September 13th, 2021

Eric Zaslow : The proper Landau-Ginzburg potential is the open mirror map

Abstract: I will discuss toric Fano surfaces in the complement of a smooth anticanonical divisor and their mirror Landau-Ginzburg theories. I will focus on relations between open Gromov-Witten invariants of fibers of the Gross fibration, relative invariants, scattering diagrams and broken lines, tropical curves, superpotentials and wall crossing. This joint work with Tim Graefnitz and Helge Ruddat builds on works of Chan, Lau, Leung, Tseng; the wall crossing takes its cue from the work Gross, Pandharipande, Siebert. Lecture Notes.

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September 20th, 2021

Renzo Cavalieri : The integral Chow ring of $M_{0,0}(\mathbb{P}^r,d)$

Abstract: We give an efficient presentation of the Chow ring with integral coefficients of the open part of the moduli space of rational maps of odd degree to projective space. A less fancy description of this space has its closed points correspond to equivalence classes of $(r+1)$-tuples of degree $d$ polynomials in one variable with no common positive degree factor. We identify this space as a $GL(2,\mathbb{C})$ quotient of an open set in a projective space, and then obtain a (highly redundant) presentation by considering an envelope of the complement. A combinatorial analyis then leads us to eliminating a large number of relations, and to express the remaining ones in generating function form for all dimensions. The upshot of this work is to observe the rich combinatorial structure contained in the Chow rings of these moduli spaces as the degree and the target dimension vary. This is joint work with Damiano Fulghesu.

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October 4th, 2021

Richard Thomas : Higher rank DT theory from curve counting

Abstract: Fix a Calabi-Yau 3-fold X. Its DT invariants count stable bundles and sheaves on X. The generalised DT invariants of Joyce-Song count semistable bundles and sheaves on X. I will describe work with Soheyla Feyzbakhsh showing these generalised DT invariants in any rank r can be written in terms of rank 1 invariants. By the MNOP conjecture the latter are determined by the GW invariants of X. Along the way we also express rank r DT invariants in terms of rank 0 invariants counting sheaves supported on surfaces in X. These invariants are predicted by S-duality to be governed by (vector-valued, mock) modular forms.

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October 18th, 2021

Ben Davison : The decomposition theorem for 2-Calabi-Yau categories

Abstract: Examples of 2CY categories include the category of coherent sheaves on a K3 surface, the category of Higgs bundles, and the category of modules over preprojective algebras or fundamental group algebras of compact Riemann surfaces. Let $p:M\rightarrow N$ be the morphism from the stack of semistable objects in a 2CY category to the coarse moduli space. I'll explain, using cohomological DT theory, formality in 2CY categories, and structure theorems for good moduli stacks, how to prove a version of the BBDG decomposition theorem for the exceptional direct image of the constant sheaf along $p$, even though none of the usual conditions for the decomposition theorem apply: $p$ isn't projective or representable, $M$ isn't smooth, the constant mixed Hodge module complex $\mathbb{Q}_M$ isn't pure... As applications, I'll explain a proof of Halpern-Leistner's conjecture on the purity of stacks of coherent sheaves on K3 surfaces, and if time permits, a (partly conjectural) way to extend nonabelian Hodge theory to Betti/Dolbeault stacks.
Video. Slides.

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October 25th, 2021

Burt Totaro : Varieties of general type with doubly exponential asymptotics

Abstract: We construct smooth projective varieties of general type with the smallest known volume and others with the most known vanishing plurigenera in high dimensions. The optimal volume bound is expected to decay doubly exponentially with dimension, and our examples achieve this decay rate. We also consider the analogous questions for other types of varieties. For example, in every dimension we conjecture the terminal Fano variety of minimal volume, and the canonical Calabi-Yau variety of minimal volume. In each case, our examples exhibit doubly exponential behavior. (Joint work with Louis Esser and Chengxi Wang.)

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November 1st, 2021

Dori Bejleri : Wall crossing for moduli of stable log varieties

Abstract: Stable log varieties or stable pairs $(X,D)$ are the higher dimensional generalization of pointed stable curves. They form proper moduli spaces which compactify the moduli space of normal crossings, or more generally klt, pairs. These stable pairs compactifications depend on a choice of parameters, namely the coefficients of the boundary divisor D. In this talk, after introducing the theory of stable log varieties, I will explain the wall-crossing behavior that governs how these compactifications change as one varies the coefficients. I will also discuss some examples and applications. This is joint work with Kenny Ascher, Giovanni Inchiostro, and Zsolt Patakfalvi.

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December 13th, 2021

Samir Canning : The Chow rings of moduli spaces of elliptic surfaces

Abstract: For each nonnegative integer N, Miranda constructed a coarse moduli space of elliptic surfaces with section over the projective line with fundamental invariant N. I will explain how to compute the Chow rings of these moduli spaces when N is at least 2. The Chow rings exhibit many properties analogous to those expected for the tautological ring of the moduli space of curves: they satisfy analogues of Faber's conjectures, and they exhibit a stability property as N goes to infinity. When N=2, these elliptic surfaces are K3 surfaces polarized by a hyperbolic lattice. I will explain how the computation of the Chow ring confirms a special case of a conjecture of Oprea and Pandharipande on the structure of the tautological rings of moduli spaces of lattice polarized K3 surfaces. This is joint work with Bochao Kong.

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