Past Algebraic and Symplectic Geometry Seminar

Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $M$ and the symplectic geometry of the cotangent bundle of $M$ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $\Lambda$ are deduced from the existence of a sheaf with microsupport $\Lambda$, which we call a quantization of $\Lambda$.

In the third talk we will see that $\Lambda$ admits a canonical quantization if it is a "conification" of a compact exact Lagrangian submanifold of a

cotangent bundle. We will see how to use this quantization to recover results of Fukaya-Seidel-Smith and Abouzaid on the topology of $\Lambda$.

Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $M$ and the symplectic geometry of the cotangent bundle of $M$ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $\Lambda$ are deduced from the existence of a sheaf with microsupport $\Lambda$, which we call a quantization of $\Lambda$.

In the second talk we will introduce a stack on $\Lambda$ by localization of the category of sheaves on $M$. We deduce topological obstructions on $\Lambda$ for the existence of a quantization.

Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $M$ and the symplectic geometry of the cotangent bundle of $M$ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $\Lambda$ are deduced from the existence of a sheaf with microsupport $\Lambda$, which we call a quantization of $\Lambda$.

In the first talk we will see that the graph of a Hamiltonian isotopy admits a canonical quantization and we deduce a new proof of Arnold's non-displaceability conjecture.

A Lagrangian submanifold of a Calabi-Yau manifold is called positive if the real part of the holomorphic volume form restricted to it is positive. A Hamiltonian isotopy class of positive Lagrangian submanifolds admits a Riemannian metric with non-positive curvature. Its universal cover

admits a functional, with critical points special Lagrangians, that is strictly convex with respect to the metric. If time permits, I'll explain

how mirror symmetry relates the metric and functional to the infinite dimensional symplectic reduction picture of Atiyah, Bott, and Donaldson in

the context of the Kobayashi-Hitchin correspondence.

A refinement of the Pandharipande-Thomas stable pair invariants for local toric Calabi-Yau threefolds is defined by what we call the virtual Bialynicki-Birula decomposition. We propose a product formula for the generating function for the refined stable pair invariants extending the motivic product formula of Morrison, Mozgovoy, Nagao, and Szendroi for local ${\bf P}^1$. I will also describe how the proposed product formula is related to the wall crossing in my first talk. This is joint work with Sheldon Katz and Albrecht Klemm.

We study the birational relationship between the moduli spaces of $\alpha$-stable pairs and the moduli space $M(d,1)$ of stable sheaves on ${\bf P}^2$ with Hilbert polynomial $dm+1$. We explicitly relate them by birational morphisms when $d=4$ and $5$, and we describe the blow-up centers geometrically. As a byproduct, we obtain the Poincare polynomials of the moduli space of stable sheaves, or equivalently the refined BPS index. This is joint work with Kiryong Chung.

Beautiful results of Deligne-Illusie, Sabbah, and Ogus-Vologodsky show that certain modifications of the de Rham complex (either the usual one, or twisted versions of it that appear in the study of the cyclic homology of categories of matrix factorizations) are formal in positive characteristic. These are the crucial steps in proving algebraic analogues of the Hodge theorem (again, either in the ordinary setting or in the presence of a twisting). I will present these results along with a new approach to understanding them using derived intersection theory. This is joint work with Dima Arinkin and Marton Hablicsek.

SEMINAR CANCELLED

How many nodal degree d plane curves are tangent to a given line? The celebrated Caporaso-Harris recursion formula gives a complete answer for any number of nodes, degrees, and all possible tangency conditions. In this talk, I will report my recent work on the generalization of the above problem to count singular curves with given tangency condition to a fixed smooth divisor on general surfaces. I will relate the enumeration to tautological integrals on Hilbert schemes of points and show the numbers of curves in question are given by universal polynomials. As a result, we can obtain infinitely many new formulas for nodal curves and understand the asymptotic behavior for all singular curves with any tangency conditions.

Let $X$ be a variety and $Z$ be a sub-variety. Can one glue vector bundles on $X-Z$ with vector bundles on some small neighborhood of $Z$? We survey two recent results on the process of gluing a vector bundle on the complement of a sub-variety with a vector bundle on some 'small' neighborhood of the sub-variety. This is joint work. The first with M. Temkin and is about gluing categories of coherent sheaves over the category of coherent sheaves on a Berkovich analytic space. The second with J. Block and is about gluing dg enhancements of the derived category of coherent sheaves.

I will explain recent results with Emanuele Macrì, in which we systematically study the birational geometry of moduli of sheaves on K3's via wall-crossing for

Bridgeland stability conditions. In particular, we obtain descriptions of their nef cones via the Mukai lattice of the K3, their moveable cones, their divisorial contractions, and obtain counter-examples to various conjectures in the literature. We also give a proof of the Lagrangian fibration conjecture (due to

Hassett-Tschinkel/Huybrechts/Sawon) via wall-crossing.

Counting nodal curves in linear systems $|L|$ on smooth projective surfaces $S$ is a problem with a long history. The G\"ottsche conjecture, now proved by several people, states that these counts are universal and only depend on $c_1(L)^2$, $c_1(L)\cdot c_1(S)$, $c_1(S)^2$ and $c_2(S)$. We present a quite general definition of reduced Gromov-Witten and stable pair invariants on S. The reduced stable pair theory is entirely computable. Moreover, we prove that certain reduced Gromov-Witten and stable pair invariants with many point insertions coincide and are both equal to the nodal curve counts appearing in the Göttsche conjecture. This can be seen as version of the MNOP conjecture for the canonical bundle $K_S$. This is joint work with R. P. Thomas.

The "de Rham-Witt complex" of Deligne and Illusie is a functorial complex of sheaves $W^*(X)$ on a smooth algebraic variety $X$ over a finite field, computing the cristalline cohomology of $X$. I am going to present a non-commutative generalization of this: even for a non-commutative ring $A$, one can define a functorial "Hochschild-Witt complex" with homology $WHH^*(A)$; if $A$ is commutative, then $WHH^i(A)=W^i(X)$, $X = Spec A$ (this is analogous to the isomorphism $HH^i(A)=H^i(X)$ discovered by Hochschild, Kostant and Rosenberg). Moreover, the construction of the Hochschild-Witt complex is actually simpler than the Deligne-Illusie construction, and it allows to clarify the structure of the de Rham-Witt complex.

I will explain an approach to study DT theory of toric CY 3-folds using $L_\infty$ algebras. Based on the construction of strong exceptional collection of line bundles on Fano toric stack of dimension two, we realize any bounded families of sheaves on local surfaces support on zero section as critical sets of the Chern-Simons functions. As a consequence of this construction, several interesting properties of DT invariants on local surfaces can be checked.

Motivated by a conjecture of Alday, Gaiotto and Tachikawa (AGT

conjecture), we construct an action of

a suitable $W$-algebra on the equivariant cohomology of the moduli

space $M_r$ of rank r instantons on $A^2$ (i.e.

on the moduli space of rank $r$ torsion free sheaves on $P^2$,

trivialized at the line at infinity). We show that

the resulting $W$-module is identified with a Verma module, and the

characteristic class of $M_r$ is the Whittaker vector

of that Verma module. One of the main ingredients of our construction

is the so-called cohomological Hall algebra of the

commuting variety, which is a certain associative algebra structure on

the direct sum of equivariant cohomology spaces

of the commuting varieties of $gl(r)$, for all $r$. Joint work with E. Vasserot.