Algebraic and Symplectic Geometry Seminar

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.

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.

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.

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.

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.

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.

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.