Seminars

Title: Mumford relations and the cohomology of Le Potier moduli spaces

Abstract: The geometry and topology of Le Potier moduli space, i.e. the moduli of 1-dimensional sheaves on the projective plane, have been studied intensively for decades. One particular interest in this moduli space arises from enumerative geometry — the perverse filtration on cohomology associated with a natural support map encodes the so-called refined BPS invariants for local P2. Thus it is important to study this cohomology. In this talk, we construct geometrically a family of tautological relations for this moduli space, originally due to Mumford on the moduli of bundles on curves, and derive several structural results on the cohomology. Based on joint works with Yakov Kononov, Woonam Lim, Miguel Moreira, and Junliang Shen.

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Title: Surfaces of general type and sl_2 triples

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Title: Metric techniques for triangulated categories

Abstract: Neeman has introduced the notion of metrics and approximable triangulated categories. These ideas have been used by Neeman and others to prove a remarkable array of results. In this talk we will discuss some of these results, and their implications in the setting of algebraic geometry. In particular, we discuss a result on passing between (left/right) admissible subcategories of the various essentially small subcategories associated to a variety.

Title: BPS-invariants for K3 surfaces and p-adic integration

Abstract: I will report on work in progress with M. Groechenig and P. Ziegler where we aim to express certain enumerative invariants, so called refined BPS-invariants, in terms of integrals on p-adic analytic manifolds naturally associated with the enumerative problem. In the case of 1-dimensional sheaves on a K3 surface, our work implies that these invariants are independent of the Euler-characteristic, confirming a conjecture of Toda.

Title: A panorama of recent results on the cone conjecture

Abstract: Following in the footsteps of the celebrated Mori cone theorem, the cone conjecture predicts the shape of the nef, movable, and effective cones of a mildly singular Calabi—Yau variety (or pair). If the conjecture holds, these cones should also capture birational and dynamical properties of the underlying variety. Since its formulation by Morrison and Kawamata in the 1990ies, the cone conjecture has been the subject of much work, notably over the past five years. In this talk, I will give an overview of the most recent developments regarding the behaviour of the cone conjecture under birational transformations, finite (and certain Galois) group actions, and deformation. I will highlight the divide between the stand-alone cases that have been settled and the reduction results that have allowed to cover more and more new cases.

Title: p-torsion Brauer classes in positive characteristic

Abstract: In this talk, we will study p-torsion Brauer classes that can be constructed from differential forms in positive characteristic p. We will then explain how these classes contribute to the Brauer–Manin obstruction. As an application, we determine the Brauer-Manin set of varieties with ‘many differential forms’ and give new qualitative results on the Brauer-Manin set of supersingular K3 surfaces.

Title: Curve contractibility via non-commutative deformations

Abstract: Deciding whether a subvariety of an algebraic variety is contractible is a deep problem of algebraic geometry. Even when the subvariety is a single smooth rational curve C, the question is extremely subtle. In this talk, I will assume moreover that the ambient variety is a Calabi-Yau threefold. When C is contractible, its Donovan-Wemyss contraction algebra (which pro-represents the deformation theory of C) governs much of the geometry. Our expectation is that deformation theory not only controls contractibility but detects it, even when C is not known to contract. To investigate the deformation theory of C, we use technology developed by Brown and Wemyss to describe a local model for C. I will introduce the key ideas and tools appearing in this problem, the leading conjectures, and I will describe the (partial) results I obtained so far in collaboration with G. Brown and M. Wemyss.

Title: On the Euler characteristic of ordinary irregular varieties

Abstract: Over the complex numbers, a useful tool to study the geometry of irregular varieties is generic vanishing. This has led to several remarkable results: characterization of abelian varieties by only fixing a few invariants, deeper understanding of the Euler characteristic of irregular varieties, study of their pluricanonical systems, and so on. These theorems rely on vanishing results of analytic nature, making this whole topic harder to reach in positive characteristic. Our goal in this talk will be to explain how generic vanishing works in positive characteristic, through the prism of the study of Euler characteristics. We will present the following theorem: if X is a smooth projective ordinary variety of maximal Albanese dimension (i.e. dim(alb(X)) = dim(X)), then the Euler characteristic of the sheaf of top forms is non-negative. If in addition this quantity is zero, then the Albanese image of X is fibered by abelian varieties. The proof uses the positive characteristic generic vanishing theory developed by Hacon-Patakfalvi, as well as our recent Witt vector version of Grauert-Riemenschneider vanishing.

Title: Stability of line bundles and vector bundles on some surfaces

Abstract: Donaldson and Uhlenbeck-Yau established the classical result that on a compact Kähler manifold, an irreducible holomorphic vector bundle admits a Hermitian metric solving the Hermitian-Yang-Mills equation if and only if the vector bundle is Mumford-Takemoto stable. A modern analog of this question was posted by Collins-Yau. In this talk, we will discuss partial answers to this modern analog for a set of line bundles and tangent/cotangent bundles on some surfaces. This is based on joint work/work in progress with Tristan Collins, Jason Lo, and Shing-Tung Yau.

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Title: Hilbert schemes of points, quantum cohomology and tropical curves

Abstract: For a smooth surface S, the Hilbert scheme of points on S gives a smooth compactification of the configuration space of n distinct points on S. Its cohomology is by now well understood and exhibits deep connections with representation theory. Understanding its quantum cohomology, a deformation of ordinary cohomology involving curve counting invariants, has since received a lot of attention. I'll explain some joint work in progress with Georg Oberdieck and Aaron Pixton about how to determine this ring for an elliptic surface, with the help of tropical geometry.

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