Tag - K3 surfaces

Emanuele Macrì: Deformations of t-structures

14th November, 2024

Bridgeland stability conditions were introduced about 20 years ago, with motivations from algebraic geometry, representation theory, and physics. One of the fundamental problems is that we currently lack methods to construct and study such stability conditions in full generality. In this talk, I will present a new technique to construct stability conditions by deformations, based on joint works with Li, Perry, Stellari, and Zhao. As an application, we can construct stability conditions on very general abelian varieties and deformations of Hilbert schemes of points on K3 surfaces, and we prove a conjecture by Kuznetsov and Shinder on quartic double solids.

Daniel Huybrechts: Brauer groups and twisted sheaves on K3 surfaces

11th September, 2023

This will be a gentle introduction into Brauer groups and twisted sheaves. The emphasis will be on geometric aspects and eventually on moduli spaces of twisted sheaves on K3 surfaces. We will study the different ways to think about Brauer groups as groups of Azumaya algebras, Brauer-Severi varieties, twisted sheaves, 𝔾m-gerbes... How to translate from one to the other, how to define Chern classes, how to split Brauer classes, etc.

Alan Thompson: The mirror Clemens-Schmid sequence

20th September, 2021

I will present a four-term exact sequence relating the cohomology of a fibration to the cohomology of an open set obtained by removing the preimage of a general linear section of the base. This exact sequence respects three filtrations, the Hodge, weight, and perverse Leray filtrations, so that it is an exact sequence of mixed Hodge structures on the graded pieces of the perverse Leray filtration. I claim that this sequence should be thought of as a mirror to the Clemens-Schmid sequence describing the structure of a degeneration and formulate a 'mirror P=W' conjecture relating the filtrations on each side. Finally, I will present evidence for this conjecture coming from the K3 surface setting.

Paul Hacking: Mirror symmetry for ℚ-Fano 3-folds

20th September, 2021

This is a report on work in progress with my student Cristian Rodriguez. The mirror of a ℚ-Fano 3-fold with b2 = 1 is a rigid K3 fibration over ℙ1 such that Hodge bundle is degree 1 and some power of the monodromy at infinity is maximally unipotent. Although prior work focused on the maximally unipotent case (without base change), perhaps a classification of such Picard-Fuchs equations is possible.

In the smooth case these fibrations were described explicitly by Przyjalkowski, and Doran-Harder-Novoseltsev-Thompson showed that they are given by etale covers of the (1-dimensional) moduli of rank 19 K3 surfaces. In the case of a single 1/2(1,1,1) singularity they are given by rigid rational curves on the (2-dimensional) moduli of rank 18 K3 surfaces, and examples suggest they are Teichmuller curves in A2 (via the Shioda-Inose correspondence relating rank 18 K3s and abelian surfaces), as studied by McMullen.

Duco van Straten: A strange Calabi-Yau degeneration

20th September, 2021

If a Calabi-Yau threefold varies in a one-parameter family and aquires some double points, a small resolution will produce a rigid space. The local monodromy at such a 'conifold transition' is of infinite order. In the talk I report on some work done with S. Cynk (Krakow), which shows similar transitions to rigid Calabi-Yaus are possible with monodromy of finite order, in sharp distinction to what can happen for K3 surfaces.

Marcello Bernardara: Fano of K3 Type: Isomorphisms and classification of Hodge structures and K3 categories

13th May, 2021

Fano varieties of (derived) K3 type are Fano varieties whose Hodge structure (derived category) contains a K3-type sub-Hodge structure (subcategory). Many examples of such varieties are known, arising as zeroes of homogeneous bundles on Grassmannians, in dimensions that grow up to 19. In this talk, I will first present joint work with Fatighenti and Manivel showing that many of these examples can be related by geometric correspondences and have actually the same K3-type Hodge structure. I will also present an ongoing project with Fatighenti, Manivel and Tanturri, whose aim is to show that in the case of Fano fourfolds, the only possible K3-type structures which are not actual K3 can arise from Gushel-Mukai, cubics and Küchle c5 fourfolds.