10:30 - 11:30 am Mauro Varesco

Title: Algebraicity of Hodge similitudes and the Hodge conjecture for Kum^2-type varieties

Abstract: In this talk, we will introduce the notion of Hodge similitudes between polarized Hodge structures of K3-type. After recalling the construction of Kuga-Satake varieties associated to polarized Hodge structures of K3-type, we will prove that it is functorial with respect to Hodge similitudes. This will be used to deduce the algebraicity of Hodge similitudes of transcendental lattices of hyperkähler manifolds of generalized Kummer type. As a corollary, we will show how this implies the Hodge conjecture for Kum^2-type varieties. This last application is product of a joint work with Floccari Salvatore.

21:00 – 22:00 pm Mirko Mauri

Title: On the geometric P=W conjecture

Abstract: The geometric P = W conjecture is a conjectural description of the asymptotic behavior of a celebrated correspondence in non-abelian Hodge theory. In a joint work with Enrica Mazzon and Matthew Stevenson, we establish the full geometric conjecture for compact Riemann surfaces of genus one, and obtain partial results in arbitrary genus: this is the first non-trivial evidence of the conjecture for compact Riemann surfaces. To this end, we employ non-Archimedean, birational and degeneration techniques to study the topology of the dual boundary complex of certain character varieties.

3:00 – 4:00 pm Reinder Meinsma

Title: Derived equivalence for elliptic K3 surfaces and Jacobians

Abstract: We present a detailed study of Fourier-Mukai partners of elliptic K3 surfaces. One way to produce Fourier-Mukai partners of elliptic K3 surfaces is by taking Jacobians. We answer the question of whether every Fourier-Mukai partner is obtained in this way. This question was raised by Hassett and Tschinkel in 2015. We fully classify elliptic fibrations on Fourier-Mukai partners in terms of Hodge-theoretic data, similar to the Derived Torelli Theorem that describes Fourier-Mukai partners. This classification has an explicit computable form in Picard rank two, building on the work of Stellari and Van Geemen. We prove that for a large class of Picard rank 2 elliptic K3 surfaces all Fourier-Mukai partners are Jacobians. However, we also show that there exist many elliptic K3 surfaces with Fourier-Mukai partners which are not Jacobians of the original K3 surface. This is joint work with Evgeny Shinder.

10:30 – 11:30 am Noah Olander

Title: Fully faithful functors and dimension

Abstract: Can one embed the derived category of a higher dimensional variety into the derived category of a lower dimensional variety? The expected answer was no. We give a simple proof and prove new cases of a conjecture of Orlov along the way.

3:00 – 4:00 pm Philip Engel

Title: Compact moduli of K3 surfaces

Abstract: For each d > 0, there is a 19-dimensional moduli space F_2d of K3 surfaces, with an ample line bundle of degree 2d. Choosing an ample divisor in a canonical way on each such K3 surface, the minimal model program provides a "KSBA" compactification of F_2d. On the other hand, the Hodge theory of K3 surfaces implies that F_2d=Γ\D is a Type IV arithmetic quotient/orthogonal Shimura variety. In this capacity, it has a variety of compactifications: Baily-Borel, toroidal, semitoroidal. Can these two types of compactifications ever be identified?
 
The first lecture will introduce K3 surfaces and their one-parameter degenerations, in particular, semistable (aka Kulikov) models. In analogy with how stable graphs encode degenerations of curves, we will describe a way to combinatorially encode the data of a degeneration, using integral-affine structures on the sphere.
 
The second lecture will focus on the geometry of moduli spaces F_2d and their compactifications, from both the Hodge-theoretic and MMP perspectives. We will discuss degeneration of the period map, and give some explicit examples of (semi)toroidal compactifications, for F₂ and for moduli of elliptic K3 surfaces.
 
The third lecture will introduce the notion of a "recognizable divisor". These are divisors chosen on the generic polarized K3 surface whose KSBA compactifications are Hodge-theoretic. We will give examples for F₂ and for moduli of elliptic K3 surfaces. Then we will discuss the general theory of recognizable divisors, and how it can be applied to compactify F_2d.

Lecture notes

3:00 - 4:00 pm Raymond Cheng

Title:  q-bic hypersurfaces

Abstract: Let’s count: 1, 2, q+1. The eponymous objects are special projective hypersurfaces of degree q+1, where q is a power of the positive ground field characteristic. This talk will sketch an analogy between the geometry of q-bic hypersurfaces and that of quadric and cubic hypersurfaces. For instance, the moduli spaces of linear spaces in q-bics are smooth and themselves have rich geometry. In the case of q-bic threefolds, I will describe an analogue of result of Clemens and Griffiths, which relates the intermediate Jacobian of the q-bic with the Albanese of its surface of lines.

1:30 - 2:30 pm Lisa Marquand

Title:  An overview of the LSV construction

10:30 - 11:30 am Genki Ouchi

Title:  Cubic fourfolds and K3 surfaces with large automorphism groups

Abstract: Relations between cubic fourfolds and K3 surfaces are described by Hodge theory and derived categories. Using Hodge theory and derived
categories, we can show that cubic fourfolds and associated K3 surfaces
share their symmetries, which are related with Mathieu groups and Conway
groups. In this talk, we find pairs of a cubic fourfold and a K3 surface
sharing large symplectic automorphism groups via Bridgeland stability
conditions on K3 surfaces.

3:00 - 4:00 pm Hyeonjun Park

Title: Shifted Symplectic Structures

Abstract: This mini-course aims to introduce shifted symplectic structures in derived algebraic geometry and their applications to Donaldson-Thomas theory of Calabi-Yau varieties.

The first lecture will cover the background on derived algebraic geometry. Heuristically, derived moduli spaces are infinitesimal thickening of moduli spaces whose cotangent complexes govern the higher-order deformation theory. We will present various derived moduli spaces and their cotangent complexes, including the moduli spaces of sheaves (or complexes), stable maps, G-bundles, and Higgs bundles.

The second lecture will focus on the shifted symplectic geometry. There are natural extensions of symplectic structures, Lagrangians, Lagrangian fibrations, and Lagrangian correspondences in the shifted symplectic setting. We will provide various examples of these structures arising in moduli spaces and local structure theorems for them. I will also explain how to pushforward symplectic fibrations along base changes, which allows us to construct symplectic quotients and symplectic zero loci.

The last lecture will provide applications to Donaldson-Thomas theory. For Calabi-Yau 3-folds, moduli spaces of sheaves are locally critical loci, and their perverse sheaves of vanishing cycles glue globally. This gives us categorical DT3 invariants, which are related to the singularity of moduli spaces. For Calabi-Yau 4-folds, moduli spaces of sheaves carry special cycle classes which are heuristically the fundamental cycles of Lagrangians. This gives us numerical DT4 invariants, which are invariant along the deformations of Calabi-Yau 4-folds for which the (0,4)-Hodge pieces of the second Chern characters remain zero.

October 14, 2023 (CEST)

October 13, 20, 27; 2023 (CEST)

November 3, 17; 2023 (CET)

December 8, 15; 2023 (CET)

10:30 – 12:00

For more informations: Schedule and Abstracts

Speaker: Nebojsa Pavic

Title: Derived categories and singularities
 
Abstract: In this mini-course, we discuss the current state of the art on semiorthogonal decompositions of derived categories of singular varieties. We define notions such as categorical absorptions and Kawamata type semiorthogonal decompositions and we give examples and obstructions to such decompositions.

In the first lecture we give a short introduction to the topic by providing explicit examples of semiorthogonal decompositions of curves and surfaces with mild isolated singularities. Along the way, we introduce the notions categorical absorptions and Kawamata type semiorthogonal decompositions. We then proceed by recalling the singularity category of a variety, state general properties about this category and discuss necessary conditions for a projective variety admitting a Kawamata type decomposition in terms of the singularity category. We then rephrase the necessary assumption on the singularity category in terms Grothendieck groups. As a consequence, we show that the defect of a projective variety with certain singularity types is an obstruction to Kawamata type decompositions.

In the second lecture, we explain the relation between the derived category of a singular variety and its resolution and we state the Bondal-Orlov localization conjecture. Moreover, we explain how the localization conjecture descends to a semiorthogonal decomposition on the singularity. We then talk about sufficient conditions on resolutions of nodal n-dimensional singularities and n-dimensional quotient singularities of type 1/n(1^n), such that a "nice" categorical absorption, respectively Kawamata type decomposition is induced on the singularity. We explain briefly the idea of the proofs and we give examples.

Abstracts

Speaker: Stefano Filipazzi

Title: On the boundedness of elliptic Calabi-Yau threefolds

Abstract:

In this talk, we will discuss the boundedness of Calabi-Yau threefolds admitting an elliptic fibration. First, we will review the notion of boundedness in birational geometry and its weak forms. Then, we will switch focus to Calabi-Yau varieties and discuss how the Kawamata-Morrison cone conjecture comes in the picture when studying boundedness properties for this class of varieties. To conclude, we will see how this circle of ideas applies to the case of elliptic Calabi-Yau threefolds. This talk is based on work joint with C.D. Hacon and R. Svaldi.