Past Algebraic and Symplectic Geometry Seminar

In the talk I plan to overview several constructions for finite dimensional represenations of DAHA: construction via quantization of Hilbert scheme of points in the plane (after Gordon, Stafford), construction via quantum Hamiltonian reduction (after Gan, Ginzburg), monodromic construction (after Calaque, Enriquez, Etingof). I will discuss the relations of the constructions to the conjectures from the first lecture.

A dimer model on a surface with punctures is an embedded quiver such that its vertices correspond to the punctures and the arrows circle round the faces they cut out. To any dimer model Q we can associate two categories: A wrapped Fukaya category F(Q), and a category of matrix factorizations M(Q). In both categories the objects are arrows, which are interpreted as Lagrangian subvarieties in F(Q) and will give us certain matrix factorizations of a potential on the Jacobi algebra of the dimer in M(Q).

We show that there is a duality D on the set of all dimers such that for consistent dimers the category of matrix factorizations M(Q) is isomorphic to the Fukaya category of its dual,  F((DQ)). We also discuss the connection with classical mirror symmetry.

Generalized Donaldson-Thomas invariants $\bar{DT}^\alpha(\tau)$ defined by Joyce and Song are rational numbers which 'count' both $\tau$-stable and $\tau$-semistable coherent sheaves with Chern character $\alpha$ on a Calabi-Yau 3-fold X, where $\tau$ denotes Gieseker stability for some ample line bundle on X. The theory of Joyce and Song is valid only over the field $\mathbb C$. We will extend it to algebraically closed fields $\mathbb K$ of characteristic zero.

We will describe the local structure of the moduli stack $\mathfrak M$ of coherent sheaves on X, showing that an atlas for $\mathfrak M$ may be written locally as the zero locus of an almost closed 1-form on an \'etale open subset of the tangent space of $\mathfrak M$ at a point, and use this to deduce identities on the Behrend

function $\nu_{\mathfrak M}$ of $\mathfrak M$. This also yields an extension of generalized Donaldson-Thomas theory to noncompact Calabi-Yau 3-folds.

Finally, we will investigate how our argument might yield generalizations of the theory to a even wider context, for example the derived framework using Toen's theory and to motivic Donaldson-Thomas theory in the style of Kontsevich and Soibelman.

There exist two constructions of families of exotic monotone Lagrangian tori in complex projective spaces and products of spheres, namely the one by Chekanov and Schlenk, and the one via the Lagrangian circle bundle construction of Biran. It was conjectured that these constructions give Hamiltonian isotopic tori. I will explain why this conjecture is true in the complex projective plane and the product of two two-dimensional spheres.

In joint work with Tim Perutz, we give a complete characterization of the Fukaya category of the punctured torus, denoted by $Fuk(T_0)$. This, in particular, means that one can write down an explicit minimal model for $Fuk(T_0)$ in the form of an A-infinity algebra, denoted by A, and classify A-infinity structures on the relevant algebra. A result that we will discuss is that no associative algebra is quasi-equivalent to the model A of the Fukaya category of the punctured torus, i.e., A is non-formal. $Fuk(T_0)$ will be connected to many topics of interest: 1) It is the boundary category that we associate to a 3-manifold with torus boundary in our extension of Heegaard Floer theory to manifolds with boundary, 2) It is quasi-equivalent to the category of perfect complexes on an irreducible rational curve with a double point, an instance of homological mirror symmetry.

The effective cone of the Hilbert scheme of points in $P^2$ has

finitely many chambers corresponding to finitely many birational models.

In this talk, I will identify these models with moduli of

Bridgeland-stable two-term complexes in the derived category of

coherent sheaves on $P^2$ and describe a

map from (a slice of) the stability manifold of $P^2$

to the effective cone of the Hilbert scheme that would explain the

correspondence. This is joint work with Daniele Arcara and Izzet Coskun.

I will discuss a conjectural Bogomolov-Gieseker type inequality for "tilt-stable" objects in the derived category of coherent sheaves on smooth projective threefolds. The conjecture implies the existence of Bridgeland stability conditions on threefolds, and also has implications to birational geometry: it implies a slightly weaker version of Fujita's conjecture on very ampleness of adjoint line bundles.

Counting the number of curves of degree $d$ with $n$ nodes (and no further singularities) going through $(d^2+3d)/2 - n$ points in general position in the projective plane is a problem which was already considered more than 150 years ago. More recently, people conjectured that for sufficiently large $d$ this number should be given by a polynomial of degree $2n$ in $d$. More generally, the Göttsche conjecture states that the number of $n$-nodal curves in a general $n$-dimensional linear subsystem of a sufficiently ample line bundle $L$ on a nonsingular projective surface $S$ is given by a universal polynomial of degree $n$ in the 4 topological numbers $L^2, L.K_S, (K_S)^2$ and $c_2(S)$. In a joint work with Vivek Shende and Richard Thomas, we give a short (compared to existing) proof of this conjecture.

I describe a conjecture equating the two items appearing in the title.

I will explain how essential information about the structure of symplectic manifolds is captured by algebraic data, and specifically by the non-commutative mixed Hodge structure on the cohomology of the Fukaya category. I will discuss computable Hodge theoretic invariants arising from twist functors, and from geometric extensions. I will also explain how the instanton-corrected Chern-Simons theory fits in the framework of normal functions in non-commutative Hodge theory and will give applications to explicit descriptions of quantum Lagrangian branes. This is a joint work with L. Katzarkov and M. Kontsevich.

If $X$ is a Fano variety with canonical bundle $O(-k)$, its derived category

has a semi-orthogonal decomposition (I will say what that means)

\[ D(X) = \langle O(-k+1), ..., O(-1), O, A \rangle, \]

where the subcategory $A$ is the "interesting piece" of $D(X)$. In the previous talk we saw that $A$ can have very rich geometry. In this talk we will see a less well-understood example of this: when $X$ is a smooth cubic in $P^5$, $A$ looks like the derived category of a K3 surface. We will discuss Kuznetsov's conjecture that $X$ is rational if and only if $A$ is geometric, relate it to Hassett's earlier work on the Hodge theory of $X$, and mention an autoequivalence of $D(Hilb^2(K3))$ that I came across while studying the problem.

There is a long-studied correspondence between intersections of two quadrics and hyperelliptic curves, first noticed by Weil and since used

as a testbed for many fashionable theories: Hodge theory, motives, and moduli of vector bundles in the '70s and '80s, derived categories in the '90s, non-commutative geometry and mirror symmetry today. The story generalizes to three, four, and more quadrics, exhibiting new geometric behaviour at each step. The case of four quadrics nicely illustrates the modern theory of flops and derivced categories and, as a special case, gives a pair of derived-equivalent Calabi-Yau 3-folds.

We first present the localized virtual cycles by cosections of obstruction sheaves constructed by Kiem and Li. This construction has two kinds of applications: one is define invariants for non-proper moduli spaces; the other is to reduce the obstruction classes. We will present two recent applications of this construction: one is the Gromov-Witten invariants of stable maps with fields (joint work with Chang); the other is studying Donaldson-Thomas invariants of Calabi-Yau threefolds (joint work with Kiem).

We first present the localized virtual cycles by cosections of obstruction sheaves constructed by Kiem and Li. This construction has two kinds of applications: one is define invariants for non-proper moduli spaces; the other is to reduce the obstruction classes. We will present two recent applications of this construction: one is the Gromov-Witten invariants of stable maps with fields (joint work with Chang); the other is studying Donaldson-Thomas invariants of Calabi-Yau threefolds (joint work with Kiem).