Past Algebraic and Symplectic Geometry Seminar

I will describe a version of the definition of stability conditions on a triangulated category to which we were led by the study of quantization of symplectic resolutions of singularities over fields of positive characteristic. Partly motivated by ideas of Tom Bridgeland, we conjectured a relation of this structure to equivariant quantum cohomology; this conjecture has been verified in some classes of examples. The talk is based on joint projects with Anno, Mirkovic, Okounkov and others

I will describe a version of the definition of stability conditions on a triangulated category to which we were led by the study of quantization of symplectic resolutions of singularities over fields of positive characteristic. Partly motivated by ideas of Tom Bridgeland, we conjectured a relation of this structure to equivariant quantum cohomology; this conjecture has been verified in some classes of examples. The talk is based on joint projects with Anno, Mirkovic, Okounkov and others

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).

Sutured manifolds are compact oriented 3-manifolds with boundary, together with a set of dividing curves on the boundary. Sutured Floer homology is an invariant of balanced sutured manifolds that is a common generalization of the hat version of Heegaard Floer homology and knot Floer homology. I will define cobordisms between sutured manifolds, and show that they induce maps on sutured Floer homology groups, providing a type of TQFT. As a consequence, one gets maps on knot Floer homology groups induced by decorated knot cobordisms.

Gravitational instantons are complete hyperkaehler 4-manifolds whose Riemann curvature tensor is square integrable. They can be viewed as Einstein geometry analogs of finite energy Yang-Mills instantons on Euclidean space. Classical examples include Kronheimer's ALE metrics on crepant resolutions of rational surface singularities and the ALF Riemannian Taub-NUT metric, but a classification has remained largely elusive. I will present a large, new connected family of gravitational instantons, based on removing fibers from rational elliptic surfaces, which contains ALG and ALH spaces as well as some unexpected geometries.

On a manifold there is the graded algebra of polyvector fields with its Lie-Schouten bracket, and the module of de Rham differentials with exterior differentiation. This package is called a "calculus". The moduli

space of sheaves (or derived category objects) on a Calabi-Yau threefold has a kind of "virtual calculus" on it, at least conjecturally. In particular, this moduli space has virtual de Rham cohomology groups, which categorify Donaldson-Thomas invariants, at least conjecturally. We describe some attempts at constructing such a virtual calculus. This is work in progress.

On a manifold there is the graded algebra of polyvector fields with its Lie-Schouten bracket, and the module of de Rham differentials with exteriour differentiation. This package is called a "calculus". The moduli space of sheaves (or derived category objects) on a Calabi-Yau threefold has a kind of "virtual calculus" on it, at least conjecturally. In particular, this moduli space has virtual de Rham cohomology groups, which categorify Donaldson-Thomas invariants, at least conjecturally. We describe some attempts at constructing such a virtual calculus. This is work in progress.

I will describe recent work on motivic DT invariants for 3-manifolds, which are expected to be a refinement of Chern-Simons theory. The conclusion will be that these should be possible to define and work with, but there will be some interesting problems along the way. There will be a discussion of the problem of upgrading the description of the moduli space of flat connections as a critical locus to the problem of describing the fundamental group algebra of a 3-fold as a "noncommutative critical locus," including a recent topological result on obstructions for this problem. I will also address the question of how a motivic DT invariant may be expected to pick up a finer invariant of 3-manifolds than just the fundamental group.

Symplectic cohomology is an invariant of symplectic manifolds with contact type boundary. For example, for disc cotangent bundles it recovers the

homology of the free loop space. The aim of this talk is to describe algebraic operations on symplectic cohomology and to deduce applications in

symplectic topology. Applications range from describing the topology of exact Lagrangian submanifolds, to proving existence theorems about closed

Hamiltonian orbits and Reeb chords.

I will show that given smooth embedded Lagrangian L in a Calabi-Yau, one can find a perturbation of L which lies in the same hamiltonian isotopy class and such that the correspondent solution to mean curvature flow develops a finite time singularity. This shows in particular that a simplified version of the Thomas-Yau conjecture does not hold.

Both parts will deal with spherical objects in the bounded derived

category of coherent sheaves on K3 surfaces. In the first talk I will

focus on cycle theoretic aspects. For this we think of the Grothendieck

group of the derived category as the Chow group of the K3 surface (which

over the complex numbers is infinite-dimensional due to a result of

Mumford). The Bloch-Beilinson conjecture predicts that over number

fields the Chow group is small and I will show that this is equivalent to

the derived category being generated by spherical objects (which

I do not know how to prove). In the second talk I will turn to stability

conditions and show that a stability condition is determined by its

behavior with respect to the discrete collections of spherical objects.

Both parts will deal with spherical objects in the bounded derived

category of coherent sheaves on K3 surfaces. In the first talk I will

focus on cycle theoretic aspects. For this we think of the Grothendieck

group of the derived category as the Chow group of the K3 surface (which

over the complex numbers is infinite-dimensional due to a result of

Mumford). The Bloch-Beilinson conjecture predicts that over number

fields the Chow group is small and I will show that this is equivalent to

the derived category being generated by spherical objects (which

I do not know how to prove). In the second talk I will turn to stability

conditions and show that a stability condition is determined by its

behavior with respect to the discrete collections of spherical objects.

The second talk will present conjectural motivic generalizations

of ADHM sheaf invariants as well as their wallcrossing formulas.

It will be shown that these conjectures yield recursive formulas

for Poincare and Hodge polynomials of moduli spaces of Hitchin

pairs. It will be checked in many concrete examples that this recursion relation is in agreement with previous results of Hitchin, Gothen, Hausel and Rodriguez-Villegas.

The first talk will present a construction of equivariant

virtual counting invariants for certain quiver sheaves on a curve, called ADHM sheaves. It will be shown that these invariants are related to the stable pair theory of Pandharipande and Thomas in a specific stability chamber. Wallcrossing formulas will be derived using the theory of generalized Donaldson-Thomas invariants of Joyce and Song.

A categorification of cycle class maps consists to define

realization functors from constructible motivic sheaves to other

categories of coefficients (e.g. constructible $l$-adic sheaves), which are compatible with the six operations. Given a field $k$, we

will describe a systematic construction, which associates,

to any cohomology theory $E$, represented in $DM(k)$, a

triangulated category of constructible $E$-modules $D(X,E)$, for $X$

of finite type over $k$, endowed with a realization functor from

the triangulated category of constructible motivic sheaves over $X$.

In the case $E$ is either algebraic de Rham cohomology (with $char(k)=0$), or $E$ is $l$-adic cohomology, one recovers in this way the triangulated categories of $D$-modules or of $l$-adic sheaves. In the case $E$ is rigid cohomology (with $char(k)=p>0$), this construction provides a nice system of $p$-adic coefficients which is closed under the six operations.

Starting from Morel and Voevodsky's stable homotopy theory of schemes, one defines, for each noetherian scheme of finite dimension $X$, the triangulated category $DM(X)$ of motives over $X$ (with rational coefficients). These categories satisfy all the the expected functorialities (Grothendieck's six operations), from

which one deduces that $DM$ also satisfies cohomological proper

descent. Together with Gabber's weak local uniformisation theorem,

this allows to prove other expected properties (e.g. finiteness

theorems, duality theorems), at least for motivic sheaves over

excellent schemes.

We report on joint work with Arend Bayer on the space of stability conditions for the canonical bundle on the projective plane.

We will describe a connected component of this space, generalizing and completing a previous construction of Bridgeland.

In particular, we will see how this space is related to classical results of Drezet-Le Potier on stable vector bundles on the projective plane. Using this, we can determine the group of autoequivalences of the derived category. As a consequence, we can identify a $\Gamma_1(3)$-action on the space of stability conditions, which will give a global picture of mirror symmetry for this example.

In the second hour we will give some details on the proof of the main theorem.

We report on joint work with Arend Bayer on the space of stability conditions for the canonical bundle on the projective plane.

We will describe a connected component of this space, generalizing and completing a previous construction of Bridgeland.

In particular, we will see how this space is related to classical results of Drezet-Le Potier on stable vector bundles on the projective plane. Using this, we can determine the group of autoequivalences of the derived category. As a consequence, we can identify a $\Gamma_1(3)$-action on the space of stability conditions, which will give a global picture of mirror symmetry for this example.

In the second hour we will give some details on the proof of the main theorem.