DOCSLIB.ORG

D-Modules and the Riemann-Hilbert Correspondence

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
D-Modules and the Riemann-Hilbert Correspondence D-MODULES AND THE RIEMANN-HILBERT CORRESPONDENCE JOSE´ SIMENTAL Abstract. These are notes for a talk given at the MIT/NEU Seminar on Hodge modules, Fall 2016. We give a quick introduction to the theory of D-modules, following mostly [BCEY, HTT, K]. 1. Differential operators 1.1. Setup. Let X be a smooth algebraic variety, and let dX := dim X. We remark that all algebraic varieties here are always assumed to be quasi-projective. By OX we will denote the structure sheaf of X, and VectX denotes the sheaf of vector fields on X, that is, the sheaf of regular sections of TX. Consider the sheaf EndCX (OX ), its sections on an open set U are HomCX (OU ; OU ). Note that we have embeddings OX ,!EndCX (OX ) and VectX ,!EndCX (OX ). Definition 1.1. The sheaf of differential operators DX is the sheaf of algebras that is generated by OX ; VectX inside of EndCX (OX ). Let us remark that DX is actually a coherent sheaf of algebras on X. This follows from the corresponding result for OX and results of Subsection 1.2 below, see Corollary 1.4. We also remark that DX is a sheaf of noncommutative algebras. Indeed, if ξ 2 VectX ; f 2 OX , then [ξ; f] = ξ(f), which is not necessarily zero. In local coordinates x1; : : : ; xn; @x1; : : : ; @xn, and element P 2 DX may be written as X α P = aα(x)@x n α2Z≥0 α1+···+αn n α @ where aα = 0 for all but a finite number of α 2 Z≥0, and @x = α1 αn . @x1 ··· @xn Remark 1.2. We remark that a local coordinate system x1; : : : ; xn;@x1 ;:::;@xn does not give an isomorphism n n from a neighborhood U to an affine subvariety of C . Rather, what we have is an ´etalemorphism U ! C . 1.2. Order of differential operators. Let P 2 DX be a differential operator. Assume that, in local P α coordinates, we have P = n aα(x)@x . We would like to say that P has order at most n if maxfjαj : α2Z≥0 aα(x) 6= 0g ≤ n. Of course, a representation of P in local coordinates is not unique, so this definition does not work as stated. However, we do have a filtration on DX that formalizes this idea. ≤m Definition 1.3. We define a filtration F (DX ) as follows: ≤m (1) F DX = 0 for m < 0. ≤0 (2) F DX = OX . ≤m ≤m−1 (3) F DX = fP 2 DX :[P; OX ] 2 F DX g for m ≥ 1. S ≤m ≤m m+1 It is clear, using a computation in local coordinates, that DX = n F DX and that F DX ⊆ F DX ≤m ≤n ≤m+n for every m. By double induction, it is easy to show that F DX · F DX ⊆ F DX , so that this is ≤m ≤n ≤m+n−1 F an algebra filtration, and that [F DX ;F DX ] ⊆ F DX , so that the associated graded gr DX is commutative. Let us describe this graded algebra explicitly. First, it is easy to check that we have F F gr0 DX = OX and gr1 DX = VectX . Note that this implies that we have a surjection: F (1) gr D Sym Vect = π O ∗ X OX X ∗ T X where π : T ∗X ! X is the canonical projection. A computation in local coordinates tells us that this is actually an isomorphism. 1 2 JOSE´ SIMENTAL Corollary 1.4. The sheaf DX is a noetherian (in particular, coherent) sheaf of algebras on X. Proof. This follows since gr DX is noetherian. We remark that the isomorphism in (1) is actually an isomorphism of Poisson algebras with respect to the F Poisson algebra structure in gr DX induced by the filtration on DX and the Poisson bracket on π∗OT ∗X induced by the canonical symplectic form on T ∗X. Using local coordinates, this is an easy exercise. 1.3. Analytic setting. When discussing the Riemann-Hilbert correspondence, we need to consider dif- ferential operators on analytic, rather than algebraic, manifolds. Most of what we have discussed so far transfers to the analytic setting, although the proofs are slightly more difficult. For example, we still have that gr D = Sym Vect . However, the latter algebra no longer coincides with π O ∗ . The coherence X OX X ∗ T X of DX here follows from the Oka-Cartan theorem, and the Hilbert basis theorem. 2. D-modules We remark that DX is quasi-coherent as a OX -module. Most of the time below, we will restrict our attention to those DX -modules which are quasi-coherent as OX -modules. We denote by DX -Modqc ⊆ DX -Mod the category of such modules. 2.1. Connections. By a DX -module we mean a left DX -module. Perhaps the easiest example of such a module is given by the sheaf OX , which is a left DX -module by definition. Note that we have an exact sequence of DX -modules: (2) DX ⊗OX VectX −! DX −! OX −! 0 where the map DX ⊗OX VectX is P ⊗ ξ 7! P ξ, and the map DX 7! OX is P 7! P · 1. Later, we will extend the sequence (2) to a complex of left DX -modules, see (7). Now let M be an OX -module. If we want to extend the OX -module structure on M to a DX -module structure, we need a compatible action of vector fields, that is a map VectX ⊗OX M!M. Taking duals, we get a connection: 1 (3) r : M ! M ⊗OX ΩX 1 Where ΩX is the sheaf of 1-forms. It is an easy exercise to check that the fact that this actually defines a 2 2 2 DX -module structure on M means that the connection (3) is flat, i.e. r = 0, where r : M ! M ⊗OX ΩX 2 is r = (r ⊗ 1 + 1 ⊗ d)r. In this sense, DX -modules are a generalization of vector bundles on X with a flat connection. 2.2. O-coherent D-modules. Following with the idea of the last paragraph, we will see in this subsection that, if a DX -module is O-coherent, then it is indeed a vector bundle with a flat connection. Proposition 2.1. Let M be a DX -module. Assume that M is OX -coherent. Then, M is locally free over OX . Proof. Let x 2 X be a closed point. It suffices to show that Mx is free over the local ring OX;x. By Nakayama's lemma, we may find generators m1; : : : ; mk 2 Mx such that m1;:::; mk form a basis of Mx=mxMx over the field OX;x=mx. We claim that m1; : : : ; mk form a basis of Mx. Indeed, assume the P contrary so there exist f1; : : : ; fk 2 OX;x such that fimi = 0. Define the order of f 2 OX;x by ord(0) = 1 ` `+1 and ord(f) = ` if f 2 m but f 62 m . Note that ord(fi) ≥ 0 for every i = 1; : : : ; n and ord(fi) > 0 for at least one i. Moreover, let `0 := minford(fi): i = 1; : : : ; n; ord(fi) > 0g. If the m1; : : : ; mk are not lineraly independent over OX;x, we may assume that `0 is minimal among such linear combinations and `0 > 0. Without loss of generality, we may assume that `0 = ord(f1). Note that there exists a vector field ξ defined locally around x such that ξ(f1) 6= 0 and ord(ξ(f1)) < ord(f1) = `0. Now, D-MODULES AND THE RIEMANN-HILBERT CORRESPONDENCE 3 0 1 X X X X (4) 0 = ξ fimi = ξ(fi)mi + fiξ(mi) = @ξ(fi) + fjajiA mi i j P P where ξ(mi) = j aijmj. Note that ord(ξ(f1) + j fjaj1) is strictly less than `0. We have arrived at a contradiction. Thus, M is locally free over OX . 2.3. Coherent D-modules. The following class of DX -modules will be important for us. Definition 2.2. A DX -module M is coherent if it is locally finitely generated, that is, for every x 2 X there ⊕N exists a neighborhood U of x and N > 0 such that we have an epimorphism (DX jU ) MjU . We denote the category of coherent DX -modules by DX -mod. Remark 2.3. In general, the definition of coherence is more complicated. In general, for a sheaf of algebras A, a A-module is coherent if it is locally finitely generated and every locally finitely generated submodule is finitely presented. In view of Corollary 1.4, this agrees with our Definition 2.2 in the case A = DX . The following result makes sure that coherent DX -modules have nice properties, and it is the key to build a bridge between the non-commutative setting of D-modules and the more familiar commutative setting of algebraic geometry, cf. Subsection 2.4. Proposition 2.4. Let M be a coherent DX -module. Then, M is globally generated (over DX ) by a coherent OX -submodule. S Proof. We can take a finite, affine open covering X = Ui such that MjUi is finitely generated over DUi . Thus, we may take a coherent OU -submodule Ni ⊆ MjU that generates MjU as a DU -module. Extend this i P i i i to a coherent OX -submodule Ni ⊆ M. Then, i Ni is OX -coherent and generates M as a DX -module. Remark 2.5. From now on, when we mention 'coherent D-module' we will always mean coherent in the sense of this subsection, not coherent as a O-module.
Load more

Recommended publications
  • NOTES on CARTIER and WEIL DIVISORS Recall: Definition 0.1. A
    NOTES ON CARTIER AND WEIL DIVISORS AKHIL MATHEW Abstract. These are notes on divisors from Ravi Vakil's book [2] on scheme theory that I prepared for the Foundations of Algebraic Geometry seminar at Harvard. Most of it is a rewrite of chapter 15 in Vakil's book, and the originality of these notes lies in the mistakes. I learned some of this from [1] though. Recall: Definition 0.1. A line bundle on a ringed space X (e.g. a scheme) is a locally free sheaf of rank one. The group of isomorphism classes of line bundles is called the Picard group and is denoted Pic(X). Here is a standard source of line bundles. 1. The twisting sheaf 1.1. Twisting in general. Let R be a graded ring, R = R0 ⊕ R1 ⊕ ::: . We have discussed the construction of the scheme ProjR. Let us now briefly explain the following additional construction (which will be covered in more detail tomorrow). L Let M = Mn be a graded R-module. Definition 1.1. We define the sheaf Mf on ProjR as follows. On the basic open set D(f) = SpecR(f) ⊂ ProjR, we consider the sheaf associated to the R(f)-module M(f). It can be checked easily that these sheaves glue on D(f) \ D(g) = D(fg) and become a quasi-coherent sheaf Mf on ProjR. Clearly, the association M ! Mf is a functor from graded R-modules to quasi- coherent sheaves on ProjR. (For R reasonable, it is in fact essentially an equiva- lence, though we shall not need this.) We now set a bit of notation.
    [Show full text]
  • Arxiv:1307.5568V2 [Math.AG]
    PARTIAL POSITIVITY: GEOMETRY AND COHOMOLOGY OF q-AMPLE LINE BUNDLES DANIEL GREB AND ALEX KURONYA¨ To Rob Lazarsfeld on the occasion of his 60th birthday Abstract. We give an overview of partial positivity conditions for line bundles, mostly from a cohomological point of view. Although the current work is to a large extent of expository nature, we present some minor improvements over the existing literature and a new result: a Kodaira-type vanishing theorem for effective q-ample Du Bois divisors and log canonical pairs. Contents 1. Introduction 1 2. Overview of the theory of q-ample line bundles 4 2.1. Vanishing of cohomology groups and partial ampleness 4 2.2. Basic properties of q-ampleness 7 2.3. Sommese’s geometric q-ampleness 15 2.4. Ample subschemes, and a Lefschetz hyperplane theorem for q-ample divisors 17 3. q-Kodaira vanishing for Du Bois divisors and log canonical pairs 19 References 23 1. Introduction Ampleness is one of the central notions of algebraic geometry, possessing the extremely useful feature that it has geometric, numerical, and cohomological characterizations. Here we will concentrate on its cohomological side. The fundamental result in this direction is the theorem of Cartan–Serre–Grothendieck (see [Laz04, Theorem 1.2.6]): for a complete arXiv:1307.5568v2 [math.AG] 23 Jan 2014 projective scheme X, and a line bundle L on X, the following are equivalent to L being ample: ⊗m (1) There exists a positive integer m0 = m0(X, L) such that L is very ample for all m ≥ m0. (2) For every coherent sheaf F on X, there exists a positive integer m1 = m1(X, F, L) ⊗m for which F ⊗ L is globally generated for all m ≥ m1.
    [Show full text]
  • 4. Coherent Sheaves Definition 4.1. If (X,O X) Is a Locally Ringed Space
    4. Coherent Sheaves Definition 4.1. If (X; OX ) is a locally ringed space, then we say that an OX -module F is locally free if there is an open affine cover fUig of X such that FjUi is isomorphic to a direct sum of copies of OUi . If the number of copies r is finite and constant, then F is called locally free of rank r (aka a vector bundle). If F is locally free of rank one then we way say that F is invertible (aka a line bundle). The group of all invertible sheaves under tensor product, denoted Pic(X), is called the Picard group of X. A sheaf of ideals I is any OX -submodule of OX . Definition 4.2. Let X = Spec A be an affine scheme and let M be an A-module. M~ is the sheaf which assigns to every open subset U ⊂ X, the set of functions a s: U −! Mp; p2U which can be locally represented at p as a=g, a 2 M, g 2 R, p 2= Ug ⊂ U. Lemma 4.3. Let A be a ring and let M be an A-module. Let X = Spec A. ~ (1) M is a OX -module. ~ (2) If p 2 X then Mp is isomorphic to Mp. ~ (3) If f 2 A then M(Uf ) is isomorphic to Mf . Proof. (1) is clear and the rest is proved mutatis mutandis as for the structure sheaf. Definition 4.4. An OX -module F on a scheme X is called quasi- coherent if there is an open cover fUi = Spec Aig by affines and ~ isomorphisms FjUi ' Mi, where Mi is an Ai-module.
    [Show full text]
  • (Quasi)Coherent Sheaves on Projective Spaces
    18.725 Algebraic Geometry I Lecture 14 Lecture 14: (Quasi)coherent sheaves on Projective Spaces First an abstract lemma. Let L : C1 !C2; R : C2 !C1 be an adjoint pair; if L is fully faithful and R is conservative, then they are inverses. The unit is Id −!Lu ◦ R and the counit is Id −!" R ◦ L. Additionally, we "()R R(u) have R −−−!R ◦ L ◦ R −−−!R = Id. Example 1. C1 = C2 = Vect. Let V be a finite dimensional vector space. Let R : U ! V ⊗ U, L : U ! E⊗δ7! E(δ) V ∗ ⊗ U = Hom(V; U). Then the operation above becomes VVVV−δ−7!−I−d−⊗!δ ⊗ ∗ ⊗ −−−−−−−! V . L is fully faithful implies Id =∼ R ◦ L. What about L ◦ R =∼ Id? It suffices to use R ! R ◦ L ◦ R ! R. Last time we showed that the set of affine maps between X and Y is the same as the set of quasicoherent sheaves of OX -algebras (which are locally finitely generated and reduced). ∼ n −1 ∼ n Definition 1. X ! Y is a vector bundle if locally = A × Y , i.e. there exists a covering f (Ui) = A × Ui n and agree on the intersection, i.e. the two copies of A × (Ui \ Uj) are glued together using GLn(k[Ui \ Uj]). The equivalence between the category of locally free sheaves and the category of vector bundles is given i by E 7! Spec(⊕iSym (E)), which is a contravariant functor. The opposite maps are from a vector bundle to the sheaf of sections of the dual bundle. Note that the total space is given by Tot(E) = Spec(Sym(E_)) where E_ = Hom(E; O).
    [Show full text]
  • Constructible Sheaves on Affine Grassmannians and Geometry of the Dual Nilpotent Cone
    CONSTRUCTIBLE SHEAVES ON AFFINE GRASSMANNIANS AND GEOMETRY OF THE DUAL NILPOTENT CONE PRAMOD N. ACHAR AND SIMON RICHE Abstract. In this paper we study the derived category of sheaves on the affine Grassmannian of a complex reductive group Gˇ, contructible with respect to the ˇ stratification by G(C[[x]])-orbits. Following ideas of Ginzburg and Arkhipov{ Bezrukavnikov{Ginzburg, we describe this category (and a mixed version) in terms of coherent sheaves on the nilpotent cone of the Langlands dual re- ductive group G. We also show, in the mixed case, that restriction to the nilpotent cone of a Levi subgroup corresponds to hyperbolic localization on affine Grassmannians. 1. Introduction 1.1. Let Gˇ be a complex connected reductive group, and let ˇ ˇ GrGˇ := G(K)=G(O) be the associated affine Grassmannian (where K = C((x)) and O = C[[x]]). The Satake equivalence is an equivalence of tensor categories ∼ SG : Rep(G) −! PGˇ(O)-eq(GrGˇ ) ˇ between the category PGˇ(O)-eq(GrGˇ ) of G(O)-equivariant perverse sheaves on GrGˇ (endowed with the convolution product) and the category Rep(G) of finite-dimen- sional G-modules (endowed with the tensor product), where G is the complex re- ductive group which is Langlands dual to Gˇ. This equivalence is \functorial with respect to restriction to a Levi" in the sense that, if L ⊂ G is a Levi subgroup and Lˇ ⊂ Gˇ a dual Levi subgroup, the restriction functor Rep(G) ! Rep(L) can be realized geometrically as a (renormalized) hyperbolic localization functor G RL : PGˇ(O)-eq(GrGˇ ) ! PLˇ(O)-eq(GrLˇ ) in the sense of Braden [Bra].
    [Show full text]
  • 3. Ample and Semiample We Recall Some Very Classical Algebraic Geometry
    3. Ample and Semiample We recall some very classical algebraic geometry. Let D be an in- 0 tegral Weil divisor. Provided h (X; OX (D)) > 0, D defines a rational map: φ = φD : X 99K Y: The simplest way to define this map is as follows. Pick a basis σ1; σ2; : : : ; σm 0 of the vector space H (X; OX (D)). Define a map m−1 φ: X −! P by the rule x −! [σ1(x): σ2(x): ··· : σm(x)]: Note that to make sense of this notation one has to be a little careful. Really the sections don't take values in C, they take values in the fibre Lx of the line bundle L associated to OX (D), which is a 1-dimensional vector space (let us assume for simplicity that D is Carier so that OX (D) is locally free). One can however make local sense of this mor- phism by taking a local trivialisation of the line bundle LjU ' U × C. Now on a different trivialisation one would get different values. But the two trivialisations differ by a scalar multiple and hence give the same point in Pm−1. However a much better way to proceed is as follows. m−1 0 ∗ P ' P(H (X; OX (D)) ): Given a point x 2 X, let 0 Hx = f σ 2 H (X; OX (D)) j σ(x) = 0 g: 0 Then Hx is a hyperplane in H (X; OX (D)), whence a point of 0 ∗ φ(x) = [Hx] 2 P(H (X; OX (D)) ): Note that φ is not defined everywhere.
    [Show full text]
  • Lecture 22 Bertini's Theorem, Coherent Sheves on Curves
    18.725 Algebraic Geometry I Lecture 22 Lecture 22: Bertini's Theorem, Coherent Sheves on Curves Let's consider some ways to construct smooth varieties. Theorem 1.1 (Bertini's Theorem). Let X ⊆ PV be a smooth subvariety. Then for a generic hyperplane H, Y = X \ H is again smooth. _ Recall that the set of hyperplanes is parametrized by the dual projective space PV . To say that a _ hyperplane is generic is equivalent to saying that there is a nonempty open subset U ⊆ PV containing the _ point in PV corresponding to that hyperplane and such that each hyperplane in U possesses the desired property. Proof. We can assume that X is irreducible. Indeed, if X has multiple irreducible components (i.e. is not connected) and if we know the claim for each irreducible component, then we have a finite set of open subsets _ in PV , whose intersection is again open and consists of hyperplanes whose intersection with X is smooth. Let d = dim(X), n = dim(PV ). For all x 2 X, we have TxX of dimension d, and TxXT⊆ xPV . If def x2 H, then HX\ will be smooth at x if TxH 6⊃ TxX. Consider the following subset Z = f(H; x) j H 3 _ x; TxH ⊃ TxXg of the product PV × X. One easily sees that is is closed. The set of H for which H \ X is _ _ singular is the image of Z under the projection PV × X ! PV . We will now proceed by dimension count. First, we want to calculate the dimension of Z.
    [Show full text]
  • Foundations of Algebraic Geometry Classes 35 and 36
    FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 35 AND 36 RAVI VAKIL CONTENTS 1. Introduction 1 2. Definitions and proofs of key properties 5 3. Cohomology of line bundles on projective space 9 In these two lectures, we will define Cech cohomology and discuss its most important properties, although not in that order. 1. INTRODUCTION As Γ(X; ·) is a left-exact functor, if 0 ! F ! G ! H ! 0 is a short exact sequence of sheaves on X, then 0 ! F(X) ! G(X) ! H(X) is exact. We dream that this sequence continues off to the right, giving a long exact se- quence. More explicitly, there should be some covariant functors Hi (i ≥ 0) from qua- sicoherent sheaves on X to groups such that H0 = Γ, and so that there is a “long exact sequence in cohomology”. (1) 0 / H0(X; F) / H0(X; G) / H0(X; H) / H1(X; F) / H1(X; G) / H1(X; H) / · · · (In general, whenever we see a left-exact or right-exact functor, we should hope for this, and in good cases our dreams will come true. The machinery behind this is sometimes called derived functor cohomology, which we will discuss shortly.) Before defining cohomology groups of quasicoherent sheaves explicitly, we first de- scribe their important properties. Indeed these fundamental properties are in some ways more important than the formal definition. The boxed properties will be the important ones. Date: Friday, February 22 and Monday, February 25, 2008. 1 Suppose X is a separated and quasicompact A-scheme. (The separated and quasicom- pact hypotheses will be necessary in our construction.) For each quasicoherent sheaf F on X, we will define A-modules Hi(X; F).
    [Show full text]
  • Equivariant K-Theory of Flag Varieties 1
    EQUIVARIANT K-THEORY OF FLAG VARIETIES 1 by Shrawan Kumar ( Notes by Valdemar Tsanov ) Contents 1 Basics of equivariant K-theory 2 1.1 K-groups of topological vector bundles . 2 1.1.1 Functoriality . 2 1.1.2 Algebra structure . 2 1.2 K-groups of algebraic vector bundles . 3 1.3 K-groups of coherent sheaves . 3 1.3.1 Functoriality . 4 G 0 1.4 Relation between K0 (X) and KG(X) ..................... 4 2 K-theory of flag varieties 5 2.1 Flag varieties of semisimple groups . 5 2.2 K-theory of complete flag varieties . 5 2.2.1 A Weyl group action on Ktop(Y ).................... 6 T0 2.2.2 Demazure operators . 6 2.3 Algebraic model for Ktop(Y ) .......................... 7 T0 2.4 K-theory of partial flag varieties . 10 2.5 A basis of Ktop(Y ) and its geometric identification . 11 T0 2.6 Positivity of the basis fτ wg ........................... 11 Introduction Let G be a connected, simply connected, semisimple linear complex algebraic group. The flag varieties of G are the coset spaces of the form G=P , where P ⊂ G is a parabolic subgroup. We present some results on the K-theory of flag varieties of G. By definition, the flag varieties carry a transitive G-action. The G-equivariant K-theory is relatively easy to understand. One gets more challenging questions and interesting results by restricting the action to subgroups of G, in particular, to a maximal torus T . We shall focus mainly on the T -equivariant K-theory of the complete flag variety G=B, where B is a Borel subgroup.
    [Show full text]
  • Lecture 3: Grassmannians (Cont.) and Flat Morphisms
    Lecture 3: Grassmannians (cont.) and flat morphisms 09/11/2019 1 Constructing Grassmannians Recall we aim to prove the following: Theorem 1. Gr(k, n) is representable by a finite type scheme over Spec Z. Proof. We will use the representability criterion from Lecture 2. It is clear Gr(k, n) is a sheaf by gluing locally free sheaves. For each subset i ⊂ f1, . , ng of size k, we will define a subfunctor Fi of the Grassman- nian functor. First, let k n si : OS !OS th th denote the the inclusion where the j direct summand is mapped by the identity to the ij direct summand. Now let Fi be defined as the subfunctor n Fi(S) = fa : OS ! V j a ◦ si is surjective g ⊂ Gr(k, n)(S). Note this is a functor since for any f : T ! S, f ∗ is right-exact. We need to show that each Fi is representable and that the collection fFig is an open cover of the functor Gr(k, n). n For any scheme S and any map S ! Gr(k, n) corresponding to the object (a : OS ! n V) 2 F(S), we have a natural morphism of finite type quasi-coherent sheaves a ◦ si : OS ! V. Now let K be the cokernel of a ◦ si. Then a ◦ si is surjective at a point x 2 S if and only if Kx = 0 if and only if x 2/ Supp(Kx). Since Supp(Kx) is closed, the set Ui where a ◦ si is surjective is open. We need to show that for any other scheme T and a morphism f : T ! S, f factors through Ui if and only if ∗ n ∗ n ∗ f (a : OS !V) = ( f a : OT ! f V) 2 Fi(T) .
    [Show full text]
  • The Geometry of Moduli Spaces of Sheaves
    The Geometry of Moduli Spaces of Sheaves Daniel Huybrechts and Manfred Lehn UniversitÈatGH Essen UniversitÈatBielefeld Fachbereich 6 Mathematik SFB 343 FakultÈatfÈur Mathematik UniversitÈatsstraûe 3 Postfach 100131 D-45117 Essen D-33501 Bielefeld Germany Germany [email protected] [email protected] v Preface The topic of this book is the theory of semistable coherent sheaves on a smooth algebraic surface and of moduli spaces of such sheaves. The content ranges from the de®nition of a semistable sheaf and its basic properties over the construction of moduli spaces to the bira- tional geometry of these moduli spaces. The book is intended for readers with some back- ground in Algebraic Geometry, as for example provided by Hartshorne's text book [98]. There are at least three good reasons to study moduli spaces of sheaves on surfaces. Firstly, they provide examples of higher dimensional algebraic varieties with a rich and interesting geometry. In fact, in some regions in the classi®cation of higher dimensional varieties the only known examples are moduli spaces of sheaves on a surface. The study of moduli spaces therefore sheds light on some aspects of higher dimensional algebraic geometry. Secondly, moduli spaces are varieties naturally attached to any surface. The understanding of their properties gives answers to problems concerning the geometry of the surface, e.g. Chow group, linear systems, etc. From the mid-eighties till the mid-nineties most of the work on moduli spaces of sheaves on a surface was motivated by Donaldson's ground breaking re- sults on the relation between certain intersection numbers on the moduli spaces and the dif- ferentiable structure of the four-manifold underlying the surface.
    [Show full text]
  • The Cohomology of Coherent Sheaves
    CHAPTER VII The cohomology of coherent sheaves 1. Basic Cechˇ cohomology We begin with the general set-up. (i) X any topological space = U an open covering of X U { α}α∈S a presheaf of abelian groups on X. F Define: (ii) Ci( , ) = group of i-cochains with values in U F F = (U U ). F α0 ∩···∩ αi α0,...,αYi∈S We will write an i-cochain s = s(α0,...,αi), i.e., s(α ,...,α ) = the component of s in (U U ). 0 i F α0 ∩··· αi (iii) δ : Ci( , ) Ci+1( , ) by U F → U F i+1 δs(α ,...,α )= ( 1)j res s(α ,..., α ,...,α ), 0 i+1 − 0 j i+1 Xj=0 b where res is the restriction map (U U ) (U U ) F α ∩···∩ Uαj ∩···∩ αi+1 −→ F α0 ∩··· αi+1 and means “omit”. Forb i = 0, 1, 2, this comes out as δs(cα , α )= s(α ) s(α ) if s C0 0 1 1 − 0 ∈ δs(α , α , α )= s(α , α ) s(α , α )+ s(α , α ) if s C1 0 1 2 1 2 − 0 2 0 1 ∈ δs(α , α , α , α )= s(α , α , α ) s(α , α , α )+ s(α , α , α ) s(α , α , α ) if s C2. 0 1 2 3 1 2 3 − 0 2 3 0 1 3 − 0 1 2 ∈ One checks very easily that the composition δ2: Ci( , ) δ Ci+1( , ) δ Ci+2( , ) U F −→ U F −→ U F is 0. Hence we define: 211 212 VII.THECOHOMOLOGYOFCOHERENTSHEAVES s(σβ0, σβ1) defined here U σβ0 Uσβ1 Vβ1 Vβ0 ref s(β0, β1) defined here Figure VII.1 (iv) Zi( , ) = Ker δ : Ci( , ) Ci+1( , ) U F U F −→ U F = group of i-cocycles, Bi( , ) = Image δ : Ci−1( , ) Ci( , ) U F U F −→ U F = group of i-coboundaries Hi( , )= Zi( , )/Bi( , ) U F U F U F = i-th Cech-cohomologyˇ group with respect to .
    [Show full text]