Past Algebraic and Symplectic Geometry Seminar

Nearly $G_2$-structures in dimension seven are, up to scaling, critical points of a geometric flow called (modified) Laplacian co-flow. Moreover, since nearly $G_2$-structures define Einstein metrics, they can also be associated to critical points of the volume-normalised Ricci flow. In this talk, we will discuss a recent joint work with Jason Lotay, showing that many of these nearly $G_2$ critical points are unstable for the modified co-flow, and giving a lower bound on the index.

I will discuss the Brill-Noether theory of a general elliptic 𝐾3 surface using wall-crossing with respect to Bridgeland stability conditions. As an application, I will provide an example of a general 𝑘-gonal curve from the perspective of Hurwitz-Brill-Noether theory. This is joint work with Gavril Farkas and Andrés Rojas.

Quiver Donaldson-Thomas invariants are integers determined by the geometry of moduli spaces of quiver representations. I will describe a correspondence between quiver Donaldson-Thomas invariants and Gromov-Witten counts of rational curves in toric and cluster varieties. This is joint work with Pierrick Bousseau.

I will describe some recent efforts to recreate the miraculous properties of perverse sheaves on complex analytic spaces in the setting of real stratified spaces.

A consequence of the works of Costello and Lurie is that the Hochschild chain complex of a Calabi-Yau category admits the structure of a framed E_2 algebra (the genus zero operations). I will describe a new algebraic point of view on these operations which admits generalizations to the setting of relative
Calabi-Yau structures, which do not seem to fit into the framework of TQFTs. In particular, we obtain a generalization of string topology to manifolds with boundary, as well as interesting operations on Hochschild homology of Fano varieties. This is joint work with Chris Brav.

We use Non-Reductive GIT to construct compactifications of Hilbert schemes of complete intersections. We then study ample line bundles on these compactifications in order to construct moduli spaces of complete intersections for certain degree types.

Given a Hamiltonian circle action on a symplectic manifold, Fukaya and Teleman tell us that we can relate the equivariant Fukaya category to the Fukaya category of a symplectic reduction.  Yanki Lekili and I have some conjectures that extend this story - in certain special examples - to singular values of the moment map. I'll also explain the mirror symmetry picture that we use to support our conjectures, and how we interpret our claims in Teleman's framework of `topological group actions' on categories.

I will review recent progress on extending the Minimal Model Program to algebraically integrable foliations, focusing on applications such as the canonical bundle formula and recent results toward the boundedness of Fano foliations.

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.

Ever since moduli spaces of polarised K3 surfaces were constructed in the 1980's, people have wondered about the question of compactification: can one make the moduli space of K3 surfaces compact by adding in some boundary components in a "nice" way? Ideally, one hopes to find a compactification that is both explicit and geometric (in the sense that the boundary components provide moduli for degenerate K3's). I will present on joint work in progress with V. Alexeev, which aims to solve the compactification problem for the moduli space of K3 surfaces of degree 2.

The Hall algebra can be constructed using the Waldhausen S-construction. We will give a systematic recipe for this and show how it extends naturally to give a bi-algebraic structure. As a result we obtain a more transparent proof of Green's theorem about the bi-algebra structure on the Hall algebra.

The study of separating invariants is a new trend in Invariant Theory and a return to its roots: invariants as a classification tool. For a finite group acting linearly on a vector space, a separating set is simply a set of invariants whose elements separate the orbits o the action. Such a set need not generate the ring of invariants. In this talk, we give lower bounds on the size of separating sets based on the geometry of the action. These results are obtained via the study of the local cohomology with support at an arrangement of linear subspaces naturally arising from the action.

(Joint with Jack Jeffries)

This is will be a progress report on our long-ongoing joint work with Bezrukavnikov on lifting the monodromy of the quantum differential equation for symplectic resolutions to automorphisms of their derived categories of coherent sheaves. I will attempt to define the ingredient that go both into the problem and into its solution.
 

Following an idea of Gaiotto, a symplectic representation of a complex Lie group G defines a complex Lagrangian subvariety inside the moduli space of G-Higgs bundles. The talk will discuss the case of G=SL(2) and its link with determinant  divisors, or equivalently Brill-Noether loci, in the moduli space of semistable SL(2)-bundles. 

Given a (-1)-shifted symplectic derived scheme or stack (X,w) over C equipped with an orientation, we explain how to construct a perverse sheaf P on the classical truncation of X so that its hypercohomology H*(P) can be regarded as a categorification of (or linearisation of) X. Given also a Lagrangian morphism L -> X equipped with a relative orientation, we outline a programme in progress to construct a natural morphism of constructible complexes on the truncation of L from the (shifted) constant complex on L to a suitable pullback of P to L. The morphisms and resulting hypercohomology classes are expected to satisfy various identities under products, composition of Lagrangian correspondences, etc. This programme will have interesting applications, such as proving associativity of a Kontsevich-Soibelman type COHA multiplication on H*(P) when X is the derived moduli stack of coherent sheaves on a Calabi-Yau 3-fold Y, and defining Lagrangian Floer cohomology and the Fukaya cat!
 egory of an algebraic or complex symplectic manifold S.

I will present a definition of symplectic homology groups for pairs of Liouville cobordisms with fillings, and explain how these fit into a formalism of homology theory similar to that of Eilenberg and Steenrod. This construction allows to understand form a unified point of view many structural results involving Floer homology groups, and yields new applications. Joint work with Kai Cieliebak.

For a holomorphic function (“superpotential”)  W: X —> C on a complex manifold X, one defines the (2-periodic) matrix factorisation category MF(X;W), which is supported on the critical locus Crit(W) of W. At a Morse singularity, MF(X;W) is equivalent to the category of modules over the Clifford algebra on the tangent space TX. It had been suggested by Kapustin and Rozansky that, for Morse-Bott W, MF(X;W) should be equivalent to the (2-periodicised) derived category of Crit(W), twisted by the Clifford algebra of the normal bundle. I will discuss why this holds when the first neighbourhood of Crit(W) splits, why it fails in general, and will explain the correct general statement.