Interactive Workshop & Hausdorff School: September 06-10, 2021

Lecture Series:

• Benjamin Bakker (University of Illinois, Chicago)

• Alexander Kuznetsov (Steklov Institute, Moscow)

• Alexander Perry (University of Michigan, Ann Arbor)

• Giulia Saccà (Columbia University, New York)

Invited lectures:

• Federico Barbacovi (University College London)

• Pieter Belmans (University of Luxembourg)

• Laura Pertusi (University of Milano)

Information & registration

Zoom link

(please use your own name and surname when connecting!)

https://uni-bonn.zoom.us/j/94636762985?pwd=YVBacDVyUTM1QVhmL01Eb0JIZE5iUT09

Schedule

(all talks take place at the Lipschitz-Saal - room 1.016, Endenicher Allee 60, Bonn)

Mon, 06/09

9 -- 10:30: Kuznetsov, 1

10:30 -- 11: Coffee Break

11 -- 12: Presentation of groups and problems

12 -- 14: Lunch

14 -- 15: Groups, 1

15 -- 15:30: Coffee Break

15:30 -- 17: Saccà, 1

17 -- 17:15: Small Break 17:15 -- 18:45: Perry, 1

Tue, 07/09

9 -- 10:30: Kuznetsov, 2 10:30 -- 11: Coffee Break 11 -- 12: Groups, 2 12 -- 14: Lunch 14 -- 15: Groups, 3 15 -- 15:30: Coffee Break (with cakes) 15:30 -- 17: Perry, 2 17 -- 17:15: Small Break 17:15 -- 18:45: Bakker, 1 Wed, 08/09 9:30 -- 10:30: Belmans 10:30 -- 11: Coffee Break 11 -- 12: Pertusi 12 -- 13: Barbacovi 13 -- 17: Lunch / Free time / Groups discussion 17 -- 18: Groups mini-reports (5mins each+questions) Thu, 09/09 9 -- 10:30: Groups, 4 10:30 -- 11: Coffee Break 11 -- 12: Groups, 5 12 -- 14: Lunch 14 -- 15: Groups, 6 15 -- 15:30: Coffee Break (with cakes) 15:30 -- 17: Saccà, 2 17 -- 17:15: Small Break 17:15 -- 18:45: Bakker, 2 Fri, 10/09 9 -- 10:30: Groups, 7 10:30 -- 11: Coffee Break 11 -- 12: Groups longer reports, 1 (20/30mins each) 12 -- 14: Lunch 14 -- 15: Groups longer reports, 2 (20/30mins each)

Abstracts

B. Bakker: Compact hyperkähler varieties: basic results

Compact hyperkähler manifolds enjoy a number of nice properties, many of which are connected to the Hodge structure on their weight 2 cohomology. Surprisingly, much of this theory extends to the case of singular compact hyperkähler varieties, which arise naturally even in the study of hyperkähler manifolds. The goal of this course is to survey some recent developments in this area. Topics will include: basic definitions and examples, Hodge theory and deformation theory, birational geometry and the global Torelli theorem, and the Beauville-Bogomolov decomposition theorem.

F. Barbacovi: Categorical dynamical systems and entropy of autoequivalences

A categorical dynamical system, as defined by Dimitrov, Haiden, Katzarkov, and Kontsevich, is given by a triangulated category together with an endofunctor. To measure the complexity the system, we compute the entropy of the endofunctor: a function from the real line to the extended real line. Of particular interest is the value of the entropy at t = 0, which takes the name of categorical entropy. Unfortunately, it is quite hard to compute the categorical entropy of an endofunctor in explicit examples. In this talk, I will explain how to compute the categorical entropy of the composition of two spherical twists around spherical objects; as a consequence, we will see how to produce new counterexamples to Kikuta-Takahashi’s conjecture. The talk is based on joint work with Jongmyeong Kim.

P. Belmans: Rozansky-Witten invariants for hyperkähler manifolds

I will give an overview of Rozansky-Witten theory, with an emphasis on Rozansky-Witten invariants. These are associated to trivalent graphs, which are constant within a deformation family, and for certain choices of trivalent graphs one recovers Chern numbers. This is a survey of work by Rozansky-Witten, Kapranov, Sawon, Roberts-Willerton, Nieper-Wisskirchen, and others.

A. Kuznetsov: Fractional Calabi-Yau properties of residual categories

Residual category is defined as the orthogonal category

to a rectangular Lefschetz collection in a triangulated category

endowed with an autoequivalence whose power is isomorphic to a shift

of the Serre functor; a typical example is the orthogonal to the

exceptional collection O, O(1), \dots, O(m-1) of line bundles on

a Fano variety of index m. In many cases residual categories have

the fractional Calabi--Yau property --- a power of their Serre

functor is isomorphic to a shift. I will explain how results

of this sort can be proved.

A. Perry: Categorical resolutions of singularities

I will discuss the definition, construction, and examples of categorical resolutions of singularities, which give a way to replace singular triangulated categories with smooth ones. Much of the interest in these resolutions stems from the fact that they often behave better than their classical counterparts.

L. Pertusi: S-invariant stability conditions and Fano threefolds of Picard rank 1

Stability conditions on the Kuznetsov component of a Fano threefold of Picard rank 1, index 1 and 2 have been constructed by Bayer, Lahoz, Macrì and Stellari, making possible to study moduli spaces of stable objects and their geometric properties. In this talk we investigate the action of the Serre functor on these stability conditions. In the index 2 case and in the case of GM threefolds, we show that they are Serre invariant. Then in almost all these cases we prove that there is a unique Serre invariant stability condition, as a consequence of a general criterion. Finally, we apply these results to the study of moduli spaces in the case of a cubic threefold X. In particular, we prove the smoothness of moduli spaces of stable objects in the Kuznetsov component of X and the irreducibility of the moduli space of stable Ulrich bundles on X. These results come from a joint work with Song Yang, and joint works in preparation with Soheyla Feyzbakhsh and with Ethan Robinett.

G. Saccà: Compactifying Lagrangian fibrations

In this course I will give a framework for compactification of Lagrangian fibrations

Articles that came out of the working groups:

Warren Cattani, Franco Giovenzana, Shengxuan Liu, Pablo Magni, Luigi Martinelli, Laura Pertusi, Jieao Song,Kernels of categorical resolutions of nodal singularities, arXiv:2209.12853