological Methods in Algebraic Geometry'' by Friedrich Hirzebruch ISBN 3-540-58663-6
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| {{DEFAULTSORT:Hirzebruch-Riemann-Roch theorem}}
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| [[Category:Topological methods of algebraic geometry]]
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| [[Category:Theorems in complex geometry]]
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| [[Category:Theorems in algebraic geometry]]
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Myrtle Benny is how I'm known as and I feel comfortable when people use the complete title. Bookkeeping is her day job now. Doing ceramics is what adore doing. For a while she's been in South Dakota.
Here is my webpage; at home std test