Abstracts

Gregory Sankaran: A family of generalised Kummers as covers of P^4

Abstract: I will describe work in progress with (at least) Samuel Boissiere and Marc Nieper-Wisskirchen in which we construct generalised Kummer varieties as certain 15 to 1 covers of P^4, up to some birational transformations. I will speculate madly about possible generalisations.

Giovanni Mongardi: Rapagnetta involution and generalized Kummers

Abstract: The natural Kummer involution on an abelian surface gives rise to a dual rapagnetta involution on the derived category of a K3 surface, which can be used to construct examples of O'Grady's manifolds. We will sketch some potential application of this to generalized Kummers. This talk is meant as a survey of these applications.

Calla Tschanz: A degeneration of generalised Kummer varieties

In this work in progress with S. Galkin and G. Kapustka, we aim to construct generalised Kummer fourfolds by constructing a maximally degenerate variety and describing a smoothing family. This is done by induction on the dimension, starting from a maximal degeneration of K3 surfaces which embeds into a maximal degeneration of Calabi-Yau threefolds, which then embeds into the maximal degeneration of generalised Kummer fourfolds.

Annalisa Grossi: On the attempt of classification of hyperähler fourfold

Abstract: Kodaira proves that all K3 surfaces are deformation equivalent, but it is unknown how many deformation families of hyperkähler manifolds do we have in a fixed dimension. Debarre-Huybrechts-Macrì-Voisin in 2022 proved that the existence of two integral degree-2 cohomology classes satisfying certain numerical intersection conditions on an hyperkähler fourfold, is sufficient to imply that the manifold is of K3^2 deformation type. They first prove a topological version of the statement, by showing that their topological assumption forces the Huybrechts–Riemann–Roch polynomial of the hyperkähler fourfold to be the same as those of K3^[2] type, and then the key part of their result is to prove the hyperkähler SYZ conjecture for hyperkähler fourfolds for divisor classes satisfying the numerical condition mentioned above. 

In this talk I will give an overview of the main steps of their proof, and I will present a joint work in progress with P. Beri about the attempt to prove a similar result for generalized Kummer fourfolds. I will show that we can find a set of topological assumptions that forces the Huybrechts–Riemann–Roch polynomial of the hyperkähler fourfold to be the same as those of hyperkähler manifolds of Kum_2 type, and I will explain which kind of obstructions we find trying to imitate the other steps of the proof of DHMV in the Kummer setting.

Pietro Beri: families of K3 surfaces associated to Kummer fourfolds

Abstract: A hyper-Kähler fourfold of generalized Kummer type admits some involutions acting trivially on its second cohomology group, whose fixed locus has a two-dimensional component which is a K3 surface. In this talk, we show that if we restrict to 2-polarized deformations of generalized Kummer fourfold, the K3 surface uniquely determines the fourfold. This is part of a joint work in progress with F. Giovenzana and A. Rios-Ortiz.

Nikon Kurnosov: Absolutely trianalytic subvarieties in the generalized Kummer and BG-manifolds.

I will discuss absolutely trianalytic subvaritieties in hyperkahler manifolds. In particular, I will outline some old results of Soldatenkov and myself on existence of absolutely trianalytic tori in the generalized Kummer and other known classes of hyperkahler manifolds. Furthermore, I will talk about the subvarieties in only one known class of simply-connected non-Kahler holomorphically symplectic manifolds, and I will explain how recursive solution for the C-symplectic Maurer-Cartan equation is used to give a generalization of Voisin's famous theorem about deformations of holomorphic Lagrangian subvarieties. The latter is work in progress with Misha Verbitsky.