Dominic Joyce (Oxford). Categorification of Donaldson-Thomas invariants of Calabi-Yau 3-folds using perverse sheaves.

Friday February 22nd. Huxley 130, 1.30-2.30pm.

Abstract:

This talk is a survey of a sprawling collection of projects in progress, joint with Lino Amorim, Dennis Borisov, Chris Brav, Vittoria Bussi, Delphine Dupont, Kobi Kremnizer, and Balazs Szendroi, funded by an EPSRC Programme Grant. Our starting point was the notion of shifted symplectic structure on derived schemes and stacks introduced by Pantev-Toen-Vaquie-Vezzosi (PTVV). As there is a great deal of material, I will just give a broad overview. Those who want to know about the details can ask questions afterwards.

We are working towards applications in extensions of Donaldson-Thomas theory of Calabi-Yau 3-folds, new sheaf-counting invariants of Calabi-Yau 4-folds, and complex symplectic geometry including a new definition of a “Fukaya category” of derived complex Lagrangians in a complex symplectic manifold.
PTVV (arXiv: 1111.3209) define a notion of “k-shifted symplectic structure” on a derived scheme or derived stack, for integers k. If a derived scheme is 0-symplectic, then it is just a classical smooth symplectic scheme. PTVV show that derived moduli stacks of (complexes of) coherent sheaves on a Calabi-Yau m-fold have a 2-m-shifted symplectic structure. In particular, for a Calabi-Yau 3-fold (the Donaldson-Thomas case), they are -1-shifted symplectic.

Our first main result shows that -1-shifted symplectic derived scheme is Zariski locally modelled, as a -1-shifted symplectic derived scheme, on the critical locus Crit(f) of a polynomial f : A^n –> A^1 on affine space. Hence, moduli schemes of (complexes of) coherent sheaves on a Calabi-Yau 3-fold are locally modelled on critical loci. A version of this result was proved in the complex analytic case for moduli of sheaves by Joyce-Song using gauge theory, and an extension to the derived category was claimed by Behrend-Getzler but has not yet appeared. Our new proof is wholly algebraic, and works over other fields.

We are working on extensions of this result to the derived Artin stack case (a corollary of this will be proof of the Behrend function identities for moduli stacks in D^bcoh(X), and hence extension of D-T invariant wall-crossing formulae to the derived category), and to k-shifted symplectic derived schemes for k \le -2 (hence, local models for Calabi-Yau m-fold moduli schemes for m \ge 4). We know what the appropriate local models for CY m-fold moduli schemes are, probably we’ll only prove they are valid etale locally rather than Zariski locally.

The next main theme is categorification of Donaldson-Thomas invariants using perverse sheaves. If V is a complex manifold and f : V –> C is a holomorphic function, it has a “perverse sheaf of vanishing cycles” PV_{V,f} supported on Crit(f) (regarded as a subscheme of V in the algebraic case, or a complex analytic subspace of V in the holomorphic case). The pointwise Euler characteristic of PV_{V,f} is the Behrend function \nu of Crit(f). The hypercohomology H^*(PV_{V,f}) is a finite-dimensional graded vector space over Q, whose graded dimension is the weighted Euler characteristic
\chi( Crit(f), \nu) of Crit(f) weighted by the Behrend function. Such weighted Euler characteristics are used to define Donaldson-Thomas invariants of Calabi-Yau 3-folds.

We know (above) that Calabi-Yau 3-fold moduli schemes (-1 shifted derived schemes) X are locally modelled on critical loci Crit(f). So we can ask: does there exist a global perverse sheaf P_X on X, which is locally modelled on PV_{V,f} when X is locally modelled on Crit (f : V –> C)? We can prove that the answer to this is yes, and P_X is unique up to canonical isomorphism, provided X is equipped with “orientation data”. Here orientation data (roughly following Kontsevich and Soibelman) is a choice of square root line bundle of the line bundle det(L_X) of the cotangent complex L_X of the derived scheme X.
It is an important open problem whether orientation data exists for CY3 moduli spaces. We have examples of -1-shifted derived schemes for which no orientation data exists, but these probably cannot arise as CY3 moduli schemes.
To prove this, we have to prove some new results on perverse sheaves of vanishing cycles (arXiv:1211.3259). Basically, we want to know: if f : V –> C, g : W –> C are holomorphic, and we have an isomorphism Crit(f) = Crit(g) of schemes / complex analytic spaces, then when is PV_{V,f} isomorphic to PV_{W,g} ? We need this because to glue the local perverse sheaves PV_{V,f} on open sets in X to get a global perverse sheaf on X, we must find isomorphisms PV_{V,f} = PV_{W,g} on overlaps between Crit(f) and Crit(g).
We show that under some natural conditions on f,g, there is an isomorphism PV_{V,f} = PV_{W,g} \otimes_{Z_2} Q, where Q is a certain principal Z_2 bundle on Crit(f) = Crit(g), which roughly parametrizes orientations for a family of nondegenerate quadratic forms. We need the “orientation data” to deal with these principal Z_2-bundles Q on overlaps.

A consequence: suppose that X is a proper moduli scheme of stable sheaves on a CY3 with no semistables, and orientation data exists for X. Then we construct a graded vector space H^*(P_X) over Q whose graded dimension is the Donaldson-Thomas invariant DT(X) — so H^*(P_X) is a “categorification” of DT(X).
We are working on extending this to Artin stacks. We hope in future to define multiplicative operations on such vector spaces H^*(P_X), and hence define “cohomological Hall algebras” by this perverse sheaf method, extending Kontsevich-Soibelman.

A third theme is applications to complex symplectic geometry. PTVV show that the (derived) intersection L \cap M of Lagrangians L,M in a complex symplectic manifold (S,\omega) has the structure of a -1-shifted derived scheme. Hence our results show that given “orientation data”, we can define a perverse sheaf P_{L,M} on L\cap M. The hypercohomology H^*(P_{L,M}) behaves very like the Lagrangian Floer cohomology HF^*(L,M).
To define “orientation data” for L\cap M, the natural thing is to assume we are given square roots K_L^{1/2}, K_M^{1/2} for the canonical bundles K_L,K_M of L,M.

Thus, we can define a “Fukaya category” F(S,\omega) of a complex symplectic manifold, whose objects are pairs (L,K_L^{1/2}) of a complex Lagrangian (not necessarily compact), and a square root K_L^{1/2} of the canonical bundle, whose morphisms Hom(L,M) are the hypercohomology H^*(P_{L,M}) of the perverse sheaf P_{L,M} on L \cap M. For Lagrangians L,M,N, I expect to define a natural morphism of perverse sheaves
P_{L,M} \otimes^L P_{M,N} –> P_{L,N}, which will induce a morphism Hom(L,M) x Hom(M,N) –> Hom(L,N), the composition in the category. I haven’t sorted the details out yet.

It is interesting that the morphisms in F(S,\omega) are finite-dimensional Q-vector spaces, rather than modules over a Novikov ring, or whatever.

In fact we can do more: we can also include “derived Lagrangians” (in the PTVV sense) in our category F(S). An advantage of this is that the composition of Lagrangian correspondences L_{12}, L_{23} in S_1 x S_2, S_2 x S_3 is only a Lagrangian correspondence in S_1 x S_3 if the intersection (L_{12} x S_3) \cap (S_1 x L_{23}) is transverse. But the intersection is always a derived Lagrangian, and compositions of derived Lagrangian correspondences are derived Lagrangian correspondences. Hence, we can use (derived) Lagrangian correspondences L_{12} in S_1 x S_2 to induce functors F(S_1) –> F(S_2), and these functors should compose nicely — it is a very clean theory.