Past Algebraic and Symplectic Geometry Seminar

Singularities are measured in different ways in characteristic zero, positive characteristic, and mixed characteristic. However, classes of singularities usually form analogous groups with similar properties, with an example of such a group being klt, strongly F-regular and BCM-regular.  In this talk we shall focus on newly introduced mixed characteristic counterparts of Du Bois and log canonical singularities and discuss their properties. 

This is joint work with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker and Jakub Witaszek. 

I am interested in studying moduli spaces and associated enumerative invariants via degeneration techniques. Logarithmic geometry is a natural language for constructing and studying relevant moduli spaces. In this talk I  will explain the logarithmic Hilbert (or more generally Quot) scheme and outline how the construction helps study enumerative invariants associated to Hilbert/Quot schemes- a story we now understand well. Time permitting, I will discuss some challenges and key insights for studying moduli of stable vector bundles/ sheaves via similar techniques - a theory whose details are still being worked out. 

Elliptic genus, and its various generalizations, is one of the simplest numerical invariants of a scheme that one can consider in elliptic cohomology. I will present a topological condition which implies that elliptic genus is invariant under wall-crossing. It is related to Krichever-Höhn’s elliptic rigidity. Many applications are possible: to GIT quotients, moduli of sheaves, Donaldson-Thomas invariants, etc.

A well-known problem in algebraic geometry is to construct smooth projective Calabi-Yau varieties $Y$. In the smoothing approach, we construct first a degenerate (reducible) Calabi-Yau scheme $V$ by gluing pieces. Then we aim to find a family $f\colon X \to C$ with special fiber $X_0 = f^{-1}(0) \cong V$ and smooth general fiber $X_t = f^{-1}(t)$. In this talk, we see how infinitesimal logarithmic deformation theory solves the second step of this approach: the construction of a family out of a degenerate fiber $V$. This is achieved via the logarithmic Bogomolov-Tian-Todorov theorem as well as its variant for pairs of a log Calabi-Yau space $f_0\colon X_0 \to S_0$ and a line bundle $\mathcal{L}_0$ on $X_0$.

Derived algebraic geometry allows us to study formal moduli problems via their tangent Lie algebras. After briefly reviewing this general paradigm, I will explain how it sheds light on deformations of Calabi-Yau varieties. 
In joint work with Taelman, we prove a mixed characteristic analogue of the Bogomolov–Tian–Todorov theorem, which asserts that Calabi-Yau varieties in characteristic $0$ are unobstructed. Moreover, we show that ordinary Calabi–Yau varieties in characteristic $p$ admit canonical (and algebraisable) lifts to characteristic $0$, generalising results of Serre-Tate for abelian varieties and Deligne-Nygaard for K3 surfaces. 
If time permits, I will conclude by discussing some intriguing questions related to our canonical lifts.  
 

Lagrangian Floer cohomology groups are extremely hard compute in most situations. In this talk I’ll describe two ways to extract information about the self-Floer cohomology of a monotone Lagrangian possessing certain kinds of symmetry, based on the closed-open string map and the Oh spectral sequence. The focus will be on a particular family of examples, where the techniques can be combined to deduce some unusual properties.

The goal of this talk will be to give an overview of recent work, joint with Kim Moore, on a short time existence problem in Lagrangian mean curvature flow. More specifically, we consider a compact initial Lagrangian submanifold with a finite number of singularities, each asymptotic to a pair of transversely intersecting planes. We show it is possible to construct a smooth Lagrangian mean curvature flow, existing for positive times, that attains the singular Lagrangian as its initial condition in a suitable weak sense.  The construction uses a family of smooth solutions whose initial conditions approximate the singular Lagrangian. In order to appeal to compactness theorems and produce the desired solution, it is necessary to first establish uniform curvature estimates on the approximating family. As time allows I hope to focus in particular on the proof of these estimates, and their role in the proof of the main theorem.

Coisotropic submanifolds arise naturally in symplectic geometry as level sets of moment maps and in algebraic geometry in the context of normal crossing divisors. In examples, the Marsden-Weinstein quotient or (Fano) complete intersections are often uniruled. 
We show that under natural geometric assumptions on a coisotropic submanifold, the symplectic quotient of the coisotropic is always geometrically uniruled. 
I will explain how to assign a Lagrangian and a hypersurface to a fibered, stable coisotropic C. The Lagrangian inherits a fibre bundle structure from C, the hypersurface captures the generalised Reeb dynamics on C. To derive the result, we then adapt and apply techniques from Lagrangian Floer theory and symplectic field theory.
This is joint work with Jonny Evans.
 

Given a Calabi-Yau threefold X, one can count curves on X using various approaches, for example using stable maps or ideal sheaves; for any curve class on X, this produces an infinite sequence of invariants, indexed by extra discrete data (e.g. by the domain genus of a stable map).  Conjecturally, however, this sequence is determined by only a finite number of integer invariants, known as Gopakumar-Vafa invariants.  In this talk, I will propose a direct definition of these invariants via sheaves of vanishing cycles, building on earlier approaches of Kiem-Li and Hosono-Saito-Takahashi.  Conjecturally, these should agree with the invariants as defined by stable maps.  I will also explain how to prove the conjectural correspondence for irreducible curves on local surfaces.  This is joint work with Yukinobu Toda.

With the purpose of examining some relevant geometric properties of the moduli space of sheaves over an algebraic surface, Le Potier conjectured some unexpected duality between the complete linear series of certain natural divisors, called Theta divisors, on the moduli space. Such conjecture is widely known as Strange Duality conjecture. After having motivated the problem by looking at certain instances of quantization in physics, we will work in the setting of surfaces. We will then sketch the proof in the case of abelian surfaces, giving an idea of the techniques that are used. In particular, we will show how the theory of discrete Heisenberg groups and fiber wise Fourier-Mukai transforms, which might be applied to other cases of interest, enter the picture. This is joint work with Alina Marian, Dragos Opera and Kota Yoshioka.

Every finite regular CW complex is, ipso facto, a cohomologically stratified space when filtered by skeleta. We outline a method to recover the canonical (i.e., coarsest possible) stratification of such a complex that is compatible with its underlying cell structure. Our construction proceeds by first localizing and then resolving a complex of cosheaves which capture local cohomology at every cell. The result is a sequence of categories whose limit recovers the desired strata via its (isomorphism classes of) objects. As a bonus, we observe that the entire process is algorithmic and amenable to efficient computations!

Given a Mori Dream Space X, we construct via tropicalisation a model dominating all the small Q-factorial modifications of X. Via this construction we recover a Minkowski basis for the Newton-Okounkov bodies of Cartier divisors on X and hence generators of the movable cone of X. 
This is joint work with Stefano Urbinati.
 

We consider two types of actions on moduli spaces of quiver representations over a field k and we decompose their fixed loci using group cohomology. First, for a perfect field k, we study the action of the absolute Galois group of k on the points of this quiver moduli space valued in an algebraic closure of k; the fixed locus is the set of k-rational points and we obtain a decomposition of this fixed locus indexed by the Brauer group of k. Second, we study algebraic actions of finite groups of quiver automorphisms on these moduli spaces; the fixed locus is decomposed using group cohomology and each component has a modular interpretation. If time permits, we will describe the symplectic and holomorphic geometry of these fixed loci in hyperkaehler quiver varieties. This is joint work with Florent Schaffhauser.

The character variety of a manifold is a moduli space of representations of its fundamental group into some fixed gauge group.  In this talk I will outline the construction of a fully extended topological field theory in dimension 4, which gives a uniform functorial quantization of the character varieties of low-dimensional manifolds, when the gauge group is reductive algebraic (e.g. $GL_N$).

I'll focus on important examples in representation theory arising from the construction, in genus 1:  spherical double affine Hecke algebras (DAHA), difference-operator q-deformations of the Grothendieck-Springer sheaf, and the construction of irreducible DAHA modules mimicking techniques in classical geometric representation theory.  The general constructions are joint with David Ben-Zvi, Adrien Brochier, and Noah Snyder, and applications to representation theory of DAHA are joint with Martina Balagovic and Monica Vazirani.

A classification of simple flops on smooth threefolds in terms of the length invariant was given by Katz and Morrison, who showed that the length must take the value 1,2,3,4,5, or 6. This classification was produced by understanding simultaneous (partial) resolutions that occur in the deformation theory of A, D, E Kleinian surface singularities. An outcome of this construction is that all simple threefold flops of length l occur by pullback from a "universal flop" of length l. Curto and Morrison understood the universal flops of length 1 and 2 using matrix factorisations. I aim to describe how these universal flops can understood for lengths >2 via noncommutative algebra.

I'll show how a simple universal property attaches a category of derived manifolds to any category with finite products and some suitable notion of "topology". Starting with the category of real Euclidean spaces and infinitely differentiable maps yields the category of derived smooth manifolds studied by D. Spivak and others, while starting with affine spaces over some ring and polynomial maps produces a flavour of the derived algebraic geometry of Lurie and Toen-Vezzosi.

I'll motivate this from the differentiable setting by showing that the universal property easily implies all of D. Spivak's axioms for being "good for intersection theory on manifolds".

One of the simplest, and yet largely still open, questions that one can ask about special Lagrangian submanifolds is whether they exist in a given homology class. One possible approach to this problem is to evolve a given Lagrangian submanifold under mean curvature flow in the hope of reaching a special Lagrangian submanifold in the same homology class. It is known, however, that even for 'nice' initial conditions the flow will develop singularities in finite time. 

I will talk about a joint work with Tom Begley, in which we prove a short time existence result for Lagrangian mean curvature flow, where the initial condition is a Lagrangian submanifold of complex Euclidean space with a certain type of singularity. This is a first step to proving, as conjectured by Joyce, that one may 'continue' Lagrangian mean curvature flow after the occurrence of singularities.

Suppose that we have a finite quotient singularity $\mathbb C^n/G$ admitting a crepant resolution $Y$ (i.e. a resolution with $c_1 = 0$). The cohomological McKay correspondence says that the cohomology of $Y$ has a basis given by irreducible representations of $G$ (or conjugacy classes of $G$). Such a result was proven by Batyrev when the coefficient field $\mathbb F$ of the cohomology group is $\mathbb Q$. We give an alternative proof of the cohomological McKay correspondence in some cases by computing symplectic cohomology+ of $Y$ in two different ways. This proof also extends the result to all fields $\mathbb F$ whose characteristic does not divide $|G|$ and it gives us the corresponding basis of conjugacy classes in $H^*(Y)$. We conjecture that there is an extension to certain non-crepant resolutions. This is joint work with Alex Ritter.

I'll discuss recent joint work with Arend Bayer and Ziyu Zhang in which we define a nef divisor class on moduli spaces of Bridgeland-stable objects in the derived category of coherent sheaves with compact support, generalising earlier work of Bayer and Macri for smooth projective varieties. This work forms part of a programme to study the birational geometry of moduli spaces of Bridgeland-stable objects in the derived category of varieties that need not be smooth and projective.