# The Hitchin map, local to global

Size: px
Start display at page:

Transcription

1 The Hitchin map, local to global Andrei Negut Let X be a smooth projective curve of genus g > 1, a semisimple group and Bun = Bun (X) the moduli stack of principal bundles on X. In this talk, we will present Hichin s construction of a middle-dimensional family of Poisson commuting functions on T Bun. This construction will be quantized in subsequent lectures to a middle-dimensional family of (twisted) differential operators on Bun. Let C = Spec(Sym g), the affine quotient of g with respect to the adjoint action. Note that (Sym g) = (Sym h) W, which is non-canonically isomorphic to a polynomial ring over C. Therefore, our C is non-canonically isomorphic to affine space. There is a natural C action on C, induced from scalar multiplication on the vector space g. Therefore, we can construct the fiber bundle on X C ΩX := Ω X C C, where the C torsor Ω X is just the canonical line bundle of X. Define the Hitchin variety to be the space of global sections of this fiber bundle: Hitch(X) = Γ(X, C ΩX ). This is an affine space of dimension (g 1) dim, so we can associate to it its ring of global functions z cl. This ring will be precisely the family of Poisson commuting functions on T Bun we are looking for. 1

2 Example 1 When = L n, any invariant function on elements g g is completely determined by the coefficients of the characteristic polynomial of g. Therefore, (Sym g) = C[e 1,..., e n ], where e i = Tr(g i ). Therefore, C = A n with a basis e 1,..., e n, and λ C acts on C by the diagonal matrix (λ, λ 2,..., λ n ) in this basis. This implies that and therefore C ΩX = Ω X Ω 2 X... Ω n X, Hitch(X) = Γ(X, Ω X ) Γ(X, Ω 2 X )... Γ(X, Ω n X ). Thus we recover the definition that Dennis presented in the first lecture. iven a principal bundle F on X, the quotient map g C induces a map g F = g F C C O X of bundles on X (note that the affine space C does not get twisted by F). Twisting this by Ω X we obtain the map g F Ω X C C Ω X = C ΩX, and passing to global sections we get the map µ F : Γ(X, g F Ω X ) Hitch(X). This is very important, because the space on the left is nothing but H 0 of the fiber of the tangent complex T Bun above F. To see this, recall that H 0 (T F Bun ) = H 1 (X, g F ), which is dual to our Γ(X, g F Ω X) by Serre duality. Therefore, the above map patches over all F to the global Hitchin map: µ : T Bun Hitch(X). (1) Passing to global sections, we get the map: h cl : z cl Γ(T Bun, O). The image of h cl will be our desired family of Poisson commuting functions on T Bun, as stated in Theorem 1 below. 2

3 Note that we have only constructed the Hitchin map µ on C points. To prove that it is a map of functors, we should construct it on S points, where S is any smooth scheme. This extra construction does not present any conceptual difficulties, but for the sake of completeness let s make it rigorous this one time. iven a scheme S and F a principal bundle on X S, the Hitchin map is T FBun (S) = Γ(X S, g F Ω X ) Γ(X S, C ΩX ) = Hitch(X)(S). Example 2 Let us again look at = L n. A point on Bun n is nothing but a rank n vector bundle M on X, which corresponds to the principal bundle F U = Isom(O n, M). Then we have the following isomorphism g F = g Isom(O n, M) = Hom(M, M), where the isomorphism is given by Then, we have (a g, φ : O n M) = φ a φ 1 : M M. (2) g F Ω X = Hom(M, M ΩX ) Γ(X, g F Ω X ) = Hom(M, M Ω X ). This just says that a cotangent vector to T Bun n at M is merely a global sheaf homomorphism f : M M Ω X. To see where is f mapped via the Hitchin map, one must just compute Tr(a i ) in the left hand side of (2). This locally equals Tr(f i ), and when we pass to global sections we must take the twist by Ω X into account. Therefore, µ(f) = Tr(f)... Tr(f n ) Γ(X, Ω X )... Γ(X, Ω n X ) = Hitch(X), where f i denotes the composition M f M Ω X f Id M Ω 2 X... M Ω i X. We thus recover Dennis description from the first lecture. The connected components of Bun are indexed by π 1 () (for example, when = C this means that line bundles in Pic(X) are distributed among the connected components according to their degrees). For γ π 1 (), let Bun γ denote the connected component of Bun corresponding to γ, and let µ γ : T Bun γ Hitch(X) denote the restriction of the Hitchin map. The main point of this lecture is the following theorem. 3

4 Theorem 1 The following hold: 1. The image of h cl consists of Poisson commuting functions. 2. Each map µ γ is surjective, and the morphism it induces on the structure rings h cl γ : z cl Γ(T Bun γ, O) is an isomorphism. In this talk, I will prove statement 1 of the theorem. The method of proof will be different from Hitchin s original one, and will involve a local-to-global principle. This principle will be used to prove the analogous statament for the quantization in future lectures. Fix a closed point x X. The local picture means replacing the curve X by the formal neighborhood of x in X. More explicitly, let O x be the local ring of X at x, and m x O x be the maximal ideal. Define Ô x = lim n O x /m n x. The space Spec Ôx is called the formal neighborhood of x. The complex analytic intuition behind this is that when X = C and x = 0, then Ôx = C[[t]] and Spec Ôx is the formal disk centered at the origin. For each n, we have a natural inclusion Spec O x /m n x X as a closed subscheme, which induces a map of schemes Spec Ôx X. Recalling the bundle C ΩX over X, we then obtain a map on sections: Hitch(X) = Γ(X, C ΩX ) Γ(Spec Ôx, C Ω ) =: Hitch x (X). (3) Here we denote by Ω the sheaf of differentials on Spec Ôx. The above map is an embedding of affine spaces, because any section of the bundle C ΩX on the smooth curve X is completely determined by its restriction to the formal neighborhood Spec Ôx (just like any homolorphic function is completely determined by its Taylor series at 0). Let z cl x be the ring of functions on Hitch x (X). Then the above embedding gives us a surjective morphism: θ cl : z cl x z cl. 4

5 Composing the global Hitchin map (1) with (3), we obtain the local Hitchin map: µ x : T Bun Hitch x (X). This induces a map on rings of functions: which naturally factors through z cl : h cl x : z cl x Γ(T Bun, O), h cl x = h cl θ cl. (4) Proof of Theorem 1, Statement 1: Since θ cl is a surjection, by (4) it is enough to prove that the image of h cl x consists of Poisson commuting functions. For this we will place a trivial Poisson structure on z cl x and show that h cl x is a morphism of Poisson algebras. Recall from Sam s lecture that Harish-Chandra pair (h, L) consists of an algebraic group L acting on the Lie algebra h, and an embedding l = Lie L h that intertwines the adjoint action of L on l and the given action on h. The Lie bracket on h induces a Poisson bracket on Sym h. We define: I cl = (Sym h)l, P cl = {x Sym h {x, I cl } I cl } I cl. The object of interest will be the Poisson algebra P cl := ( P cl /I cl ) π 0(L) = (Sym (h/l)) L. (5) We say that a Harish-Chandra pair (h, L) acts on a scheme Y if we are given an action of L on Y and a L equivariant map of Lie algebras h Γ(Y, T Y ) which restricts to the infinitesimal action on l h. This map of Lie algebras induces the following commutative diagram, where the horizontal arrows are 5

6 maps of Poisson algebras: Sym(h) Sym(h/l) P cl = (Sym(h/l)) L Ã Γ(T Y, O) B Γ(T Y Y Y, O) C Γ(T Y, O) The vertical maps are the standard inclusions/projections. eometrically, the above induces the following commutative diagram: T Y T Y Y Y T Y A B Spec(Sym(h)) Spec(Sym(h/l)) C Spec(Sym(h/l)) L We will seek apply the above framework to h = g K x, L = (Ôx), Y = and Y = Bun. Then we have the map of Poisson algebras: Bun ( x) C : P cl = (Sym(g K x /Ôx)) (Ôx) Γ(T Bun, O). Our theorem then reduces to the following two claims: 1. There exists an map χ : z cl x P cl such that 2. The Poisson bracket on Im( χ) P cl is trivial. C χ = h cl x. (6) This would conclude the proof, since then Im(h cl ) = C(Im( χ)) and C preserves the Poisson bracket. Since this bracket is trivial on Im( χ), it is also trivial on Im(h cl x ). 6

7 To decipher what the maps A, B, C look like in our situation, take a point F Bun. By definition, a tangent vector (deformation) to Bun at F is a principal bundle F ε on X Spec (C[ε]/ε 2 ), which restricts to F when ε = 0. If we take a faithfully flat affine cover {U i X}, then F ε is determined by the glueing data: ϕ ij : U ij g F, (7) satisfying the appropriate cocycle condition. Note that the we twist the vector space g by F in order to eliminate any non-canonical choices in the trivializations of F itself. This gives us a Cech cocycle in Z 1 (X, g F ), which is a coboundary in B 1 (X, g F ) precisely when the deformation is trivial. This implies that: T F Bun = H 1 (X, g F ) T FBun = Γ(X, g F Ω X ), by Serre duality. Recall that: Bun ( x) = {(F, ψ)} = lim{(f, ψ (n) )} = lim n n Bun (nx), where ψ (respectively ψ (n) ) is a trivialization of F on Spec Ôx (respectively Spec O x /m n x). By a similar argument with the previous paragraph, the tangent space to Bun (nx) at (F, ψ (n) ) is H 1 (X, g F ( nx)). Taking the projective limit, we obtain a description for the tangent space to Bun ( x) at the point (F, ψ): again by Serre duality. T (F,ψ) Bun ( x) T (F,ψ)Bun ( x) = lim H 1 (X, g F ( nx)) n = (lim n H 1 (X, g F ( nx))) = = lim n Γ(X, g F(nx) Ω X ) = Γ(X x, g F Ω X ), There is a natural action of ( K x ) on Bun ( x) : trivialize F on the cover X x Spec Ôx, then F will be determined by a cocycle Spec K x, and 7

8 let ( K x ) act on this cocycle by multiplication. As in the general framework discussed earlier, we have the commutative diagram: T Bun ( x) T Bun Bun Bun ( x) T Bun A Spec(Sym(g K x )) B Spec(Sym(g K x /Ôx)) C Spec(Sym(g K x /Ôx)) (Ôx). (8) The diagram is naturally commutative. The two top vertical arrows are closed embeddings, whereas the bottom two vertical arrows are dominant maps. The map A is explicitly given by: A(F, ψ, f) = (z Res x f Spec Kx, z ) (g K x ), for any (F, ψ) Bun ( x) and f Γ(X x, g F Ω X). To make sense of the above pairing, note that any z g K x can be perceived as a function Spec K x g. This can further be perceived as a function Spec K x g F via the trivialization ψ. Then pairing this with f Spec gives an element Kx in Γ(Spec K x, Ω X ), i.e. a differential on the punctured formal disk whose residue we can take. The diagram (8) can be completed with the following: Spec(Sym(g K x )) Spec(Sym(g K x /Ôx)) Spec(Sym(g K x /Ôx)) (Ôx) χ Γ(Spec K x, g Ω ) χ Γ(Spec Ôx, g Ω ) χ Γ(Spec Ôx, C Ω ) = Hitch x (X). The vertical maps are the standard inclusions/projections. To define the maps χ, χ, χ, note that Ω is the sheaf of differentials. The bilinear form (f, ω) Res x (fω) represents a perfect pairing between elements f K x (9) 8

9 and global differentials ω Γ(Spec K x, Ω) on the punctured formal disk. This produces a canonical isomorphism: Γ(Spec K x, Ω) = K x Γ(Spec K x, g Ω) χ = (g Kx ). (10) The isomorphism χ is defined similarly (it turns out to be (Ôx) equivariant), and χ is the map canonically induced on the IT quotient. We will soon show that it is also an isomorphism. As in diagram (8), the upper two vertical maps in (9) are closed embeddings, while the lower two vertical maps are dominant. By unraveling the definitions, we note that: χ A(F, ψ, f) = f Spec Kx, where g = g F via the trivialization ψ. Since the top vertical maps are all closed embeddings, it follows that: χ B(F, ψ, f) = f Spec Ô x, since f has no more poles at x now. Finally, since the lower vertical maps are dominant, we obtain: χ C(F, f) = (f Spec Ô x )//(Ôx) χ C = µ cl x. Passing to rings of global functions, we obtain statement (6). Remark 1 Let s actually show that χ is an isomorphism. It is easy to see that it is dominant. To show it is a closed embedding, we need to show that any (Ôx) invariant function on Γ(Spec Ôx, g Ω ) comes from a function on Γ(Spec Ôx, C Ω ). But restricting to an open subset whose complement has codimension > 1 does not affect rings of global functions. Therefore, one can replace g with g reg (the codimension 3 locus of regular elements of g) and C by g reg /. Then the desired statement is immediate, since the action on g reg is smooth and transitive on fibers. Finally, we need to prove that the Poisson bracket on Im( χ) is trivial. 9

10 This follows from the commutative diagram: Spec(Sym(g K x /Ôx)) (Ôx) Spec(Sym(g K x ) g K x) π 0 () χ Γ(Spec Ôx, C Ω ) Γ(Spec K x, C Ω ) The lower horizontal arrow is just the restriction map from the closed disk to the punctured disk, obviously a closed embedding. The vertical right morphism is the analogue of χ for K x, defined by pushing the completion of χ down to the affine ( K x ) quotient. The upper horizontal map is induced by the quotient map. The diagram is naturally commutative. Passing to rings of global functions in the above diagram, the bottom horizontal map becomes a surjection. Therefore Im( χ) is contained in the image of the top horizontal map: D := Sym(g K x ) g K x Sym(g Kx /Ôx) (Ôx) = P cl. But the above is just the natural quotient map, and thus a map of Poisson algebras. So it is enough to show that the Poisson bracket on D is trivial. This is clear, because the Poisson bracket is trivial on the non-completed Sym(g K x ) g K x, which is dense in D. This concludes our proof. 10

### QUANTIZATION VIA DIFFERENTIAL OPERATORS ON STACKS

QUANTIZATION VIA DIFFERENTIAL OPERATORS ON STACKS SAM RASKIN 1. Differential operators on stacks 1.1. We will define a D-module of differential operators on a smooth stack and construct a symbol map when

### An Atlas For Bun r (X)

An Atlas For Bun r (X) As told by Dennis Gaitsgory to Nir Avni October 28, 2009 1 Bun r (X) Is Not Of Finite Type The goal of this lecture is to find a smooth atlas locally of finite type for the stack

### SEMINAR NOTES: QUANTIZATION OF HITCHIN S INTEGRABLE SYSTEM AND HECKE EIGENSHEAVES (SEPT. 8, 2009)

SEMINAR NOTES: QUANTIZATION OF HITCHIN S INTEGRABLE SYSTEM AND HECKE EIGENSHEAVES (SEPT. 8, 2009) DENNIS GAITSGORY 1. Hecke eigensheaves The general topic of this seminar can be broadly defined as Geometric

### Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo. Alex Massarenti

Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo Alex Massarenti SISSA, VIA BONOMEA 265, 34136 TRIESTE, ITALY E-mail address: alex.massarenti@sissa.it These notes collect a series of

### We can choose generators of this k-algebra: s i H 0 (X, L r i. H 0 (X, L mr )

MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 43 5.3. Linearisations. An abstract projective scheme X does not come with a pre-specified embedding in a projective space. However, an ample line bundle

### Conformal blocks for a chiral algebra as quasi-coherent sheaf on Bun G.

Conformal blocks for a chiral algebra as quasi-coherent sheaf on Bun G. Giorgia Fortuna May 04, 2010 1 Conformal blocks for a chiral algebra. Recall that in Andrei s talk [4], we studied what it means

### Representations and Linear Actions

Representations and Linear Actions Definition 0.1. Let G be an S-group. A representation of G is a morphism of S-groups φ G GL(n, S) for some n. We say φ is faithful if it is a monomorphism (in the category

### BRIAN OSSERMAN. , let t be a coordinate for the line, and take θ = d. A differential form ω may be written as g(t)dt,

CONNECTIONS, CURVATURE, AND p-curvature BRIAN OSSERMAN 1. Classical theory We begin by describing the classical point of view on connections, their curvature, and p-curvature, in terms of maps of sheaves

### PICARD GROUPS OF MODULI PROBLEMS II

PICARD GROUPS OF MODULI PROBLEMS II DANIEL LI 1. Recap Let s briefly recall what we did last time. I discussed the stack BG m, as classifying line bundles by analyzing the sense in which line bundles may

### Math 249B. Nilpotence of connected solvable groups

Math 249B. Nilpotence of connected solvable groups 1. Motivation and examples In abstract group theory, the descending central series {C i (G)} of a group G is defined recursively by C 0 (G) = G and C

### Introduction to Chiral Algebras

Introduction to Chiral Algebras Nick Rozenblyum Our goal will be to prove the fact that the algebra End(V ac) is commutative. The proof itself will be very easy - a version of the Eckmann Hilton argument

### 1. Algebraic vector bundles. Affine Varieties

0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasi-projective varieties over a field k Affine Varieties 1.

### ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES

ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES HONGHAO GAO FEBRUARY 7, 2014 Quasi-coherent and coherent sheaves Let X Spec k be a scheme. A presheaf over X is a contravariant functor from the category of open

### THE HITCHIN FIBRATION

THE HITCHIN FIBRATION Seminar talk based on part of Ngô Bao Châu s preprint: Le lemme fondamental pour les algèbres de Lie [2]. Here X is a smooth connected projective curve over a field k whose genus

### HARTSHORNE EXERCISES

HARTSHORNE EXERCISES J. WARNER Hartshorne, Exercise I.5.6. Blowing Up Curve Singularities (a) Let Y be the cusp x 3 = y 2 + x 4 + y 4 or the node xy = x 6 + y 6. Show that the curve Ỹ obtained by blowing

### Chern classes à la Grothendieck

Chern classes à la Grothendieck Theo Raedschelders October 16, 2014 Abstract In this note we introduce Chern classes based on Grothendieck s 1958 paper [4]. His approach is completely formal and he deduces

### BEZOUT S THEOREM CHRISTIAN KLEVDAL

BEZOUT S THEOREM CHRISTIAN KLEVDAL A weaker version of Bézout s theorem states that if C, D are projective plane curves of degrees c and d that intersect transversally, then C D = cd. The goal of this

### ARITHMETICALLY COHEN-MACAULAY BUNDLES ON HYPERSURFACES

ARITHMETICALLY COHEN-MACAULAY BUNDLES ON HYPERSURFACES N. MOHAN KUMAR, A. P. RAO, AND G. V. RAVINDRA Abstract. We prove that any rank two arithmetically Cohen- Macaulay vector bundle on a general hypersurface

### DONAGI-MARKMAN CUBIC FOR HITCHIN SYSTEMS arxiv:math/ v2 [math.ag] 24 Oct 2006

DONAGI-MARKMAN CUBIC FOR HITCHIN SYSTEMS arxiv:math/0607060v2 [math.ag] 24 Oct 2006 DAVID BALDUZZI Abstract. The Donagi-Markman cubic is the differential of the period map for algebraic completely integrable

### Notes on p-divisible Groups

Notes on p-divisible Groups March 24, 2006 This is a note for the talk in STAGE in MIT. The content is basically following the paper [T]. 1 Preliminaries and Notations Notation 1.1. Let R be a complete

### Symplectic varieties and Poisson deformations

Symplectic varieties and Poisson deformations Yoshinori Namikawa A symplectic variety X is a normal algebraic variety (defined over C) which admits an everywhere non-degenerate d-closed 2-form ω on the

### MATH 8253 ALGEBRAIC GEOMETRY WEEK 12

MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y -scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f

### Cohomological Formulation (Lecture 3)

Cohomological Formulation (Lecture 3) February 5, 204 Let F q be a finite field with q elements, let X be an algebraic curve over F q, and let be a smooth affine group scheme over X with connected fibers.

### DERIVED HAMILTONIAN REDUCTION

DERIVED HAMILTONIAN REDUCTION PAVEL SAFRONOV 1. Classical definitions 1.1. Motivation. In classical mechanics the main object of study is a symplectic manifold X together with a Hamiltonian function H

### THE MODULI OF SUBALGEBRAS OF THE FULL MATRIX RING OF DEGREE 3

THE MODULI OF SUBALGEBRAS OF THE FULL MATRIX RING OF DEGREE 3 KAZUNORI NAKAMOTO AND TAKESHI TORII Abstract. There exist 26 equivalence classes of k-subalgebras of M 3 (k) for any algebraically closed field

### 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

### D-MODULES: AN INTRODUCTION

D-MODULES: AN INTRODUCTION ANNA ROMANOVA 1. overview D-modules are a useful tool in both representation theory and algebraic geometry. In this talk, I will motivate the study of D-modules by describing

### Math 797W Homework 4

Math 797W Homework 4 Paul Hacking December 5, 2016 We work over an algebraically closed field k. (1) Let F be a sheaf of abelian groups on a topological space X, and p X a point. Recall the definition

### Lectures on Galois Theory. Some steps of generalizations

= Introduction Lectures on Galois Theory. Some steps of generalizations Journée Galois UNICAMP 2011bis, ter Ubatuba?=== Content: Introduction I want to present you Galois theory in the more general frame

### Geometric Class Field Theory

Geometric Class Field Theory Notes by Tony Feng for a talk by Bhargav Bhatt April 4, 2016 In the first half we will explain the unramified picture from the geometric point of view, and in the second half

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24 RAVI VAKIL CONTENTS 1. Vector bundles and locally free sheaves 1 2. Toward quasicoherent sheaves: the distinguished affine base 5 Quasicoherent and coherent sheaves

### Factorization of birational maps for qe schemes in characteristic 0

Factorization of birational maps for qe schemes in characteristic 0 AMS special session on Algebraic Geometry joint work with M. Temkin (Hebrew University) Dan Abramovich Brown University October 24, 2014

### Preliminary Exam Topics Sarah Mayes

Preliminary Exam Topics Sarah Mayes 1. Sheaves Definition of a sheaf Definition of stalks of a sheaf Definition and universal property of sheaf associated to a presheaf [Hartshorne, II.1.2] Definition

### INTERSECTION THEORY CLASS 12

INTERSECTION THEORY CLASS 12 RAVI VAKIL CONTENTS 1. Rational equivalence on bundles 1 1.1. Intersecting with the zero-section of a vector bundle 2 2. Cones and Segre classes of subvarieties 3 2.1. Introduction

### Systems of linear equations. We start with some linear algebra. Let K be a field. We consider a system of linear homogeneous equations over K,

Systems of linear equations We start with some linear algebra. Let K be a field. We consider a system of linear homogeneous equations over K, f 11 t 1 +... + f 1n t n = 0, f 21 t 1 +... + f 2n t n = 0,.

### Algebraic v.s. Analytic Point of View

Algebraic v.s. Analytic Point of View Ziwen Zhu September 19, 2015 In this talk, we will compare 3 different yet similar objects of interest in algebraic and complex geometry, namely algebraic variety,

### ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES.

ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES. ANDREW SALCH 1. Affine schemes. About notation: I am in the habit of writing f (U) instead of f 1 (U) for the preimage of a subset

### Three Descriptions of the Cohomology of Bun G (X) (Lecture 4)

Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) February 5, 2014 Let k be an algebraically closed field, let X be a algebraic curve over k (always assumed to be smooth and complete), and

### Hecke modifications. Aron Heleodoro. May 28, 2013

Hecke modifications Aron Heleodoro May 28, 2013 1 Introduction The interest on Hecke modifications in the geometrical Langlands program comes as a natural categorification of the product in the spherical

### Constructible isocrystals (London 2015)

Constructible isocrystals (London 2015) Bernard Le Stum Université de Rennes 1 March 30, 2015 Contents The geometry behind Overconvergent connections Construtibility A correspondance Valuations (additive

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37 RAVI VAKIL CONTENTS 1. Motivation and game plan 1 2. The affine case: three definitions 2 Welcome back to the third quarter! The theme for this quarter, insofar

### 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

### Introduction and preliminaries Wouter Zomervrucht, Februari 26, 2014

Introduction and preliminaries Wouter Zomervrucht, Februari 26, 204. Introduction Theorem. Serre duality). Let k be a field, X a smooth projective scheme over k of relative dimension n, and F a locally

### Connecting Coinvariants

Connecting Coinvariants In his talk, Sasha taught us how to define the spaces of coinvariants: V 1,..., V n = V 1... V n g S out (0.1) for any V 1,..., V n KL κg and any finite set S P 1. In her talk,

### Fundamental Lemma and Hitchin Fibration

Fundamental Lemma and Hitchin Fibration Gérard Laumon CNRS and Université Paris-Sud May 13, 2009 Introduction In this talk I shall mainly report on Ngô Bao Châu s proof of the Langlands-Shelstad Fundamental

### Synopsis of material from EGA Chapter II, 3

Synopsis of material from EGA Chapter II, 3 3. Homogeneous spectrum of a sheaf of graded algebras 3.1. Homogeneous spectrum of a graded quasi-coherent O Y algebra. (3.1.1). Let Y be a prescheme. A sheaf

### MATH 233B, FLATNESS AND SMOOTHNESS.

MATH 233B, FLATNESS AND SMOOTHNESS. The discussion of smooth morphisms is one place were Hartshorne doesn t do a very good job. Here s a summary of this week s material. I ll also insert some (optional)

### ABSTRACT NONSINGULAR CURVES

ABSTRACT NONSINGULAR CURVES Affine Varieties Notation. Let k be a field, such as the rational numbers Q or the complex numbers C. We call affine n-space the collection A n k of points P = a 1, a,..., a

### LOCAL VS GLOBAL DEFINITION OF THE FUSION TENSOR PRODUCT

LOCAL VS GLOBAL DEFINITION OF THE FUSION TENSOR PRODUCT DENNIS GAITSGORY 1. Statement of the problem Throughout the talk, by a chiral module we shall understand a chiral D-module, unless explicitly stated

### COMPLEX ALGEBRAIC SURFACES CLASS 9

COMPLEX ALGEBRAIC SURFACES CLASS 9 RAVI VAKIL CONTENTS 1. Construction of Castelnuovo s contraction map 1 2. Ruled surfaces 3 (At the end of last lecture I discussed the Weak Factorization Theorem, Resolution

### Porteous s Formula for Maps between Coherent Sheaves

Michigan Math. J. 52 (2004) Porteous s Formula for Maps between Coherent Sheaves Steven P. Diaz 1. Introduction Recall what the Thom Porteous formula for vector bundles tells us (see [2, Sec. 14.4] for

### Wild ramification and the characteristic cycle of an l-adic sheaf

Wild ramification and the characteristic cycle of an l-adic sheaf Takeshi Saito March 14 (Chicago), 23 (Toronto), 2007 Abstract The graded quotients of the logarithmic higher ramification groups of a local

### LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS

LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS IVAN LOSEV 16. Symplectic resolutions of µ 1 (0)//G and their deformations 16.1. GIT quotients. We will need to produce a resolution of singularities for C 2n

### EXERCISES IN POISSON GEOMETRY

EXERCISES IN POISSON GEOMETRY The suggested problems for the exercise sessions #1 and #2 are marked with an asterisk. The material from the last section will be discussed in lecture IV, but it s possible

### Math 248B. Applications of base change for coherent cohomology

Math 248B. Applications of base change for coherent cohomology 1. Motivation Recall the following fundamental general theorem, the so-called cohomology and base change theorem: Theorem 1.1 (Grothendieck).

### Vector Bundles on Algebraic Varieties

Vector Bundles on Algebraic Varieties Aaron Pribadi December 14, 2010 Informally, a vector bundle associates a vector space with each point of another space. Vector bundles may be constructed over general

### Factorization of birational maps on steroids

Factorization of birational maps on steroids IAS, April 14, 2015 Dan Abramovich Brown University April 14, 2015 This is work with Michael Temkin (Jerusalem) Abramovich (Brown) Factorization of birational

### Cohomology jump loci of local systems

Cohomology jump loci of local systems Botong Wang Joint work with Nero Budur University of Notre Dame June 28 2013 Introduction Given a topological space X, we can associate some homotopy invariants to

### MATH 8254 ALGEBRAIC GEOMETRY HOMEWORK 1

MATH 8254 ALGEBRAIC GEOMETRY HOMEWORK 1 CİHAN BAHRAN I discussed several of the problems here with Cheuk Yu Mak and Chen Wan. 4.1.12. Let X be a normal and proper algebraic variety over a field k. Show

### LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL

LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL In this lecture we discuss a criterion for non-stable-rationality based on the decomposition of the diagonal in the Chow group. This criterion

### Introduction to Arithmetic Geometry Fall 2013 Lecture #18 11/07/2013

18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #18 11/07/2013 As usual, all the rings we consider are commutative rings with an identity element. 18.1 Regular local rings Consider a local

### LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES

LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES 1. Vector Bundles In general, smooth manifolds are very non-linear. However, there exist many smooth manifolds which admit very nice partial linear structures.

### APPENDIX 3: AN OVERVIEW OF CHOW GROUPS

APPENDIX 3: AN OVERVIEW OF CHOW GROUPS We review in this appendix some basic definitions and results that we need about Chow groups. For details and proofs we refer to [Ful98]. In particular, we discuss

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43 RAVI VAKIL CONTENTS 1. Facts we ll soon know about curves 1 1. FACTS WE LL SOON KNOW ABOUT CURVES We almost know enough to say a lot of interesting things about

### QUANTIZATION OF HITCHIN S INTEGRABLE SYSTEM AND HECKE EIGENSHEAVES

QUANTIZATION OF HITCHIN S INTEGRABLE SYSTEM AND HECKE EIGENSHEAVES A. BEILINSON AND V. DRINFELD 1991 Mathematics Subject Classification. Subject classes here. This research was partially supported by INTAS

### Geometry of Conformal Field Theory

Geometry of Conformal Field Theory Yoshitake HASHIMOTO (Tokyo City University) 2010/07/10 (Sat.) AKB Differential Geometry Seminar Based on a joint work with A. Tsuchiya (IPMU) Contents 0. Introduction

### 9. Birational Maps and Blowing Up

72 Andreas Gathmann 9. Birational Maps and Blowing Up In the course of this class we have already seen many examples of varieties that are almost the same in the sense that they contain isomorphic dense

### The Hecke category (part II Satake equivalence)

The Hecke category (part II Satake equivalence) Ryan Reich 23 February 2010 In last week s lecture, we discussed the Hecke category Sph of spherical, or (Ô)-equivariant D-modules on the affine grassmannian

### h M (T ). The natural isomorphism η : M h M determines an element U = η 1

MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 7 2.3. Fine moduli spaces. The ideal situation is when there is a scheme that represents our given moduli functor. Definition 2.15. Let M : Sch Set be a moduli

### Direct Limits. Mathematics 683, Fall 2013

Direct Limits Mathematics 683, Fall 2013 In this note we define direct limits and prove their basic properties. This notion is important in various places in algebra. In particular, in algebraic geometry

### Geometry 9: Serre-Swan theorem

Geometry 9: Serre-Swan theorem Rules: You may choose to solve only hard exercises (marked with!, * and **) or ordinary ones (marked with! or unmarked), or both, if you want to have extra problems. To have

### Tamagawa Numbers in the Function Field Case (Lecture 2)

Tamagawa Numbers in the Function Field Case (Lecture 2) February 5, 204 In the previous lecture, we defined the Tamagawa measure associated to a connected semisimple algebraic group G over the field Q

### A p-adic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1

A p-adic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1 ALEXANDER G.M. PAULIN Abstract. The (de Rham) geometric Langlands correspondence for GL n asserts that to an irreducible rank n integrable connection

### Geometric invariant theory

Geometric invariant theory Shuai Wang October 2016 1 Introduction Some very basic knowledge and examples about GIT(Geometric Invariant Theory) and maybe also Equivariant Intersection Theory. 2 Finite flat

### Introduction to Arithmetic Geometry Fall 2013 Lecture #17 11/05/2013

18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #17 11/05/2013 Throughout this lecture k denotes an algebraically closed field. 17.1 Tangent spaces and hypersurfaces For any polynomial f k[x

### On algebraic index theorems. Ryszard Nest. Introduction. The index theorem. Deformation quantization and Gelfand Fuks. Lie algebra theorem

s The s s The The term s is usually used to describe the equality of, on one hand, analytic invariants of certain operators on smooth manifolds and, on the other hand, topological/geometric invariants

### 1 Notations and Statement of the Main Results

An introduction to algebraic fundamental groups 1 Notations and Statement of the Main Results Throughout the talk, all schemes are locally Noetherian. All maps are of locally finite type. There two main

### The Hecke category (part I factorizable structure)

The Hecke category (part I factorizable structure) Ryan Reich 16 February 2010 In this lecture and the next, we will describe the Hecke category, namely, the thing which acts on D-modules on Bun G and

### 1 Existence of the Néron model

Néron models Setting: S a Dedekind domain, K its field of fractions, A/K an abelian variety. A model of A/S is a flat, separable S-scheme of finite type X with X K = A. The nicest possible model over S

### GALOIS DESCENT AND SEVERI-BRAUER VARIETIES. 1. Introduction

GALOIS DESCENT AND SEVERI-BRAUER VARIETIES ERIC BRUSSEL CAL POLY MATHEMATICS 1. Introduction We say an algebraic object or property over a field k is arithmetic if it becomes trivial or vanishes after

### CHAPTER 0 PRELIMINARY MATERIAL. Paul Vojta. University of California, Berkeley. 18 February 1998

CHAPTER 0 PRELIMINARY MATERIAL Paul Vojta University of California, Berkeley 18 February 1998 This chapter gives some preliminary material on number theory and algebraic geometry. Section 1 gives basic

### Algebraic varieties and schemes over any scheme. Non singular varieties

Algebraic varieties and schemes over any scheme. Non singular varieties Trang June 16, 2010 1 Lecture 1 Let k be a field and k[x 1,..., x n ] the polynomial ring with coefficients in k. Then we have two

### On the Virtual Fundamental Class

On the Virtual Fundamental Class Kai Behrend The University of British Columbia Seoul, August 14, 2014 http://www.math.ubc.ca/~behrend/talks/seoul14.pdf Overview Donaldson-Thomas theory: counting invariants

### 3. The Sheaf of Regular Functions

24 Andreas Gathmann 3. The Sheaf of Regular Functions After having defined affine varieties, our next goal must be to say what kind of maps between them we want to consider as morphisms, i. e. as nice

### 10. Smooth Varieties. 82 Andreas Gathmann

82 Andreas Gathmann 10. Smooth Varieties Let a be a point on a variety X. In the last chapter we have introduced the tangent cone C a X as a way to study X locally around a (see Construction 9.20). It

### LECTURE 3 MATH 261A. Office hours are now settled to be after class on Thursdays from 12 : 30 2 in Evans 815, or still by appointment.

LECTURE 3 MATH 261A LECTURES BY: PROFESSOR DAVID NADLER PROFESSOR NOTES BY: JACKSON VAN DYKE Office hours are now settled to be after class on Thursdays from 12 : 30 2 in Evans 815, or still by appointment.

### Del Pezzo surfaces and non-commutative geometry

Del Pezzo surfaces and non-commutative geometry D. Kaledin (Steklov Math. Inst./Univ. of Tokyo) Joint work with V. Ginzburg (Univ. of Chicago). No definitive results yet, just some observations and questions.

### mult V f, where the sum ranges over prime divisor V X. We say that two divisors D 1 and D 2 are linearly equivalent, denoted by sending

2. The canonical divisor In this section we will introduce one of the most important invariants in the birational classification of varieties. Definition 2.1. Let X be a normal quasi-projective variety

### Quaternionic Complexes

Quaternionic Complexes Andreas Čap University of Vienna Berlin, March 2007 Andreas Čap (University of Vienna) Quaternionic Complexes Berlin, March 2007 1 / 19 based on the joint article math.dg/0508534

### Exercises on chapter 0

Exercises on chapter 0 1. A partially ordered set (poset) is a set X together with a relation such that (a) x x for all x X; (b) x y and y x implies that x = y for all x, y X; (c) x y and y z implies that

### Classification of Complex Algebraic Surfaces

ALGANT Master Thesis in Mathematics Classification of Complex Algebraic Surfaces Alberto Corato Advised by Prof. Dajano Tossici Università degli Studi di Padova Université de Bordeaux Academic year 2017/2018

### 3.1. Derivations. Let A be a commutative k-algebra. Let M be a left A-module. A derivation of A in M is a linear map D : A M such that

ALGEBRAIC GROUPS 33 3. Lie algebras Now we introduce the Lie algebra of an algebraic group. First, we need to do some more algebraic geometry to understand the tangent space to an algebraic variety at

### THE QUANTUM CONNECTION

THE QUANTUM CONNECTION MICHAEL VISCARDI Review of quantum cohomology Genus 0 Gromov-Witten invariants Let X be a smooth projective variety over C, and H 2 (X, Z) an effective curve class Let M 0,n (X,

### where Σ is a finite discrete Gal(K sep /K)-set unramified along U and F s is a finite Gal(k(s) sep /k(s))-subset

Classification of quasi-finite étale separated schemes As we saw in lecture, Zariski s Main Theorem provides a very visual picture of quasi-finite étale separated schemes X over a henselian local ring

### Mini-Course on Moduli Spaces

Mini-Course on Moduli Spaces Emily Clader June 2011 1 What is a Moduli Space? 1.1 What should a moduli space do? Suppose that we want to classify some kind of object, for example: Curves of genus g, One-dimensional

### VERLINDE ALGEBRA LEE COHN. Contents

VERLINDE ALGEBRA LEE COHN Contents 1. A 2-Dimensional Reduction of Chern-Simons 1 2. The example G = SU(2) and α = k 5 3. Twistings and Orientations 7 4. Pushforward Using Consistent Orientations 9 1.

### Lecture 9 - Faithfully Flat Descent

Lecture 9 - Faithfully Flat Descent October 15, 2014 1 Descent of morphisms In this lecture we study the concept of faithfully flat descent, which is the notion that to obtain an object on a scheme X,

### A SHORT PROOF OF ROST NILPOTENCE VIA REFINED CORRESPONDENCES

A SHORT PROOF OF ROST NILPOTENCE VIA REFINED CORRESPONDENCES PATRICK BROSNAN Abstract. I generalize the standard notion of the composition g f of correspondences f : X Y and g : Y Z to the case that X

### Noncommutative compact manifolds constructed from quivers

Noncommutative compact manifolds constructed from quivers Lieven Le Bruyn Universitaire Instelling Antwerpen B-2610 Antwerp (Belgium) lebruyn@wins.uia.ac.be Abstract The moduli spaces of θ-semistable representations

### Synopsis of material from EGA Chapter II, 4. Proposition (4.1.6). The canonical homomorphism ( ) is surjective [(3.2.4)].

Synopsis of material from EGA Chapter II, 4 4.1. Definition of projective bundles. 4. Projective bundles. Ample sheaves Definition (4.1.1). Let S(E) be the symmetric algebra of a quasi-coherent O Y -module.