Past Algebraic and Symplectic Geometry Seminar

Quasimaps provide compactifications, depending on a stability parameter epsilon, for moduli spaces of maps from nonsingular algebraic curves to a large class of GIT quotients. These compactifications enjoy good properties and in particular they carry virtual fundamental classes. As the parameter epsilon varies, the resulting invariants are related by wall-crossing formulas. I will present some of these formulas in genus zero, and will explain why they can be viewed as generalizations (in several directions) of Givental's toric mirror theorems. I will also describe extensions of wall-crossing to higher genus, and (time permitting) to orbifold GIT targets as well.
The talk is based on joint works with Bumsig Kim, and partly also with Daewoong Cheong and with Davesh Maulik.

The concept of a matrix factorization was originally introduced by Eisenbud to study syzygies over local rings of singular hypersurfaces. More recently, interactions with mathematical physics, where matrix factorizations appear in quantum field theory, have provided various new insights. I will explain how matrix factorizations can be studied in the context of noncommutative algebraic geometry based on differential graded categories. We will see the relevance of the noncommutative analogue of de Rham cohomology in terms of classical singularity theory. Finally, I will outline how the Kapustin-Li formula for the noncommutative Serre duality pairing (originally computed via path integral methods) can be mathematically explained using a combination of homological perturbation theory and local duality.
Partly based on joint work with Daniel Murfet.

The concept of a matrix factorization was originally introduced by Eisenbud to study syzygies over local rings of singular hypersurfaces. More recently, interactions with mathematical physics, where matrix factorizations appear in quantum field theory, have provided various new insights. I will explain how matrix factorizations can be studied in the context of noncommutative algebraic geometry based on differential graded categories. We will see the relevance of the noncommutative analogue of de Rham cohomology in terms of classical singularity theory. Finally, I will outline how the Kapustin-Li formula for the noncommutative Serre duality pairing (originally computed via path integral methods) can be mathematically explained using a combination of homological perturbation theory and local duality.
Partly based on joint work with Daniel Murfet.

I will speak about recent work with Michael Wemyss (arXiv:1309.0698), applying noncommutative deformation theory to study the birational geometry of 3-folds. In particular, I will explain how every flippable or floppable rational curve in a 3-fold has a naturally associated algebra of noncommutative deformations, even in the singular setting. We investigate the properties of this algebra, and indicate how to calculate it in examples using quiver techniques. This gives new information about the (commutative) geometry of 3-folds, and in particular provides a new tool to differentiate between flops.

As a further application, we show how the noncommutative deformation algebra controls the homological properties of a floppable curve. In this setting, work of Bridgeland and Chen yields a Fourier-Mukai flop-flop functor which acts on the derived category of the 3-fold (assuming any singularities are at worst Gorenstein terminal). We show that this functor can be described as a spherical twist about the universal family over the noncommutative deformation algebra.

In the second part, I will talk about further work in progress, and explain some more technical details, such as the use of noncommutative deformation functors, and the categorical mutations of Iyama and Wemyss. If there is time, I will also give some higher-dimensional examples, and discuss situations involving chains of intersecting floppable curves. In this latter case, deformations, intersections and homological properties are encoded by the path algebra of a quiver, generalizing the algebra of noncommutative deformations.

I will speak about recent work with Michael Wemyss (arXiv:1309.0698), applying noncommutative deformation theory to study the birational geometry of 3-folds. In particular, I will explain how every flippable or floppable rational curve in a 3-fold has a naturally associated algebra of noncommutative deformations, even in the singular setting. We investigate the properties of this algebra, and indicate how to calculate it in examples using quiver techniques. This gives new information about the (commutative) geometry of 3-folds, and in particular provides a new tool to differentiate between flops.

As a further application, we show how the noncommutative deformation algebra controls the homological properties of a floppable curve. In this setting, work of Bridgeland and Chen yields a Fourier-Mukai flop-flop functor which acts on the derived category of the 3-fold (assuming any singularities are at worst Gorenstein terminal). We show that this functor can be described as a spherical twist about the universal family over the noncommutative deformation algebra.

In the second part, I will talk about further work in progress, and explain some more technical details, such as the use of noncommutative deformation functors, and the categorical mutations of Iyama and Wemyss. If there is time, I will also give some higher-dimensional examples, and discuss situations involving chains of intersecting floppable curves. In this latter case, deformations, intersections and homological properties are encoded by the path algebra of a quiver, generalizing the algebra of noncommutative deformations.

We review the theory of $E_n$-algebras (roughly, algebras with $n$ compatible multiplications) and discuss $E_n$-deformation theory in the sense of Lurie. We then describe, to the best of our ability, the use of $E_n$-deformation theory in the on-going work of Calaque, Pantev, Toen, Vezzosi, and Vaquie about deformation quantization of derived stacks with shifted Poisson structure.

We review the theory of $E_n$-algebras (roughly, algebras with $n$ compatible multiplications) and discuss $E_n$-deformation theory in

the sense of Lurie. We then describe, to the best of our ability, the use of $E_n$-deformation theory in the on-going work of Calaque, Pantev, Toen,

Vezzosi, and Vaquie about deformation quantization of derived stacks with shifted Poisson structure.

Many interesting examples of singular symplectic algebraic varieties and their symplectic resolutions are built by Hamiltonian reduction. There is a corresponding construction of "quantum Hamiltonian reduction" which is of substantial interest to representation theorists. It starts from a twisted-equivariant D-module, an analogue of an algebraic vector bundle (or coherent sheaf) on a moment map fiber, and produces an object on the quantum analogue of the symplectic resolution. In order to understand how far apart the quantisation of the singular symplectic variety and its symplectic resolution can be, one wants to know "what gets killed by quantum Hamiltonian reduction?" I will give a precise answer to this question in terms of effective combinatorics. The answer has consequences for exactness of direct images, and thus for t-structures, which I will also explain. The beautiful geometry behind the combinatorics is that of a stratification of a GIT-unstable locus called the "Kirwan-Ness stratification." The lecture will not assume familiarity with D-modules, nor with any previous talks by the speaker or McGerty in this series. The new results are joint work with McGerty.

Given a Lie group $G$, one can construct a principal $G$-bundle on a manifold $M$ by taking a cover $U\to M$, specifying a transition cocycle on the cover, and then descending the trivialized bundle $U \times G$ along the cocycle. We demonstrate the existence of an analogous construction for local $n$-bundles for general $n$. We establish analogues for simplicial Lie groupoids of Moore's results on simplicial groups; these imply that bundles for strict Lie $n$-groupoids arise from local $n$-bundles. We conclude by constructing a simple finite dimensional model of the Lie 2-group String($n$) using cohomological data.

A Landau-Ginzburg B-model is a smooth scheme $X$, equipped with a global function $W$. From $(X,W)$ we can construct a category $D(X,W)$, which is called by various names, including ‘the category of B-branes’. In the case $W=0$ it is exactly the derived category $D(X)$, and in the case that $X$ is affine it is the category of matrix factorizations of $W$. There has been a lot of foundational work on this category in recent years, I’ll describe the most modern and flexible approach to its construction.

I’ll then interpret Nick Addington’s thesis in this language. We’ll consider the case that $W$ is a quadratic form on a vector bundle, and the corresponding global version of Knorrer periodicity. We’ll see that interesting gerbe structures arise, related to the bundle of isotropic Grassmannians.

I will explain some recent joint work with Georgios Dimitroglou Rizell in which we use moduli spaces of holomorphic discs with boundary on a monotone Lagrangian torus in ${\mathbb C}^n$ to prove that all such tori are smoothly isotopic when $n$ is odd and at least 5

Consider non-negative integers assigned to the vertexes of an oriented graph. To this combinatorial data we associate a so-called quiver representation. We will study the geometry and the algebra of this representation, when the underlying un-oriented graph is of Dynkin type ADE.

A remarkable object we will consider is Kazarian's equivariant cohomology spectral sequence. The edge homomorphism of this spectral sequence defines the so-called quiver polynomials. These polynomials are generalizations of remarkable polynomials in algebraic combinatorics (Giambelli-Thom-Porteous, Schur, Schubert, their double, universal, and quantum versions). Quiver polynomials measure degeneracy loci of maps among vector bundles over a common base space. We will present interpolation, residue, and (conjectured) positivity properties of these polynomials.

The quiver polynomials are also encoded in the Cohomological Hall Algebra (COHA) associated with the oriented graph. This is a non-commutative algebra defined by Kontsevich and Soibelman in relation with Donaldson-Thomas invariants. The above mentioned spectral sequence has a structure identity expressing the fact that the sequence converges to explicit groups. We will show the role of this structure identity in understanding the structure of the COHA. The obtained identities are equivalent to Reineke's quantum dilogarithm identities associated to ADE quivers and certain stability conditions.

Consider non-negative integers assigned to the vertexes of an oriented graph. To this combinatorial data we associate a so-called quiver representation. We will study the geometry and the algebra of this representation, when the underlying un-oriented graph is of Dynkin type ADE.

A remarkable object we will consider is Kazarian's equivariant cohomology spectral sequence. The edge homomorphism of this spectral sequence defines the so-called quiver polynomials. These polynomials are generalizations of remarkable polynomials in algebraic combinatorics (Giambelli-Thom-Porteous, Schur, Schubert, their double, universal, and quantum versions). Quiver polynomials measure degeneracy loci of maps among vector bundles over a common base space. We will present interpolation, residue, and (conjectured) positivity properties of these polynomials.

The quiver polynomials are also encoded in the Cohomological Hall Algebra (COHA) associated with the oriented graph. This is a non-commutative algebra defined by Kontsevich and Soibelman in relation with Donaldson-Thomas invariants. The above mentioned spectral sequence has a structure identity expressing the fact that the sequence converges to explicit groups. We will show the role of this structure identity in understanding the structure of the COHA. The obtained identities are equivalent to Reineke's quantum dilogarithm identities associated to ADE quivers and certain stability conditions.

The bounded coherent dg-category on (suitable versions of) the Steinberg stack of a reductive group G is a categorification of the affine Hecke algebra in representation theory.  We discuss how to describe the center and universal trace of this monoidal dg-category.  Many of the techniques involved are very general, and the description makes use of the notion of "odd micro-support" of coherent complexes.  This is joint work with Ben-Zvi and Nadler.

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.