Ball quotients and algebraic geometry

Abstract:About 27 years ago Allcock found a ball quotient of dimension 13, which is intriguing for two reasons: its expected connections with some of the sporadic finite simple groups (such as the Monster group) and the likelihood of it having a modular interpretation. I shall first review some known examples of ball quotients with a modular interpretation (all of dimension at most 10) and then describe recent progress on the 'moonshine properties' of the Allcock ball quotient.

Algebraic versus homological equivalence of algebraic cycles

Abstract: A cycle on a smooth projective variety is algebraically trivial if it can be deformed to zero. This implies that its cohomology class is zero; in 1969 Griffiths showed that the converse is false for many hypersurfaces. A different example is constructed from a curve C embedded in its Jacobian JC : the cycle [C] - [(-1)*C] in JC is not algebraically trivial if C is general (Ceresa, 1983), while it is if C is hyperelliptic. In the last three years a number of approaches have been developed to find non-hyperelliptic curves for which this cycle is algebraically trivial. In the talk I will survey the history of the problem, then discuss these recent examples of non-hyperelliptic curves, in particular the approach of Laga and Shnidman (2024).

On the Chow ring of double EPW sextics TBA

Abstract: The Chow rings of hyperKaehler varieties are expected to satisfy special properties. Specifically, Beauville and Voisin have conjectured that the subring (of the Chow ring) generated by divisors and Chern classes should inject into cohomology. Building on the work of many other people, I proved in 2023 that this conjecture is true for double EPW sextics - this is the second locally complete family of hyperKaehler varieties for which the conjecture is known. My talk will present and motivate the Beauville-Voisin conjecture, discuss what is known and what not, and sketch a proof of my result.

Vanishing Brauer Classes

Abstract: A specialization of a K3 surface with Picard rank one to a K3 with rank two defines vanishing classes of order p in the Brauer group of the general K3 surface, for any prime p. An element of order p in the Brauer group defines a sublattice of the transcendental lattice. Given a K3 surface S, we study those elements for which this sublattice is Hodge-isometric to the transcendental lattice of another K3 surface X. Using vanishing Brauer classes, we show how the Picard lattice of X determines the Picard lattice of S in the case that Picard number of X, and of S, is two. We then discuss some applications when p=2 and the K3 of rank two is a double plane defined by a cubic fourfold with a plane. This is a joint work with Bert van Geemen.

Cohomologically trivial automorphisms of elliptic surfaces

Abstract: Cohomologically trivial automorphisms form a classical topic in the area of compact complex surfaces. I will report on joint work with Catanese, Frapporti, Gleißner and Liu which attacks the case of elliptic surfaces of Kodaira dimension one. Along the way, we will in particular revisit the work of Chris.

Enriques varieties and applications to the cone conjecture

Abstract: Enriques surfaces are special quotients of K3 surfaces. In higher dimension the notion can be generalized and one can introduce Enriques manifolds and construct exemples as quotients of irreducible holomorphic symplectic manifolds and Calabi--Yau manifolds. The definition of Enriques manifold I will give is slightly different from the original definition given independetly by Oguiso, Schroer and by Boissière, Nieper-Wisskirchen and myself in 2011. A reason is that one wants to extend the definition to singular Enriques varieties, trying to generlize the notion of Log Enriques surfaces. In the talk I will introduce the notion of Enriques manifold and its properties. As an application I will discuss the famous Morrison--Kawamanta cone conjecture for Enriques manifolds and time permitting I will say some words about the singular setting.

Codimension 4 Gorenstein constructions

Abstract: I lectured in Leiden in 1977 on Godeaux surfaces with 3-torsion, based on my 1978 Tokyo paper, my entry point into the research topic of codimension 4 Gorenstein rings. Our Tom and Jerry unprojection methods now give several hundred constructions of new families of algebraic surfaces, 3-folds and higher dimensional varieties. A current challenge is to apply these methods to 3-torsion Godeaux surfaces in mixed characteristic at 3.