DOCSLIB.ORG

Serre Duality

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Serre Duality Serre Duality Carl Lian June 17, 2016 Abstract Notes for a talk on Serre Duality following Vakil’s notes [V] with appeal to [H] and [M] toward the end. A feature of this proof seems to be that it generally pushes the hardest ingredients to the end. In particular, the existence of the dualizing sheaf is relatively cheap, and the work goes into identifying it with the canonical bundle on smooth varieties. Note that there are no assumptions on the ground field. 1 Preliminaries on Ext and Ext Let X be a ringed space and F an OX -module. Recall that we have left-exact functors Hom(F; −) and Hom(F; −); the former lands in Γ(OX )-modules and the latter in OX -modules. The category of OX -modules has enough injectives, so we get the right derived functors Exti (global Ext) and Exti (sheaf Ext) satisfying the usual properties. In particular, they can be computed via injective resolutions. The following facts are not hard; the key points are that the tensor product of an injective sheaf with a locally free sheaf is injective, and the restriction of an injective sheaf to an open subset is injective. Proposition 1.1 If E is locally free, Exti(F ⊗ E_; G) =∼ Exti(F; G ⊗ E) =∼ Exti(F; G) ⊗ E and Exti(F ⊗ E_; G) =∼ Exti(F; G ⊗ E) i i Proposition 1.2 If U ⊂ X is open, Ext (F; G)(U) = Ext (FjU ; GjU ). Hence “sheaf Ext is local.” (b’) The functors Ext and Ext are genuinely different, aside from landing in i i i different categories. For instance, Ext (OX ; F) = H (X; F), but Ext (OX ; F) = 0 0 unless i = 0, in which case Ext (OX ; F) = F. However, the two functors are related in the following way. Theorem 1.3 (Local-to-global spectral sequence) There is a spectral se- p;q j i i+j quence with E2 = H (X; Ext (F; G)) converging to Ext (F; G). 1 This follows (after checking the needed hypothesis) from the Grothendieck spectral sequence: Hom is the composition of Hom and the global sections functor Γ. In the category of abelian groups, Ext can also be computed with projective resolutions, but this is no longer true for sheaves: there are not enough projec- tives (for example, free sheaves may not be projective)! However, Ext can be computed with locally free resolutions: this follows from acyclicity of the trivial sheaf (and hence of locally free sheaves) and a spectral sequence. (Note how this fails for Ext.) A consequence of this is that if X is a locally Noetherian scheme and F; G are coherent, then Exti(F; G) is coherent (not true for quasicoherent sheaves, even for i = 0!). 2 Statements Theorem 2.1 (Serre Duality) Let X be a Cohen-Macaulay (CM) projective k-scheme of pure dimension n. Then, there is a unique (up to unique isomor- phism) coherent sheaf !X such that: (a) For any vector bundle F, there is a natural perfect pairing Hi(X; F) × n−i _ H (X; F ⊗ !X ) ! k. i (b) For any coherent sheaf F, there is a natural perfect pairing Ext (F;!X )× Hn−i(X; F) ! k. (b’) For any coherent sheaf F, there is a natural perfect pairing Hom(F;!X )× Hn(X; F) ! k. (The CM hypothesis is not needed for this statement.) ∼ Vn (c) If X is smooth, then !X = KX := ΩX . The uniqueness of !X is a formal consequence of Yoneda’s Lemma. Note that (b’) is just (b) for i = 0, but we state it separately because it needs to be proven first. The CM hypothesis will mostly be black-boxed: it is used only to construct a finite flat morphism X ! Pn, which is possible by the “Miracle Flatness” theorem. This hypothesis is necessary: when X is the union of two general 2-planes in P4, Serre Duality fails [V, Exercise 30.1.E]. In fact, statement (b) implies X must be Cohen-Macaulay, see [H, Theorem 7.6]. n A consequence of (b’) is that we get a trace map t : H (X; !X ) ! k. Indeed, taking F = !X , the perfect pairing gives us a distinguished element n _ t 2 H (X; !X ) corresponding to id 2 Hom(!X ;!X ). Then given an element n of Hom(F;!X ), we get an induced map on H , yielding the map Hom(F;!X )× n n H (X; F) ! H (X; !X ); one checks that composing with t recovers the original pairing. In other words, we have a diagram n n Hom(F;!X ) × H (X; F) / H (X; !X ) t ) k 2 In fact, the perfect pairings of (a) and (b) factor through t as well, but we will not prove this. The strategy of the proof is as follows: first, prove all statements for Pn by direct calculation. Then, prove statement (b) by considering a finite flat map n ! n π : X ! P ; the dualizing sheaf !X will be π !P . Finally, prove statement (c) by considering a regular embedding i : X ! PN ; we will show that the dualizing sheaf (canonically) satisfies the same adjunction formula as the canonical sheaf and hence these sheaves agree. Then, part (a) follows from part (b): Proposition 2.2 (b) implies (a). Proof We have (functorially) i _ ∼ n−i H (X; F) = Ext (F;!X ) ∼ n−i _ = Ext (OX ; F ⊗ !X ) ∼ n−i _ = H (X; F ⊗ !X ): 3 Serre Duality holds for projective space n We prove statement (b) with !P = O(−n − 1); hence we get statement (c) as well. By Proposition 2.2, we get (a) for free. n Lemma 3.1 Let X = P and !X = O(−n − 1). Then (b’) holds. Proof As discussed in the previous section, there is a canonical map Hom(F;!X )× n n n ∼ H (X; F) ! H (X; !X ). Then, choose any isomorphism H (X; !X ) = k. First, if F =∼ O(m) is a line bundle, our pairing is Hom(O(m); O(−n − 1)) × Hn(O(m)) ! Hn(O(−n − 1)) =∼ k. But recall that Hom(O(m); O(−n − 1)) = H0(O(−m−n−1)) can be identified with the space of monomials of degree −m− n−1 in n+1 variables with non-negative exponents, and Hn(O(m));Hn(O(−n− 1)) with the spaces of monomials of degrees m; −n − 1 in n + 1 variables with negative exponents. Then, one checks that the pairing map is multiplication, which is easily seen to be perfect. This proves (b’) for direct sums of line bundles E. Now, let F be arbitrary. Then there exists a short exact sequence 0 !G!E!F! 0 where E is a direct sum of line bundles. Hence we have a diagram 0 / Hom(F;!X ) / Hom(E;!X ) / Hom(G;!X ) 0 / Hn(X; F)_ / Hn(X; E)_ / Hn(X; G)_ where the rows are exact (note that Hn+1(X; G) = 0 because X = Pn has dimension n). The middle vertical arrow is an isomorphism, from which we conclude the left vertical arrow is injective. But then this is true for all F, so the right vertical arrow is injective too. Diagram chasing gives the conclusion. 3 n Proposition 3.2 Let X = P and !X = O(−n − 1). Then (b) holds. i n−i _ Proof We will show that the functors Ext (−;!X );H (X; −) are universal δ-functors. Because we have a isomorphism of functors for i = 0 (part (b’)), this is enough. We have left-exactness for i = 0 and we have long exact sequences, so we only need to check coeffaceability. Let F be arbitrary; then there is a surjec- tion O(−m)N !F for m >> 0 and some N. Indeed, Hn−i(Pn; O(−m)N ) = 0 i for all i > 0, and Ext (O(−m)N ;!) = Hi(Pn; O(m − n − 1))N = 0 for all i > 0. 4 The dualizing sheaf exists in general Lemma 4.1 Let X be an equidimensional projective k-scheme. Then, there exists a finite morphism π : X ! Pn. If X is CM then π is flat. Proof Embed X in PN , and find a codimension n + 1 linear space H missing X. Projecting away from H yields π, and if X is CM then Miracle Flatness [V, Theorem 26.2.11]. This proof only works when k is infinite, but can be modified for k finite [V, Exercise 30.3.D]. Now, let π : X ! Y be a finite morphism and G a quasicoherent OY -module. ! We will define a quasicoherent OX -module π G as follows. First, if Y = Spec B is affine and X = Spec A, G = N, then let M = HomB(A; N), which is a priori a B-module, but in fact an A-module as well. This construction behaves well under localization at an element of B, so it will define a quasicoherent sheaf on ! X. Globally, we thus define the OX -module π G = HomOY (π∗OX ; G). This is quasicoherent, and in fact coherent if G is coherent and π is finite. Recall (Frobenius reciprocity!) that we have the adjunction ∼ HomA(M; HomB(A; N)) = HomB(MB;N): ! ∼ This globalizes to a natural isomorphism π∗HomOX (F; π G) = HomOY (π∗F; G), ! ∼ and thus by taking global sections we get the adjunction HomOX (F; π G) = HomOY (π∗F; G). We can now prove statements (b’) and (b) for X. n ! Lemma 4.2 Let π : X ! Y = P be as above, and set !X = π !Y .
Load more

Recommended publications
  • Curves, Surfaces, and Abelian Varieties
    Curves, Surfaces, and Abelian Varieties Donu Arapura April 27, 2017 Contents 1 Basic curve theory2 1.1 Hyperelliptic curves.........................2 1.2 Topological genus...........................4 1.3 Degree of the canonical divisor...................6 1.4 Line bundles.............................7 1.5 Serre duality............................. 10 1.6 Harmonic forms............................ 13 1.7 Riemann-Roch............................ 16 1.8 Genus 3 curves............................ 18 1.9 Automorphic forms.......................... 20 2 Divisors on a surface 22 2.1 Bezout's theorem........................... 22 2.2 Divisors................................ 23 2.3 Intersection Pairing.......................... 24 2.4 Adjunction formula.......................... 27 2.5 Riemann-Roch............................ 29 2.6 Blow ups and Castelnuovo's theorem................ 31 3 Abelian varieties 34 3.1 Elliptic curves............................. 34 3.2 Abelian varieties and theta functions................ 36 3.3 Jacobians............................... 39 3.4 More on polarizations........................ 41 4 Elliptic Surfaces 44 4.1 Fibered surfaces........................... 44 4.2 Ruled surfaces............................ 45 4.3 Elliptic surfaces: examples...................... 46 4.4 Singular fibres............................. 48 4.5 The Shioda-Tate formula...................... 51 1 Chapter 1 Basic curve theory 1.1 Hyperelliptic curves As all of us learn in calculus, integrals involving square roots of quadratic poly- nomials can be evaluated by elementary methods. For higher degree polynomi- als, this is no longer true, and this was a subject of intense study in the 19th century. An integral of the form Z p(x) dx (1.1) pf(x) is called elliptic if f(x) is a polynomial of degree 3 or 4, and hyperelliptic if f has higher degree, say d. It was Riemann who introduced the geometric point of view, that we should really be looking at the algebraic curve Xo defined by y2 = f(x) 2 Qd in C .
    [Show full text]
  • Orientations for DT Invariants on Quasi-Projective Calabi-Yau 4-Folds
    Orientations for DT invariants on quasi-projective Calabi{Yau 4-folds Arkadij Bojko∗ Abstract For a Calabi{Yau 4-fold (X; !), where X is quasi-projective and ! is a nowhere vanishing section of its canonical bundle KX , the (derived) moduli stack of compactly supported perfect complexes MX is −2-shifted symplectic and thus has an orientation ! bundle O !MX in the sense of Borisov{Joyce [11] necessary for defining Donaldson{ Thomas type invariants of X. We extend first the orientability result of Cao{Gross{ Joyce [15] to projective spin 4-folds. Then for any smooth projective compactification Y , such that D = Y nX is strictly normal crossing, we define orientation bundles on the stack MY ×MD MY and express these as pullbacks of Z2-bundles in gauge theory, constructed using positive Dirac operators on the double of X. As a result, we relate the ! orientation bundle O !MX to a gauge-theoretic orientation on the classifying space of compactly supported K-theory. Using orientability of the latter, we obtain orientability of MX . We also prove orientability of moduli spaces of stable pairs and Hilbert schemes of proper subschemes. Finally, we consider the compatibility of orientations under direct sums. 1 Introduction 1.1 Background and results Donaldson-Thomas type theory for Calabi{Yau 4-folds has been developed by Borisov{ Joyce [11], Cao{Leung [22] and Oh{Thomas [73] . The construction relies on having Serre duality Ext2(E•;E•) =∼ Ext2(E•;E•)∗ : for a perfect complex E•. This gives a real structure on Ext2(E•;E•), and one can take 2 • • its real subspace Ext (E ;E )R.
    [Show full text]
  • Section 3.7 - the Serre Duality Theorem
    Section 3.7 - The Serre Duality Theorem Daniel Murfet October 5, 2006 In this note we prove the Serre duality theorem for the cohomology of coherent sheaves on a projective scheme. First we do the case of projective space itself. Then on an arbitrary projective ◦ scheme X, we show that there is a coherent sheaf ωX , which plays the role in duality theory similar to the canonical sheaf of a nonsingular variety. In particular, if X is Cohen-Macaulay, it gives a duality theorem just like the one on projective space. Finally, if X is a nonsingular variety ◦ over an algebraically closed field we show that the dualising sheaf ωX agrees with the canonical sheaf ωX . n Theorem 1. Suppose X = Pk for some field k and n ≥ 1 and set ω = ωX/k. Then for any coherent sheaf of modules F on X we have (a) There is a canonical isomorphism of k-modules Hn(X, ω) =∼ k. (b) There is a canonical perfect pairing of finite dimensional vector spaces over k τ : Hom(F , ω) × Hn(X, F ) −→ k (c) For i ≥ 0 there is a canonical isomorphism of k-modules natural in F η : Exti(F , ω) −→ Hn−i(X, F )∨ Proof. (a) By (DIFF,Corollary 22) there is a canonical isomorphism of sheaves of modules ω =∼ O(−n−1). Composing Hn(X, ω) =∼ Hn(X, O(−n−1)) with the isomorphism Hn(X, O(−n−1)) =∼ k of (COS,Theorem 40)(b) gives the required canonical isomorphism of k-modules. (b) The k-modules Hom(F , ω) and Hn(X, F ) are finitely generated by (COS,Corollary 44) and (COS,Theorem 43), therefore both are free of finite rank.
    [Show full text]
  • An Algebraic Proof of Riemann-Roch
    BABY ALGEBRAIC GEOMETRY SEMINAR: AN ALGEBRAIC PROOF OF RIEMANN-ROCH RAVI VAKIL Contents 1. Introduction 1 2. Cohomology of sheaves 2 3. Statements of Riemann-Roch and Serre Duality; Riemann-Roch from Serre Duality 4 4. Proof of Serre duality 5 4.1. Repartitions, I(D), and J(D)5 4.2. Differentials enter the picture 7 4.3. Proving duality 7 References 8 1. Introduction I’m going to present an algebraic proof of Riemann-Roch. This is a hefty task, especially as I want to say enough that you can genuinely fill in all the details yourself. There’s no way I can finish in an hour without making this less self- contained, so what I’ll do is explain a bit about cohomology, and reduce Riemann- Roch to Serre duality. That should be do-able within an hour. Then I’ll give people a chance to leave, and after that I’ll prove Serre duality in the following half-hour. Also, these notes should help. And you should definitely stop me and ask ques- tions. For example, if I mention something I defined last semester in my class, and you’d like me to refresh your memory as to what the definition was, please ask. The proof I’ll present is from Serre’s Groupes alg´ebriques et corps de classes,Ch. 2, [S]. I found out this past Tuesday that this proof is originally due to Weil. Throughout, C is a non-singular projective algebraic curve over an algebraically closed field k. Date: Friday, February 11, 2000.
    [Show full text]
  • Residues, Duality, and the Fundamental Class of a Scheme-Map
    RESIDUES, DUALITY, AND THE FUNDAMENTAL CLASS OF A SCHEME-MAP JOSEPH LIPMAN Abstract. The duality theory of coherent sheaves on algebraic vari- eties goes back to Roch's half of the Riemann-Roch theorem for Riemann surfaces (1870s). In the 1950s, it grew into Serre duality on normal projective varieties; and shortly thereafter, into Grothendieck duality for arbitrary varieties and more generally, maps of noetherian schemes. This theory has found many applications in geometry and commutative algebra. We will sketch the theory in the reasonably accessible context of a variety V over a perfect field k, emphasizing the role of differential forms, as expressed locally via residues and globally via the fundamental class of V=k. (These notions will be explained.) As time permits, we will indicate some connections with Hochschild homology, and generalizations to maps of noetherian (formal) schemes. Even 50 years after the inception of Grothendieck's theory, some of these generalizations remain to be worked out. Contents Introduction1 1. Riemann-Roch and duality on curves2 2. Regular differentials on algebraic varieties4 3. Higher-dimensional residues6 4. Residues, integrals and duality7 5. Closing remarks8 References9 Introduction This talk, aimed at graduate students, will be about the duality theory of coherent sheaves on algebraic varieties, and a bit about its massive|and still continuing|development into Grothendieck duality theory, with a few indications of applications. The prerequisite is some understanding of what is meant by cohomology, both global and local, of such sheaves. Date: May 15, 2011. 2000 Mathematics Subject Classification. Primary 14F99. Key words and phrases. Hochschild homology, Grothendieck duality, fundamental class.
    [Show full text]
  • SERRE DUALITY and APPLICATIONS Contents 1
    SERRE DUALITY AND APPLICATIONS JUN HOU FUNG Abstract. We carefully develop the theory of Serre duality and dualizing sheaves. We differ from the approach in [12] in that the use of spectral se- quences and the Yoneda pairing are emphasized to put the proofs in a more systematic framework. As applications of the theory, we discuss the Riemann- Roch theorem for curves and Bott's theorem in representation theory (follow- ing [8]) using the algebraic-geometric machinery presented. Contents 1. Prerequisites 1 1.1. A crash course in sheaves and schemes 2 2. Serre duality theory 5 2.1. The cohomology of projective space 6 2.2. Twisted sheaves 9 2.3. The Yoneda pairing 10 2.4. Proof of theorem 2.1 12 2.5. The Grothendieck spectral sequence 13 2.6. Towards Grothendieck duality: dualizing sheaves 16 3. The Riemann-Roch theorem for curves 22 4. Bott's theorem 24 4.1. Statement and proof 24 4.2. Some facts from algebraic geometry 29 4.3. Proof of theorem 4.5 33 Acknowledgments 34 References 35 1. Prerequisites Studying algebraic geometry from the modern perspective now requires a some- what substantial background in commutative and homological algebra, and it would be impractical to go through the many definitions and theorems in a short paper. In any case, I can do no better than the usual treatments of the various subjects, for which references are provided in the bibliography. To a first approximation, this paper can be read in conjunction with chapter III of Hartshorne [12]. Also in the bibliography are a couple of texts on algebraic groups and Lie theory which may be beneficial to understanding the latter parts of this paper.
    [Show full text]
  • Lectures on the Geometric Langlands Conjecture and Non-Abelian Hodge Theory
    Lectures on the geometric Langlands conjecture and non-abelian Hodge theory Ron Donagi Tony Pantev University of Pennsylvania Department of Mathematics Philadelphia, PA 19104-6395 2 Contents 1 Introduction 5 2 Review of the geometric Langlands conjecture 9 2.1 The geometric Langlands conjecture in general . .9 2.2 The geometric Langlands conjecture for GLn(C)................ 14 3 Higgs bundles, the Hitchin system, and abelianization 17 3.1 Higgs bundles and the Hitchin map . 17 3.2 Using abelianization . 19 4 The classical limit 27 4.1 The classical limit conjecture . 27 4.2 Duality of Hitchin systems . 29 5 Non-abelian Hodge theory 33 5.1 Results from non-abelian Hodge theory . 33 5.2 Using non-abelian Hodge theory . 36 6 Parabolic Higgs sheaves on the moduli of bundles 43 6.1 Wobbly, shaky, and unstable bundles . 43 6.2 On functoriality in non-abelian Hodge theory . 45 A Ind-schemes and smoothness 47 3 4 Contents Chapter 1 Introduction In these lectures which are a slightly expanded version of the survey [DP09a] we will focus on the aspects of the geometric Langlands Conjecture and that relate it to non abelian Hodge theory. This relation was unravelled in works by many mathematicians and physicists, but we will emphasize the point of view that evolved in a series of works by the authors, starting from the outline in [Don89], through the recent proof of the classical limit conjecture in [DP06], and leading to the works in progress [DP09b], [DPS09b], and [DPS09a]. The Langlands program is the non-abelian extension of class field theory.
    [Show full text]
  • Hypercohomologies of Truncated Twisted Holomorphic De Rham Complexes 3
    HYPERCOHOMOLOGIES OF TRUNCATED TWISTED HOLOMORPHIC DE RHAM COMPLEXES LINGXU MENG Abstract. We investigate the hypercohomologies of truncated twisted holomorphic de Rham complexes on (not necessarily compact) complex manifolds. In particular, we general- ize Leray-Hirsch, K¨unneth and Poincar´e-Serre duality theorems on them. At last, a blowup formula is given, which affirmatively answers a question posed by Chen, Y. and Yang, S. in [3]. 1. Introduction All complex manifolds mentioned in this paper are connected and not necessarily compact unless otherwise specified. Let X be a complex manifold. The de Rham theorem says that the cohomology Hk(X, C) with complex coefficients can be computed by the hypercohomology Hk • • (X, ΩX ) of the holomorphic de Rham complex ΩX . Moreover, if X is a compact K¨ahler F pHk X, C Hr,k−r X, C manifold, the Hodge filtration dR( )= ∂¯ ( ) on the de Rham cohomology rL≥p k C Hk ≥p HdR(X, ) is just the hypercohomology (X, ΩX ) of the truncated holomorphic de Rham ≥p complex ΩX , which plays a significant role in the Hodge theory. The Deligne cohomology is an important object in the research of algebraic cycles, since it connects with the intermediate Jacobian and the integeral Hodge class group. It has a strong relation with the singular ∗ Z H∗ ≤p−1 cohomology H (X, ) and the hypercohomology (X, ΩX ) of the truncated holomorphic ≤p−1 de Rham complex ΩX via the long exact sequence ··· / k−1 Z / Hk−1 ≤p−1 / k Z / k Z / Hk ≤p−1 / ··· H (X, ) (X, ΩX ) HD (X, (p)) H (X, ) (X, ΩX ) .
    [Show full text]
  • 6 Serre Duality
    6 Serre duality Recall the famous Poincar´eduality, which is a basic result on the structure of singular homology and cohomology groups of manifolds: Let X be an n-dimensional orientable closed manifold. Then for any integer i i,wehaveH (X, R) Hn i(X, R). ' − n i Thanks to the universal coefficient theorem, the right-hand-side is isomorphic to H − (X, R)_, the dual of the (n i)-th cohomology group. Assume that X is a smooth manifold. Then − one may interpret this duality as a perfect pairing as follows. Applying de Rham’s the- i i orem, one may identify H (X, R)=HdR(X, R), and hence, the composition of the cup product map and the integration map Hi(X, R) n i n ⇥ H − (X, R) H (X, R) R defined by (⌘,⇠) (⌘ ⇠) gives a perfect ! ! 7! X ^ pairing. R Since the sheaf cohomology generalizes the singular cohomology, we may expect there is an analogous “duality” theorem for varieties/schemes – at least, under mild assumptions on the underlying space. Serre duality is a special case of the duality called the coherent duality in a much general setting. It is based on earlier works in several complex variables, however, we will observe it algebraically. One di↵erence between the Poincar´eduality and the Serre duality is the role of the “dualizing sheaf”. We will see that the dualizing sheaf coincides with the canonical sheaf when the underlying space X is a nonsingular variety over an algebraically closed field. n First, we begin with the duality on the projective space.
    [Show full text]
  • Nonabelian Hodge Theory and Vector Valued Modular Forms
    Contemporary Mathematics Nonabelian Hodge theory and vector valued modular forms Cameron Franc and Steven Rayan Abstract. We examine the relationship between nonabelian Hodge theory for Rie- mann surfaces and the theory of vector valued modular forms. In particular, we explain how one might use this relationship to prove a conjectural three-term inequal- ity on the weights of free bases of vector valued modular forms associated to complex, finite dimensional, irreducible representations of the modular group. This conjecture is known for irreducible unitary representations and for all irreducible representations of dimension at most 12. We prove new instances of the three-term inequality for cer- tain nonunitary representations, corresponding to a class of maximally-decomposed variations of Hodge structure, by considering the same inequality with respect to a new type of modular form, called a \Higgs form", that arises naturally on the Dol- beault side of nonabelian Hodge theory. The paper concludes with a discussion of a strategy for reducing the general case of nilpotent Higgs bundles to the case under consideration in our main theorem. Contents 1. Introduction 1 2. Filtered objects 4 3. Vector valued modular forms 9 4. Nonabelian Hodge theory 10 5. New cases of the three-term inequality for PSL2(Z) 18 References 21 1. Introduction Suppose that Y is a hyperbolic orbifold curve uniformized by the complex upper half plane H, so that Y can be identified with a quotient Y = Γn H of H under the action of a discrete group of linear fractional transformations Γ. Assume that Γ is a Fuchsian group of the first kind, so that Y has finite volume and a finite number of cusps.
    [Show full text]
  • 18.726 Algebraic Geometry Spring 2009
    MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry (K.S. Kedlaya, MIT, Spring 2009) Higher Riemann-Roch In this lecture, we discuss some higher-dimensional versions of the Riemann-Roch theo­ rem: the Riemann-Roch theorem for surfaces, the Hirzebruch-Riemann-Roch theorem, and the Grothendieck-Riemann-Roch theorem. For the first, see Hartshorne V.1; for the others, see Chapter 15 of Fulton’s Intersection Theory (a well-deserved winner of the Steele Prize for mathematical exposition). 1 Surfaces Let X be a smooth irreducible projective surface over an algebraically closed field k. Let K be a canonical divisor on X. As in the case of curves, the Riemann-Roch theorem combines an input from Serre duality with an Euler characteristic calculation. The input from Serre duality is that for any divisor D, 0 ∨ ∼ 2 ∨ H (X, L(D) ⊗ ωX ) = H (X, L(D)) . We can thus write the Euler characteristic χ(X, L(D)) as 0 1 0 dimk H (X, L(D)) − dimk H (X, L(D)) + dimk H (X, L(K − D)). 1 Unfortunately, we can’t do much with the term dimk H (X, L(D)) other than give it a name: it’s called the superabundance of D. However, we do at least know that it is nonnegative, and this turns out to be surprisingly useful. The Euler characteristic calculation is made as follows. Write D as the difference between two effective divisors C − E with no common components.
    [Show full text]
  • Global Topology of the Hitchin System
    Global topology of the Hitchin system Tamás Hausel Abstract. Here we survey several results and conjectures on the cohomology of the total space of the Hitchin system: the moduli space of semi-stable rank n and degree d Higgs bundles on a complex algebraic curve C. The picture emerging is a dynamic mixture of ideas originating in theoretical physics such as gauge theory and mirror symmetry, Weil conjectures in arithmetic algebraic geometry, representation theory of finite groups of Lie type and Langlands duality in number theory. Contents 1 Introduction 30 1.1 Mirror symmetry 33 1.2 Langlands duality 34 1.3 Hitchin system 36 2 Higgs bundles and the Hitchin system 37 2.1 The moduli space of vector bundles on a curve 37 2.2 The Hitchin system 38 2.3 Hitchin systems for SLn and PGLn 40 2.4 Cohomology of Higgs moduli spaces 41 3 Topological mirror symmetry for Higgs bundles 41 3.1 Spectral curves 42 3.2 Generic fibres of the Hitchin map 42 3.3 Symmetries of the Hitchin fibration 44 3.4 Duality of the Hitchin fibres 45 3.5 Gerbes on Mˇ and Mˆ 46 3.6 The stringy Serre polynomial of an orbifold 47 3.7 Topological mirror test 48 3.8 Topological mirror symmetry for n = 2 51 4 Topological mirror symmetry for character varieties 52 4.1 An arithmetic technique to calculate Serre polynomials 54 4.2 Arithmetic harmonic analysis on MB 55 4.3 The case n = 2 for TMS-B 56 2000 Mathematics Subject Classification. Primary 14D20; Secondary 14J32 14C30 20C33 .
    [Show full text]