Talks and schedule

Arend Bayer: Kuznetsov components of Fano threefolds.

I will give an overview of recent results on Kuznetsov components of Fano threefolds. I will emphasize how they help organise the classification and moduli spaces of prime Fano threefolds. The results are based on Bridgeland stability conditions, Brill-Noether loci, and enhanced equivariant categories.

Olivier Benoist: Cohomology of real algebraic varieties and sums of squares.

I will describe relations between Chern classes and constant cohomology classes in the cohomology of real algebraic varieties. I will give applications to sums of squares problems. This is joint work with Olivier Wittenberg.

Jean-Louis Colliot-Thélène: Sur l'arithmétique des intersections de deux quadriques.

On considère les intersections complètes lisses de deux quadriques dans l'espace projectif de dimension n sur un corps quelconque. Pour n au moins 5, on conjecture que le principe de Hasse vaut pour les points rationnels de telles variétés sur un corps de nombres. C'est connu pour n au moins 8 depuis 1987, et pour n = 7 depuis 2018. Je passerai en revue ce qu'on sait sur le sujet, et montrerai comment des résultats récents sur les points de degré 2 dans le cas n=4 sur un corps p-adique mènent à une démonstration relativement courte du principe de Hasse pour n=7. Il s'agit de discuter l'existence de coniques sur ces variétés, lorsque le corps de base est soit un corps p-adique, soit un corps de nombres.

Gabi Farkas: Resonance, Koszul modules and Chen invariants.

Inspired from ideas in topology, Koszul modules turned out to have important algebro-geometric applications for instance to (i) Green's Conjecture on syzygies of canonical curves, (ii) stabilization of cohomology of projective varieties in arbitrary characteristics and (iii) a resolution of an effective form of an important conjecture of Suciu's on Chen invariants of hyperplane arrangements. I will discuss new developments related to this circle of ideas obtained in joint work with Aprodu, Raicu and Suciu.

Andreas Höring: Canonical extensions and positivity of the tangent bundle.

Given a complex projective manifold X and a Kähler class on X, we can associate an extension V of T_X by the structural sheaf. This canonical extension gives an embedding P(T_X) \subset P(V) which allows us to investigate the positivity of the tangent bundle in a more geometric way. We will see that the study of the complement P(V) \setminus P(T_X) naturally leads us to projective manifolds with pseudoeffective tangent bundle. So far these manifolds are poorly understood, but I will explain the tools that are available for their classification. This talk is based on joint work with Thomas Peternell.

Daniel Huybrechts: Nodal quintic surfaces and the Fano varieties of lines

I will survey old and new results concerning this classical class of surfaces and discuss further open questions.

Bruno Klingler: Hodge loci: where we stand.

Given a smooth family of projective varieties on a quasi projective base S, its Hodge locus is the set of closed points of S where the fiber admits "exceptional" Hodge classes. In 1995 Cattani, Deligne and Kaplan showed that it is a countable union of algebraic subvarieties of S, as predicted by the Hodge conjecture. In this talk I will discuss the recent progress in understanding the geometry and arithmetic of the Hodge locus: in "most cases" it is actually algebraic (rather than a countable union of algebraic subvarieties of S), defined over a number field if the family is. Based on works of Baldi-Klingler-Ullmo, Klingler-Otwinowska-Urbanik, Kreutz

Alexander Kuznetsov: Higher-dimensional del Pezzo varieties.

A del Pezzo variety of dimension n \ge 3 is a terminal variety X such that K_X = (1 - n)A for an ample Cartier divisor class A. The theory of del Pezzo varieties is unexpectedly rich and beautiful, and I will try to overview some of its aspects. This is a joint work with Yuri Prokhorov.

Rob Lazarsfeld: Measures of association between algebraic varieties.

I will talk about joint work with Olivier Martin that attempts to measure "how far from birationally isomorphic" two given varieties X and Y of the same dimension may be. The idea is to study the minimal complexity of correspondences between them. Besides presenting a few results, I will discuss some open problems and directions for further work.

Davesh Maulik: Cohomology of Higgs moduli via positive characteristic.

In this talk, I will discuss structures on the cohomology of the moduli space of Higgs bundles on a curve, in particular the P=W conjecture relating them to the character variety of the curve . I will then explain how certain symmetries of this cohomology, which are predictions of this conjecture, can be constructed using techniques from non-abelian Hodge theory in positive characteristic. Based on joint work with Mark de Cataldo, Junliang Shen, and Siqing Zhang.

Kieran O'Grady: Hyperkähler fourfolds of Kummer type: modular sheaves and projective models.

We describe a construction of stable rank 4 vector bundles on a general fourfold of Kummer type with polarization of divisibility 2 and square congruent to -6

modulo 16. These vector bundles are fixed by any automorphism acting trivially on 2 cohomology. We discuss the relevance of this property in the quest for an explicit locally complete family of polarized fourfolds of Kummer type.

John Ottem: A Tale of Two Coniveau Filtrations.

A cohomology class of a smooth complex variety of dimension n is said to be of "coniveau" at least c if it vanishes on the complement of a closed subvariety of codimension at least c, and of "strong coniveau" at least c if it comes by proper pushforward from the cohomology of a smooth variety of dimension at most n-c. These notions give rise to two filtrations on the cohomology groups of a variety, which are known to coincide in many cases (for instance, they agree on the rational cohomology of any smooth projective variety). However, we show that they differ in general, both for integral classes on smooth projective varieties and for rational classes on smooth open varieties. The difference between the two filtrations also give rise to new birational invariants. The talk will give an introduction to the circle of ideas surrounding these two filtrations, with lots of explicit examples and constructions. This is joint work with Olivier Benoist.

Colleen Robles: Global asymptotic structure of period mappings.

The problem of generalizing the Satake-Baily-Borel compactification and Borel’s extension theorem to arbitrary period mappings raises questions about the global behavior of period mappings at infinity. I will discuss two of those questions, and what the answers tells us about the motivating problem.

Giulia Saccà: Moduli spaces as Irreducible Symplectic Varieties.

Recent developments by Druel, Greb-Guenancia-Kebekus, Horing-Peternell have led to the formulation of a decomposition theorem for singular (klt) projective varieties with numerical trivial canonical class. Irreducible symplectic varieties are one the building blocks provided by this theorem, and the singular analogue of irreducible hyper-Kahler manifolds. In this talk I will show that moduli spaces of Bridgeland stable objects on the Kuznetsov component of a cubic fourfold with respect to a generic stability condition are always projective irreducible symplectic varieties. This builds on the recent work of Bayer-Lahoz-Macri-Neuer-Perry-Stellari, which, ending a long series of results by several authors, proved the analogue statement in the smooth case.

Christian Schnell: Finiteness for self-dual classes in variations of Hodge structure.

I will talk about a new finiteness theorem for variations of Hodge structure. It is a generalization of the Cattani-Deligne-Kaplan theorem from Hodge classes to so-called self-dual (and anti-self-dual) classes. For example, among integral cohomology classes of degree 4, those of type (4,0) + (2,2) + (0,4) are self-dual, and those of type (3,1) + (1,3) are anti-self-dual. The result is suggested by considerations in theoretical physics, and the proof uses o-minimality and the definability of period mappings. This is joint work with Benjamin Bakker, Thomas Grimm, and Jacob Tsimerman.

Stefan Schreieder: Speculations on integral cycle conjectures.

Atiyah and Hirzebruch showed 1962 that the integral Hodge conjecture fails by constructing torsion classes in cohomology that are non-algebraic. Since then, most conjectures on algebraic cycles have been formulated rationally. In this talk we speculate to which extent some of these conjectures may hold integrally and in fact on torsion cycles.

Burt Totaro: Varieties of general type with small volume

In trying to classify algebraic varieties, the fundamental invariant is the "volume", generalizing the genus of a curve. In higher dimensions, the volume need not be an integer, and so it is a fascinating challenge to construct varieties with the smallest possible positive volume. These can be considered the varieties that are hardest to find. (Joint work with Louis Esser and Chengxi Wang.)

Qizheng Yin: Perverse-Hodge symmetry for Lagrangian fibrations.

For a Lagrangian fibration from a projective irreducible symplectic variety, the perverse numbers of the fibration are equal to the Hodge numbers of the source variety. I will try to explain this fact from two perspectives: its relation with the LLV algebra (under the hyper-Kähler SYZ conjecture), and possible enhancements to the level of sheaves/categories. The second part uses no hyper-Kähler geometry and makes sense for non-proper source varieties. Joint work with Junliang Shen.

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