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1 Lecture 1 2 Lecture 2 1 Lecture 1 Possible Topics: 1. Hodge Conjecture. Probably about 5 lectures. 2. Geometric Langlands. Perhaps 5 lectures, and might contain geometry of derived categories 3. Geometry of Derived Categories 4. Grothendieck-Riemann-Roch? Maybe 2 lectures 5. Mirror Symmetry/Calabi-Yau's. As many lectures as needed. 2 Lecture 2 Geometric Langlands Program. Rather than describe the program right out, we will discuss examples first. This is essentially a nonabelian phenomenon. Input: A curve (smooth, compact, genus g) C and G a reductive group. Examples of reductive groups are GL(n); SL(n); SO(n); Sp(n), products, etc. Baby case: Abelian Geometric Langlands Conjecture. We have G = (C∗)r, and will take r = 1. This will roughly correspond to Class Field Theory in Number Theory, and the full GLC corresponds to nonabelian class field theory. Roughly we have objects G-bundles versus G local systems. In the case G = GL(n), we have that a G-bundle is a rank n vector bundle and a G-local system is a homomorphism π1(C) ! G which is the same as a rank n vector bundle with a flat connection. (All connections are flat, because curvature is a 2-form, and we are on a 1-dimensional object) 1 The moduli of rank n vector bundles is H (C; GLn(O)) = Bun and the 1 moduli space of G-local systems is H (C; GLn(C)) = Loc. 1 We know both that TV Bun = H (C; EndO (V )) and that T(V;r)Loc = 1 H (C; EndC(V )). We want to compute these dimensions, and we will use Riemann-Roch: h0 − h1(V ⊗ V ∗) = deg −rank(g − 1). If n = 1 then V = L . End(V ) = V ⊗ V ∗ = L ⊗ L −1 = O, so deg is zero. −1 −1 If n = 2 and split, then V = L1 ⊕ L2. So EndV = L1 ⊗ L1 ⊕ L1 ⊗ L2 ⊕ −1 −1 L2 ⊗ L1 ⊕ L2 ⊗ L2 which is 0 + d1 − d2 + d2 − d1 + 0 = 0. And so, deg(End(V )) = 0 for all V . First we note that deg V = deg det V , 2 and det(End V ) = Vn (End(V )). 1 2 0 And so dimV Bun = h (C; End(V )) = n (g − 1) + h (C; End(V )). 0 For a "general" vector bundle, h = 1, and so generically dimV Bun = n2(g − 1) + 1 The dimension of T(V;r)Loc is 2 dimV Bun. Rough Statement of the Geometric Langlands Program: Any G local system E on C determines a local system cE on a Zariski open subset of BunC;G, the 1 moduli space of bundles with structure group G on C. The key property is that this is an automorphic sheaf. If we extend to the whole space, we get a "perverse sheaf." We will not be talking about perverse sheaves and D-modules, but will fo- cus on automorphic sheaves and restrict ourselves to open subsets whenever necessary. ∗ Example 2.1. Case where n = 1, so G = GL(1) = C . On this, BunC;GL(1) = Pic(C). Abelian Langlands tells us that every C∗ local system on C determines a C∗ local system on Pic(C). ∗ ∗ A local system on C means a map ρ : π1(C) ! C .A C -local system 0 ∗ on Pic (C) = Jac(C) is cρ : π1(Jac(C)) ! C . We note that π1(Jac(C)) = H1(Jac(C); Z) = H1(C; Z), which is the abelianization of π1(C). ∗ ∗ So a map ρ : π1(C) ! C factors through π1(Jac(C)) because C is abelian, ∗ and so gives us cρ : π1(Jac(C)) ! C , a local system of Jac(C). This naturally extends to all of Pic(C). Choose L 2 Picd(C). Then ⊗L−1 : Picd(C) ! Pic0(C), and we pull back along this map. This doesn't depend on L, because the induced automorphism on π1(Jac(C)) given by translation by L0 is trivial. Example 2.2. If we take a rank 2 local system on C, then we get a local system on and open subset of Bun whose rank is 23g−3. In general, rank n gives 23g−335g−5 : : : n(2n−1)(g−1). We need to discuss Hecke correspondences now. A correspondence between varieties X,Y is a subvariety H ⊂ X × Y . If H is proper over Y , then H induces a map on cohomology H∗(X) ! H∗(H) ! H∗(Y ) by first pulling back and then pushing forward. Example 2.3. Let f : X ! Y be any morphism and H be the graph of f. Then the operator induced by H is f∗. Example 2.4. Let g : Y ! X be a morphism and H it's graph. Then the operator induced by H is g∗. Hecke Correspondence: Hecke ?? ?? ? BunC;G BunC;G ×C The Hecke Correspondence is f(V; V 0; x)jV ⊂ V 0 ⊂ V (x)g where V (x) = V ⊗ O(x), and the inclusions are inclusions of coherent sheaves. An automorphic sheaf is an eigensheaf of Hecke. 3 Lecture 3 More on the Hecke Correspondence. 2 Denote Bun = Bunn;C. Then Hecke ⊂ Bun × Bun × C is the locus f(V 0; V; x)jV ⊂ V 0 ⊂ V (x)g. We call p1 : Hecke ! Bun the first projection and p2 : Hecke ! Bun × C the second and third projections taken together. −1 What is p2 (V; x)? 0 Example 3.1. x 2 C and V = OC ⊕ OC . What is the set of all V such that 0 OC ⊕ OC = V ⊂ C ⊂ V (x) = OC (x) ⊕ OC (x). 0 V and V work, as do OC ⊕ OC (x) and OC (x) ⊕ OC . Is there anything else? Yes! You can take other coordinates. So in fact we get two points and a projective line. 0 Choose any linear function f : Vx ! C. Let V = ker f, and this gives the answer. −1 Back to the question: what is p2 (V; x)? It is disconnected. To see this we look at `(V 0=V ), the length of this sheaf. 0 0 Because V =V is supported at x, and it's fiber there is Vx=Vx, then (V=V )x ⊂ 0 Vx=Vx ⊂ (V (x)=V )x, so the first is zero and the last can be anything up to n. −1 `n So the answer is that p2 (V; x) = i=0 Gr(i; Vx), where n = dim(Vx). 0 We saw that the vector bundle V determines a subspace of Vx. Conversely, 0 given a subspace W ⊂ Vx, Γ(U; V ) = fx 2 Γ(U; V (x))jpole(s) ⊂ W g. Summary: Given a vector bundle V and a subspace W ⊂ Vx, we get the "elementary transform" V 0 = 0 0 V is a line bundle, and V ⊂ V ⊂ V (x), 0 ! W ! Vx ! Q ! 0 and 0 0 ! Q ! Vx ! W ! 0 are short exact `n i i p2 Hecke = i=0 Hecke , where Hecke ! Bun × C is a fiber bundle with fiber Gr(i; n). i i i ∗ i i i∗ Hecke Operators: H := (p2)∗(p1) . Explicitly, H (F ) := p2∗p1 F . Let F be a sheaf on X and G a sheaf on Y , then F G is defined to be ∗ ∗ (pX F ) ⊗ (pY G ). Conjecture 3.1 (Baby Version of GLC). For each local system E on C, there i Vi exists a "local system" cE on Bun such that H (cE) = cE E. `1 d Case n = 1, then i = 0; 1. Bun = Bun1;C = Pic(C) = d=−∞ Pic (C). Then Hecke = f(L0; L; x)jL; L0 2 Pic(C);L ⊂ L0 ⊂ L(x)g. So Hecke0 = f(L0; L; x)jL0 = Lg and Hecke1 = f(L0; L; x)jL0 = L(x)g. As abstract varieties, Hecke0 = Pic(C) × C and Hecke1 = Pic(C) × C. What does it mean to be a Hecke0 eigensheaf? Nothing. The condition is ∗ ∗ p1cE = cE 1 = p1cE. 1 What does it mean to be a Hecke eigensheaf? Want cE on Pic(C) with ∗ ∼ 1 the condition A@ cE = cE E where A@ : Hecke ! Pic(C) and takes (L; x) to L0. N It must be E on C, then on C × C we have E E. So in general E on CN . 3 4 Lecture 4 We have been looking at GLCC for the case G = GL(1), the abelian case. Then we have E a local system on C and it induces natural local systems on ×d d [d] ×d C = C × ::: × C and Sym C = C = C =Sd. ×d We have E ::: E on C whose fiber at (x1; : : : ; xd) = Ex1 ⊗ ::: ⊗ Exd . We claim that there exists a natural local system on Symd(C) whose pull- ×d d back to C is E . The fiber at an unordered d-tuple fx1; : : : ; xdg = D is Bij(f1; : : : ; dg;D). ×d d d We do have a map π : C ! Sym (C), but we can't take π∗(E ), because it has rank d!. We have two actions of Sd, permute the sheets of π and get an action on F and we can act directly on the fiber e1 ⊗:::⊗ed 7! eσ(1) ⊗:::⊗eσ(d). d S d So we look at (π∗E ) d , the sheaf of the Sd-equivariant sections of E . The fiber at D is the assignment f : Bij ! (π∗E)jD such that f(σx) = σf(x). So this gives a sheaf on Symd C. It is a local system for n = 1 but not otherwise. ×n Topologically, ρ : π1(C) ! GL(1), and π1(C ) has maps to π1(C) and ×n GL(1) such that they all commute, and π1(C ) ! GL(1) factors through n ×n π1(Sym (C)), because as GL(1) is abelian, the representation of π1(C ) fac- n tors through π1(Sym (C)).
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