# Trimester Seminar Series

**Venue:** HIM lecture hall, Poppelsdorfer Allee 45, Bonn

Organizers: Raymond Cheng, Sarah Frei, Mirko Mauri, Laura Pertusi

Degeneration Seminar

Friday December 8, 10:30-11:30, HIM lecture hall

Speaker: Mauro Varesco

Title: Degenerations of Type III

SAG

Thursday December 7, 10:30-11:30, HIM lecture hall

Speaker: Roberto Fringuelli

Title: The automorphism group of the moduli space of G-bundles over a curve

Abstract: For any almost-simple group G over an algebraically closed field k characteristic zero, we describe the automorphism group of the moduli space of semistable G-bundles over a curve of genus at least 4. The result is achieved by studying the singular fibers of the Hitchin fibration. Time permitting, we also provide a proof of a Torelli-type theorem for these moduli spaces.

STReTCH

Wednesday December 6, 13:30-14:30, HIM lecture hall

Speaker: Sarah Frei

MAGHI

Tuesday December 5, 15:00-16:00, HIM lecture hall

Speaker: Mark de Cataldo

Title: Geometry of moduli of t-connections

Abstract: I will review some of my recent work on moduli spaces of Higgs bundles, of connections and of t-connections (which subsums both). These moduli spaces are objects of interest in the Non Abelian Hodge Theory of projective manifolds --warning: I will only discuss the case of curves-- over the complex numbers and they have been intensively studied in the last thirty plus years. The situation over fields of positive characteristic is less explored and has also become the focus of what people call NAHT in characteristic p. While the moduli spaces of Higgs bundles and of connections are homeomorphic over the complex numbers, the situation over fields of positive characteristic is less clear. I will focus on explaining how a suitable compactification of these moduli spaces allows to bypass the lack of a homeomorphism to yield a canonical isomorphism of cohomology rings. Along the way, I will discuss some of the new phenomena, absent over the complex numbers, that emerge in positive characteristic. I will be short on technical details and my plan is to make the talk accessible to non-experts. For example, in illustrating the compactification technique, I will use as a guide the Ehresmann Lemma from differential topology. The talk contains some joint work with Siqing Zhang and with Davesh Maulik, Junliang Shen and Siqing Zhang.

Degeneration Seminar

Friday December 1, 10:30-11:30, HIM lecture hall

Speaker: Evgeny Shinder

Title: Degenerations of HK varieties

STReTCH

Wednesday November 29, 13:30-14:30, HIM lecture hall

Speaker: Pablo Magni

ABSTRACT: We will recall Hassett's association between K3 surfaces and cubic fourfolds, both in the sense of Hodge theory and derived categories. Afterwards we discuss descent of this association to number fields with a look towards arithmetic applications.

MINI-COURSE

Tuesday November 28, Wednesday November 29, & Thursday November 30, 10:30-11:30 , HIM lecture hall

Speaker: Alessio Bottini

Title: Sheaves on HKs and deformation theory

Abstract: In this course, I will describe the state of the art of the theory of sheaves on high dimensional hyper-Kähler manifolds. It started with the work of Kobayashi and Verbitsky, and it has recently received a lot of attention thanks to the work of O’Grady, Beckmann and Markman. The ultimate goal is to construct hyper-Kähler manifolds as moduli spaces of such sheaves.

I will start by reviewing the main properties of sheaves on K3 surfaces, especially I will try to highlight the main properties that we use to prove that moduli spaces of stable sheaves are hyper-Kähler. In higher dimension, these properties are no longer true and one needs to isolate a good class of sheaves. There are various notions available, and I will discuss in particular hyperholomorphic bundles (introduced by Verbitsky) and atomic sheaves (studied by Beckmann and Markman). The first ones have the advantage that their moduli spaces are symplectic, and the latter are preserved by derived equivalences. In passing, I will also recall the necessary background on the cohomology of hyper-Kähler manifolds, especially focusing on Taelman's work. Lastly, I will describe an example of a moduli space parametrizing hyperholomorphic bundles on a HK manifold of dimension four, which is itself a HK manifold of dimension ten.

MINI-COURSE

Monday November 27, Tuesday November 28, & Thursday November 30, 15:00-16:00 , HIM lecture hall

Speaker: Emma Brakkee

Title: K3 surfaces and modular curves

Abstract: Even indefinite lattices of rank 3 give rise to modular curves, or more generally Shimura curves. These show up naturally when studying K3 surfaces of low or high Picard rank: For example, moduli spaces of K3 surfaces with Picard rank 19 are unions of Shimura curves. In this lecture series, we explain how modular curves can be used to approach questions about K3 surfaces of both arithmetic and of derived nature.

The structure will be as follows: We start by giving a reminder on modular curves, and on lattices and period domains. We will also recall the construction of moduli spaces of lattice polarized K3 surfaces. For rank 19 lattices, these moduli spaces are Shimura curves, and we will explain how to study them via the even Clifford algebra of the lattice. We will then focus on applications of this theory: first on Brauer classes on 1-parameter families of K3 surfaces over number fields, and second on derived auto-equivalences of complex (twisted) K3 surfaces.

SAG

Thursday November 27, 10:30-11:30, HIM lecture hall

Speaker: Olivier Martin

Title: Isotrivial Lagrangian fibrations of compact hyper-Kähler manifolds

Abstract: I will present a recent exploration of the geometry of isotrivial Lagrangian fibrations conducted with Y. Kim and R. Laza. We show that the smooth fiber of such a fibration is isogenous to the power of an elliptic curve. Moreover, we introduce a dichotomy arising from the Kodaira dimension of the minimal Galois cover of the base trivializing monodromy. We also classify up to birational equivalence isotrivial fibrations with a section in the case where this cover has Kodaira dimension 0.

MINI-COURSE

Friday November 17, 14:15-16:00 (with 15 minutes break), HIM lecture hall

Speaker: Nebojsa Pavic

Title: Derived categories and singularities (second lecture)

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.

Degeneration Seminar

Friday November 17, 10:30-11:30, HIM lecture hall

Speaker: Andres Fernandez Herrero

Title: P=W for compact hyperkahler manifolds

SAG

Thursday November 16, 10:30-11:30, HIM lecture hall

Speaker: Andres Fernandez Herrero

Title: Towards curve counting on the classifying stack BGL_n

STReTCH

Wednesday November 15, 13:30-14:30, HIM lecture hall

Speaker: Richard Haburcak

MAGHI

Tuesday 14 November, 15:00-16:00, HIM lecture hall

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.

MINI-COURSE

Monday 13 November and Wednesday 15 November, 10:30-11:30, 15:00-16:00, HIM lecture hall

Speaker: Tudor Padurariu and Yukinobu Toda

Title: Quasi-BPS categories

MINI-COURSE

Friday November 3 and November 17, 14:15-16:00 (with 15 minutes break), HIM lecture hall

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.

Degeneration Seminar

Friday October 27, 10:30-11:30, HIM lecture hall

Speaker: Anna Abasheva

Title: Dual complexes of degenerations of hyperkahler manifolds

SAG

Thursday November 2, 10:30-11:30, HIM lecture hall

Speaker: Franco Rota

Title: Non-commutative deformations and contractibility of rational curves

Abstract: When can we contract a rational curve C? The situation is much more complicated for threefolds than it is for surfaces: Jimenez gives examples of (-3,1)-rational curves neither contract nor move. Their behaviour is controlled by the functor of non-commutative deformations of C, which conjecturally controls exactly their contractibility.

I will report on work in progress with M. Wemyss, and reinterpret some of Jimenez's examples in terms of non-commutative deformations.

STReTCH

Wednesday November 2, 13:30-14:30, HIM lecture hall

Speaker: Xuqiang Qin

Title: LLSvS Eightfolds

**MAGHI****Tuesday October 31, 15:00-16:00, HIM lecture hall ****Speaker:** Jack Petok **Title:** Zeta function of the K3 category of a cubic

Abstract: We study the arithmetic of the K3 category associated to a cubic fourfold over a non-algebraically closed field k. We start by constructing the Mukai structure of this K3 category with a natural action of Galois. For k a finite field, this lets us define the zeta function of a K3 category, an invariant under FM-equivalence of K3 categories. We provide a characterization of those cubic fourfolds whose K3 category has zeta function arising from a K3 surface defined over k. One interesting outcome is that the zeta function does not always detect the geometricity of the K3 category. This is joint work with Asher Auel.

MAGHI

Friday October 27, 15:00-16:00, HIM lecture hall

Speaker: Shengxuan Liu

Title: A note on spherical bundles on K3 surfaces

Abstract: Let S be a K3 surface with the bounded derived category D^b(S). Let E be a spherical object in D^b(S). Then there always exists a non-zero object F satisfying RHom(E,F)=0. Further, there exists a spherical bundle E on some K3 surfaces that is unstable with respect to all polarization on S. Also we “count” spherical bundles with a fixed Mukai vector. These provide (partial) answers to some questions of Huybrechts. This is a joint work with Chunyi Li.

Degeneration seminar

Friday October 27, 10:30-11:30, HIM lecture hall

Speaker: Dominique Mattei

Title: Degenerations of type I

STReTCH

Wednesday October 25, 13:30-14:30, HIM lecture hall

Speaker: Moritz Hartlieb

Title: LSV: Pfaffians and OG10 type

MAGHI

Monday October 23, 15:00-16:00, HIM lecture hall

Speaker: Fumiaki Suzuki

Title: Maximal linear spaces for pencils of quadrics and rationality

Abstract: Over an arbitrary field k of odd characteristic, let X be a smooth complete intersection of two quadrics in P^{2g+1}. For every g at least 2, we show that the existence of a (g-1)-plane, defined over k, on X may be characterized by k-rationality of a certain 3-dimensional subvariety of the Fano scheme of (g-2)-planes on X, generalizing the g = 2 case due to Hassett-Tschinkel and Benoist-Wittenberg. We also present a related result on k-rationality of the Fano schemes of non-maximal linear spaces on X. This is joint work in progress with Lena Ji.

MAGHI

Friday October 20, 15:00-16:00, HIM lecture hall

Speaker: Fei Xie

Title: Quadric bundles over smooth surfaces

Abstract: For a flat quadric bundle of relative even dimension with fibers of corank at most 1, there is a well established relation between its derived category and its relative Hilbert scheme of maximal isotropic subspaces (or its relative moduli of spinor bundles). For a smooth 2m-fold with the structure of a quadric bundle over a smooth surface, there is a finite number of fibers with corank 2 and this relation fails. I will discuss how to fix the relation in this case.

Degeneration seminar

Friday October 20, 10:30-11:30, HIM lecture hall

Speaker: Mirko Mauri

Title: Mixed Hodge structure of log symplectic pairs

STReTCH

Wednesday October 18, 13:30-14:30, HIM lecture hall

Speaker: Jack Petok

Title: LSV: Hyperkahler structure

MAGHI

Monday October 16, 15:00-16:00, HIM lecture hall

Speaker: Tudor Ciurca

Title: Irrationality of cubic threefolds in characteristic 2

Abstract: In 1972 Clemens and Griffiths gave a formidable proof that a smooth cubic threefold over C is irrational. The proof was soon after adapted to any algebraically closed field of characteristic not 2 using algebraic methods. I will finish the story by extending the proof to the case of characteristic 2. As arithmetic applications, we answer a question of Deligne regarding arithmetic Torelli maps and establish the Shafarevich conjecture for cubic threefolds over function fields of characteristic 2.

**Degeneration seminar**

**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

Degeneration seminar

October 14, 2023 (CEST)

10:30 - 11:30 am Philip Engel

Title: Limiting mixed Hodge structure

Mini-course

October 9, 11, 13; 2023 (CEST)

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.

SAG

October 12, 2023 (CEST)

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.

STReTCH

October 11, 2023 (CEST)

1:30 - 2:30 pm Lisa Marquand

Title: An overview of the LSV construction

MAGHI

October 10, 2023 (CEST)

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.

**Mini-course**

**September 25, 27, 29; 2023 (CEST)**

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 F_{2} 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 F_{2} 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}.

**SAG**

**September 28, 2023 (CEST)**

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.

**Organizational Meeting for Reading Group**

**September 27, 2023 (CEST)**

2:00 – 3:00 pm

**MAGHI**

**September 26, 2023 (CEST)**

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.

**Exceptional Evening Talk **

**September 21, 2023 (CEST)**

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.

**SAG**

**September 21, 2023 (CEST)**

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.