Hirzebruch–Riemann–Roch theorem

Hirzebruch–Riemann–Roch theorem Hirzebruch–Riemann–Roch theorem Field Algebraic geometry First proof by Friedrich Hirzebruch First proof in 1954 Generalizations Atiyah–Singer index theorem Grothendieck–Riemann–Roch theorem Consequences Riemann–Roch theorem Riemann–Roch theorem for surfaces In mathematics, the Hirzebruch–Riemann–Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruch's 1954 result generalizing the classical Riemann–Roch theorem on Riemann surfaces to all complex algebraic varieties of higher dimensions. The result paved the way for the Grothendieck–Hirzebruch–Riemann–Roch theorem proved about three years later.

Contents 1 Statement of Hirzebruch–Riemann–Roch theorem 2 Riemann Roch theorem for curves 3 Riemann Roch theorem for surfaces 4 Asymptotic Riemann–Roch 5 See also 6 References 7 External links Statement of Hirzebruch–Riemann–Roch theorem The Hirzebruch–Riemann–Roch theorem applies to any holomorphic vector bundle E on a compact complex manifold X, to calculate the holomorphic Euler characteristic of E in sheaf cohomology, namely the alternating sum {displaystyle chi (X,E)=sum _{i=0}^{n}(-1)^{i}dim _{mathbb {C} }H^{i}(X,E)} of the dimensions as complex vector spaces, where n is the complex dimension of X.

Hirzebruch's theorem states that χ(X, E) is computable in terms of the Chern classes ck(E) of E, and the Todd classes {displaystyle operatorname {td} _{j}(X)} 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 {displaystyle H^{2n}(X).} The Hirzebruch formula asserts that {displaystyle chi (X,E)=sum operatorname {ch} _{n-j}(E)operatorname {td} _{j}(X),} where the sum is taken over all relevant j (so 0 ≤ j ≤ n), using the Chern character ch(E) in cohomology. In other words, the products are formed in the cohomology ring of all the 'matching' degrees that add up to 2n. Formulated differently, it gives the equality {displaystyle chi (X,E)=int _{X}operatorname {ch} (E)operatorname {td} (X)} where {displaystyle operatorname {td} (X)} is the Todd class of the tangent bundle of X.

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 classes are in some sense reciprocals of characteristic classes.

Riemann Roch theorem for curves For curves, the Hirzebruch–Riemann–Roch theorem is essentially the classical Riemann–Roch theorem. To see this, recall that for each divisor D on a curve there is an invertible sheaf O(D) (which corresponds to a line bundle) such that the linear system of D is more or less the space of sections of O(D). For curves the Todd class is {displaystyle 1+c_{1}(T(X))/2,} and the Chern character of a sheaf O(D) is just 1+c1(O(D)), so the Hirzebruch–Riemann–Roch theorem states that {displaystyle h^{0}({mathcal {O}}(D))-h^{1}({mathcal {O}}(D))=c_{1}({mathcal {O}}(D))+c_{1}(T(X))/2 } (integrated over X).

But h0(O(D)) is just l(D), the dimension of the linear system of D, and by Serre duality h1(O(D)) = h0(O(K − D)) = l(K − D) where K is the canonical divisor. Moreover, c1(O(D)) integrated over X is the degree of D, and c1(T(X)) integrated over X is the Euler class 2 − 2g of the curve X, where g is the genus. So we get the classical Riemann Roch theorem {displaystyle ell (D)-ell (K-D)={text{deg}}(D)+1-g.} For vector bundles V, the Chern character is rank(V) + c1(V), so we get Weil's Riemann Roch theorem for vector bundles over curves: {displaystyle h^{0}(V)-h^{1}(V)=c_{1}(V)+operatorname {rank} (V)(1-g).} Riemann Roch theorem for surfaces Main article: Riemann–Roch theorem for surfaces For surfaces, the Hirzebruch–Riemann–Roch theorem is essentially the Riemann–Roch theorem for surfaces {displaystyle chi (D)=chi ({mathcal {O}})+((D.D)-(D.K))/2.} combined with the Noether formula.

If we want, we can use Serre duality to express h2(O(D)) as h0(O(K − D)), but unlike the case of curves there is in general no easy way to write the h1(O(D)) term in a form not involving sheaf cohomology (although in practice it often vanishes).

Asymptotic Riemann–Roch Let D be an ample Cartier divisor on an irreducible projective variety X of dimension n. Then {displaystyle h^{0}left(X,{mathcal {O}}_{X}(mD)right)={frac {(D^{n})}{n!}}.m^{n}+O(m^{n-1}).} More generally, if {displaystyle {mathcal {F}}} is any coherent sheaf on X then {displaystyle h^{0}left(X,{mathcal {F}}otimes {mathcal {O}}_{X}(mD)right)=operatorname {rank} ({mathcal {F}}){frac {(D^{n})}{n!}}.m^{n}+O(m^{n-1}).} See also Grothendieck–Riemann–Roch theorem - contains many computations and examples Hilbert polynomial - HRR can be used to compute Hilbert polynomials References Friedrich Hirzebruch,Topological Methods in Algebraic Geometry ISBN 3-540-58663-6 External links The Hirzebruch-Riemann-Roch Theorem hide vte Bernhard Riemann Cauchy–Riemann equationsGeneralized Riemann hypothesisGrand Riemann hypothesisGrothendieck–Hirzebruch–Riemann–Roch theoremHirzebruch–Riemann–Roch theoremLocal zeta functionMeasurable Riemann mapping theoremRiemannRiemann Xi functionRiemann curvature tensorRiemann hypothesisRiemann integralRiemann invariantRiemann mapping theoremRiemann formRiemann problemRiemann series theoremRiemann solverRiemann sphereRiemann sumRiemann surfaceRiemann zeta functionRiemann's differential equationRiemann's minimal surfaceRiemannian circleRiemannian connection on a surfaceRiemannian geometryRiemann–Hilbert correspondenceRiemann–Hilbert problemsRiemann–Lebesgue lemmaRiemann–Liouville integralRiemann–Roch theoremRiemann–Roch theorem or smooth manifoldsRiemann–Siegel formulaRiemann–Siegel theta functionRiemann–Silberstein vectorRiemann–Stieltjes integralRiemann–von Mangoldt formula Category Categories: Topological methods of algebraic geometryTheorems in complex geometryTheorems in algebraic geometryBernhard Riemann

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