Titles and Abstracts

• Pieter Belmans (Utrecht University) - Moduli spaces of semiorthogonal decompositions

Abstract: If you have a semiorthogonal decomposition of a smooth projective variety, and you have a deformation of this variety, what can be said about the derived category of the deformation? More generally, how do semiorthogonal decompositions behave in smooth projective families? These questions lead to the construction of a moduli space of semiorthogonal decompositions, which has an intriguing geometry. I will describe the construction and some of its properties, and explain the open problems related to it. This is joint work with Shinnosuke Okawa and Andrea Ricolfi.

• Ana Botero (Bielefeld) - On an arithmetic BKK theorem for toric vector bundles

Abstract: The classical Newton polyhedra theory gives formulas for discrete geometric and topological invariants of complete intersections in the algebraic torus defined by generic Laurent polynomial equations. For example, the Bernštein-Kušnirenko-Khovanskii theorem (BKK theorem) states that the number of isolated common zeros (counted with multiplicities) of a family of Laurent polynomials is bounded above by the mixed volume of its Newton polytopes. In this talk, we will see both geometric and arithmetic versions of this result for vector-valued Laurent polynomials using the theory of toric vector bundles and their connections to affine Bruhat-Tits buildings. This is joint work with José Burgos, Kiumars Kaveh and Vivek Mallik.

• Yajnasena Dutta (Leiden University) - Sheaves associated to Lagrangian fibrations

In this talk I will introduce a few analytic sheaves of abelian groups that keep track of Tate-Shafarevich twists, Mordell-Weil groups as well as dual fibrations for Lagrangian fibration of hyperkähler manifolds. I will calculate the cohomological invariants associated to these sheaves in some example(s). This is based on a joint work with Mattei and Shinder.

• David Holmes (Leiden University) - Modular forms for universal abelian varieties

Modular forms can be seen as sections of line bundles on moduli spaces of elliptic curves. These generalise naturally to Siegel-Jacobi forms, which are sections of line bundles on the universal abelian variety over a moduli space of abelian varieties. Finite generation (or otherwise) of rings of these forms has various applications, such as to understanding projective embeddings of these moduli spaces. In this talk we will spend quite some time setting up the background, after which we will explain how techniques from tropical geometry can help determine the (non)finite generation of various rings of Siegel-Jacobi forms. This is joint work with José Burgos, Ana Maria Botero, and Robin de Jong. 

• Paul Kiefer (University of Antwerp) - A $\Lambda$-Adic Family of Funke-Millson Cycles and a $\Lambda$-Adic Funke-Millson Lift

Generalizing the construction of the Kudla-Millson geometric theta function, Funke-Millson showed that generating series of cohomology classes of Funke-Millson cycles with values in a local system are modular. We show that these Funke-Millson cycles can be put into a $\Lambda$-adic family and show that the corresponding generating series is a $\Lambda$-adic family of modular forms. In particular, we obtain a $\Lambda$-adic Funke-Millson lift. This is work in progress with Lennart Gehrmann.

• Franco Rota (Paris-Saclay) - Towards curve contractibility via non-commutative deformations

Deciding whether a subvariety of an algebraic variety is contractible is a classical problem of algebraic geometry. Even when the subvariety is an integral 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 obtained so far in collaboration with G. Brown and M. Wemyss.

• Navid Nabijou (Queen Mary University of London) - Logarithms, orbifolds, negative tangencies

Abstract: Logarithmic and orbifold structures provide two different paths to the enumeration of algebraic curves with fixed tangencies along a normal crossings divisor. Simple examples demonstrate that the resulting systems of invariants differ, but a more structural explanation of this defect has remained elusive.

I will explain how the two systems of invariants can be identified by passing to an appropriate blowup. This identifies “birational invariance” as the key property distinguishing the two theories. Our proof hinges on a technique – rank reduction – for reducing questions about normal crossings divisors to questions about smooth divisors. Time permitting, I will discuss extensions of this result to the setting of negative tangencies, where the pathological geometry of the moduli space is controlled using tropical geometry.

This is joint work with Luca Battistella and Dhruv Ranganathan

• Cecilia Salgado (University of Groningen) - Hilbert Property: From K3 Surfaces to Campana Points

Let $X$ be a smooth projective variety over a number field $k$, and let $D_m$ be a divisor on $X$ with multiplicities encoded by a weight vector $m$. Campana points on $X$ are rational points that are integral with respect to this weighted boundary divisor. They generalize classical integral points and allow one to interpolate between the theories of rational and integral points in a systematic way. Several conjectures and theorems about rational points on varieties have natural analogues in the Campana setting. For example, it is conjectured that if the pair $(X,D_m)$ is log Fano, then the set of Campana points is Zariski dense—echoing the expected potential density of rational points on Fano varieties. In this talk, we explore a finer arithmetic question, namely of whether the set of Campana points on log Fano pairs is not thin (over some finite extension of $k$), i.e., satisfies the (potential) Hilbert property. We focus on the case of log Fano pairs of the form $(\mathbb{P}^2,D_m)$, and discuss how their (potential) Campana Hilbert property is connected to the classical (potential) Hilbert property for K3 surfaces over number fields. This is based on work in progress with Marta Pieropan and Soumya Sankar.

• Alessandra Sarti (University of Poitiers) - Log Enriques varieties from Hyperkähler and Calabi-Yau manifolds

Enriques surfaces are special free quotients of K3 surfaces by a fixed point free involution. In higher dimension the notion can be generalized and one can introduce Enriques manifolds and in the singular setting, Log Enriques vaieties. 

In this talk I will explain general properties of Enriques manifolds and of Log Enriques varieties. I will then provide and discuss several examples in the singular setting, in particular  I will talk about Log Enriques varieties that arise as quotients of generalized Fermat manifolds. These manifolds were studied recently by Hidalgo, Hughes and Leyton-Alvarez. The results I will present are contained in several joint works with S. Boissière, C. Camere, M. Nieper-Wisskirchen and in a recent work in progress with A. Palomino. 

• Ed Segal (University College London) - The symmetric square of the Kuznetsov component

The derived category of a cubic 4-fold contains a semi-orthogonal component which famously behaves as a `non-commutative K3 surface'. The symmetric square of this component is thus a `non-commutative hyperkaehler 4-fold'. Galkin conjectured that it should be equivalent to the derived category of an actual hyperkaehler 4-fold: the Fano of lines in the cubic. I will explain my recent proof of this conjecture with Kimoi Kemboi.

Funders

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