Past Algebraic and Symplectic Geometry Seminar

We introduce a higher rank analog of Pandharipande-Thomas theory of stable pairs. Given a Calabi-Yau threefold $X$, we define the higher
rank stable pairs (which we call frozen triples) given by the data $(F,\phi)$ where $F$ is a pure coherent sheaf with one dimensional support over $X$ and $\phi:{\mathcal O}^r\rightarrow F$ is a map. We compute the Donaldson-Thomas type invariants associated to the frozen triples using the wall-crossing formula of Joyce-Song and Kontsevich-Soibelman. This work is a sequel to arXiv:1011.6342, where we gave a deformation theoretic construction of a higher rank enumerative theory of stable pairs over a Calabi-Yau threefold, and we computed similar invariants using Graber-Pandharipande virtual localization technique.

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.

We show how to compute the symplectic homology of a 4-dimensional Weinstein manifold from a diagram of the Legendrian link which is the attaching locus of its 2-handles. The computation uses a combination of a generalization of Chekanov's description of the Legendrian homology of links in standard contact 3-space, where the ambient contact manifold is replaced by a connected sum of $S^1\times S^2$'s, and recent results on the behaviour of holomorphic curve invariants under Legendrian surgery.

Lagrangian submanifolds are an important class of objects in symplectic geometry. They arise in diverse settings: as vanishing cycles in complex algebraic geometry, as invariant sets in integrable systems, as Heegaard tori in Heegaard-Floer theory and of course as "branes" in the A-model of mirror symmetry. We ask the difficult question: when are two Lagrangian submanifolds isotopic? Restricting to the simplest case of Lagrangian spheres in rational surfaces we will give examples where this question has a complete answer. We will also give some very pictorial examples (due to Seidel) illustrating how two Lagrangians can fail to be isotopic.