SLIDE 1 Classical (co)homology theory
◮ Let X be a topological space. Can define abelian groups H∗(X; Z) and H∗(X; Z) given by very cute construction of the so-called chains and co-chains. Very useful invariants. ◮ Grothendieck taught us it’s desirable to view things relatively; would like a theory not only for X → pt but also for f : X → Y . ◮ This is achieved by thinking about sheaves: can define H∗(X; Z) using sheaf cohomology. ◮ Theorem. If X is locally contractible and paracompact, then singular cohomology and sheaf cohomology agree. ◮ Now to generalize this for f : X → Y we look at the sheaves Rif∗ZX. ◮ For various reasons, particularly for composition f ◦ g, the object Rf∗ZX is desired. It lies in Db(Y ), the (bounded) derived category
- f sheaves of abelian groups on Y .
SLIDE 2
Poincar´ e duality
◮ In topology, it is desirable to intersect cycles. While this can be represented by cup product in cohomology, the most useful result is probably the Poincar´ e duality: If X is a compact orientable real n-dimensional manifold, then we have perfect pairing H∗(X, Z) × Hn−∗(X, Z) → Hn(X, Z) ∼ = Z. ◮ When X is no longer a manifold, Poincar´ e duality no longer holds. On the other hand, it’s unclear how to intersect (co)chains in X nicely. ◮ Let us bring our attention to the case when X is an n-dimensional complex variety. So the non-smoothness of X may be studied as follows: we have a stratification X = X2n ⊃ X2n−2 ⊃ ... ⊃ X2 ⊃ X0 such that each X2i−2 ⊂ X2i is closed and each X2i − X2i−2 is a topological orientable real manifold of dimension 2i.
SLIDE 3
Intersection homology
◮ Goresky-MacPherson came up with the idea of intersection (co)homology, as follows: a k-chain C on X is allowable if C ∩ X2n−2i has dimension at most k − i − 1 for any i ≥ 1. ◮ Let IH∗(X; Q) be the homology groups of the complex of allowable chains with allowable boundaries on X, called the intersection homology groups. Goresky-MacPherson showed that IH∗(X; Q) does not depend on the choice of the stratification. ◮ For example let X be the nodal curve X = (y 2 = x3 − x2) ⊂ CP2. We have H1(X; Q) = Q. However, IH1(X; Q) = 0 because an allowable 1-chain must not touch the singularity. ◮ We still have IH2(X; Q) ∼ = IH0(X; Q) ∼ = Q, so the intersection homology of X is the same as the ordinary homology its normalization ˜ X ∼ = CP1. In fact there is always a natural isomorphism IH∗(X; Q) ∼ = H∗( ˜ X; Q) for any algebraic curve X. ◮ Theorem. Suppose X is proper. We have a perfect pairing IH∗(X; Q) × IH2n−∗(X; Q) → IH2n(X; Q) = H2n(X; Q) ∼ = Q for coefficients in Q.
SLIDE 4 Perverse sheaves
Theorem. We have again a perfect pairing IH∗(X; Q) × IH2n−∗(X; Q) → IH2n(X; Q) = H2n(X; Q) ∼ = Q for coefficient in Q.
◮ As we saw to have (co)homology theory relatively (for X → Y rather than X → pt) we would like a sheaf version. With Goresky and MacPherson, Deligne worked out the sheaf version: there exists a subcategory Perv(X) of the bounded derived category Db(X) of sheaves of Q-vector spaces on X, called the subcategory of perverse sheaves. ◮ Perv(X) is the full subcategory of Db(X) whose objects are those F such that
- 1. dimR supp(H−k(F)) ≤ 2k (in particular is empty if k < 0).
- 2. dimR supp(H−k(DXF)) ≤ 2k, for the Verdier dual DXF of F.
◮ The category Perv(X) can be proved to have an object ICX whose restriction to the smooth part X ′ := X2n − X2n−2 of X is just QX ′[n], and such that there is a canonical isomorphism IHn−∗(X; Q) ∼ = H∗(X; ICX). ◮ In fact, ICX can be characterized as the unique simple object in Perv(X) with this property. ◮ The Poincar´ e duality can be interpreted as that ICX is Verdier self-dual.
SLIDE 5 An abelian category
Perv(X) is the full subcategory of Db(X) whose objects are those F such that
- 1. dim supp(H−k(F)) ≤ k (in particular Hk(F) is trivial for k > 0).
- 2. dim supp(H−k(DX F)) ≤ k, for the Verdier dual DX F of F.
◮ The category Db(X) is an additive category that is basically never abelian; because the non-abelian-ness of Db(X) reflects why Sh(X) (the category of sheaves of Q-vector spaces on X) has to be derived. ◮ Miracle happens, that the subcategory Perv(X) is abelian! ◮ Well, I guess the idea is really that Perv(X) is secretly a way to modify the abelian category Sh(X). In fact, the bounded derived category of Perv(X) is again Db(X). ◮ The fact that the category Perv(X) is abelian is somewhat behind its numerous applications in representation theory, typically in the form of equivalences of category that for some stack X we have Perv(X) = some abelian category of representations or characters.
SLIDE 6
That Perv(X) is great for representation theory
Perv(X) = some abelian category of representations or characters.
(This slide is a supplement and is not needed for future slides.) ◮ Basically, representation theory studies how non-commutative a group and/or algebra is. In a twisted sense it attaches invariants to groups/algebras that detect how non-commutative they are. ◮ On the other hand, consider for an algebraic group G the quotient stack [G/G] where G acts on G by conjugation. That G is non-commutative is reflected by how different element in G have different centralizer, or how stacky [G/G] is. ◮ This gives a reason, at heuristically, how Perv([G/G]) detects representation theory of G. ◮ This is particularly fruitful for representation theory of G(Fq) where G is a reductive group over a finite field Fq; Lusztig describes G(Fq) via various subcategories of Perv([G/G]).
SLIDE 7
Another example: Geometric Satake
Perv(X) = some abelian category of representations or characters.
(This slide is a supplement and is not needed for future slides.) ◮ Let G be a connected reductive group over C. Consider LG = Hom(Spec C((t)), G) the loop group and L+G = Hom(Spec C[[t]], G) the arc group. Then we have an equivalence of category Perv(L+G\LG/L+G) = Rep(G ∨) where G ∨ is the dual reductive group (a reductive group over C whose combinatorial datum is opposite to that of G). This is the Geometric Satake, and is of fundamental importance to Langlands program as Perv(L+G\LG/L+G) describes representations of LG with a fixed LG-vector. In the mixed characteristic setting, it’s representations of G(Qp) with a G(Zp)-fixed vector.
SLIDE 8 A toy example in Minimal Model Program
◮ In minimal model program for 3-folds, Mori connected minimal models with flops. ◮ A flop is a pair of birational proper surjections: X Y Z
- f 3-folds with certain properties. In particular, X and Y are similar
to being smooth (terminal singularity) and we will pretend they are smooth. ◮ The morphisms f , g are small contractions; outside a few curves on X and Y and a few points on Z they are isomorphic, and the preimage of a point in Z is at worst curves. ◮ In general, a morphism f : X → Z to an equi-dimensional variety Z is called small to representation theorists (not algebraic geometers unless dim Z = 3) if codim{z ∈ Z | dim f −1(z) ≥ i} ≥ 2i + 1. ◮ Do you agree this looks like a relative version of allowable chains?
SLIDE 9
A toy example in Minimal Model Program, cont.
codim{y ∈ Y | dim f −1(y) ≥ i} ≥ 2i + 1. Do you agree this looks like a relative version of allowable chains?
◮ It is indeed the case that the machinery of perverse sheaves is able to treat small proper morphisms as if they are smooth of dimension 0, i.e. ´ etale. ◮ Birational ´ etale morphisms are isomorphisms. For us this means we have Rf∗ICX = ICZ = Rg∗ICY . But X and Y are (almost) smooth! We have H∗(X; Q) = H∗(Z; Rf∗QX) = H∗(Z; Rf∗ICX[−3]) = H∗(Z; ICZ[−3]) = IH∗(Z; Q). ◮ Same for Y , so H∗(X; Q) = IH∗(Z; Q) = H∗(Y ; Q). ◮ For Mori, this proved that birationally equivalent minimal models in 3d have isomorphic (co)homology groups in Q-coefficients. Sweet?
