Hyperkähler manifolds

Time: April 11-14, 2012

Main speakers:  Daniel Greb (Freiburg)
Misha Verbitsky (Moscow) --- cancelled

Location: Banach Center IMPAN, Warsaw

Organizers:  Grzegorz Kapustka (grzegorz.kapustka at uj.edu.pl)
Jarosław Buczyński (jabu at mimuw.edu.pl)

!!! IMPORTANT !!! Due to problems beyond our control Misha Verbitsky had to cancel his arrival.

Hyperkähler manifolds are important and intensively investigated building block in the classification of algebraic varieties. Some of the recent development will be presented during our research group.

We dedicate the meeting in memory of Andrey Todorov, who unexpectedly died in March.

Main series of lectures

Singularities in hyperkähler geometry. (Misha Verbitsky, lectures cancelled)

A "singular hyperkähler variety" can be defined, following Deligne and Simpson, in terms of the twistor fibration. Such a structure arises naturally when one deals with the moduli spaces of subvarieties or stable vector bundles on a hyperkähler manifold. Also, a trianalytic subvariety of a hyperkähler manifold is singular hyperkähler. I will present some examples and then give a proof of the desingularization theorem for singular hyperkähler varieties.

Structure theory of varieties with trivial canonical class: Fibrations and Foliations (Daniel Greb)

After a short general introduction to hyperkähler manifolds, I will focus on the following question of Beauville: If a hyperkähler manifold X contains a complex torus L as a Lagrangian submanifold, does X admit a Lagrangian fibration with fibre L? In my talk, I will describe joint work with Christian Lehn and Sönke Rollenske giving a positive answer to Beauville's question in the non-projective case. Furthermore, I will derive a criterion for the existence of an almost holomorphic Lagrangian fibration in the projective case, which has recently been used by Hwang and Weiss to provide a positive answer to Beauville's question for projective hyperkähler manifolds. Finally, I will explain how the Minimal Model Program together with arguments of Matsushita can be used to see that also in the projective case the fibration produced by Hwang-Weiss can be made holomorphic via a modification of the base space.

Contributed talks

The following people will give a contributed talk:

  • Grzegorz Kapustka, IMPAN, Warsaw, Jagiellonian University, Cracow, and University of Zurich.
    Title: On the O'Grady conjecture

    Abstract: We disscuss the difficulties in the proof of the conjecture of O'Grady about the classification of irreducible symplectic 4-fold numerically equivalent to the Douady space (K3)^[2].

  • Michał Kapustka, Jagiellonian University, Cracow, and University of Zurich.
    Title: The two point hilbert schemes of K3 surfaces of genus 10

    Abstract: We present known explicit constructions of locally complete deformation families of polarised hyperkähler fourfolds. Next we present several descriptions of the Hilbert scheme of two points on a K3 surface of genus 10 and discuss the possibility of using them for finding a new explcitly described family.

  • Christian Lehn, University of Grenoble.
    Title: The second cohomology of a hyperkähler manifold and deformations

    Abstract: We discuss deformation problems of hyperkähler manifolds and their cohomological description. After explaining some general facts we will turn to applications to a geometric approach to the Beauville problem addressed in D. Greb's lectures (joint work with D. Greb and S. Rollenske) and also to the study of singular fibers of Lagrangian fibrations.

  • Piotr Pragacz, IMPAN, Warsaw.
    Title: Lagrangian Thom polynomials
  • Andrey Soldatenkov, Higher School of Economics, Moscow.
    Title: Subvarieties of hypercomplex SL(n,H)-manifolds

    Abstract: A hypercomplex structure on a smooth manifold is a triple of integrable almost-complex structures I, J, K, that satisfy quaternionic relations. I will describe some recent results concerning compact complex subvarieties of a special class of hypercomplex manifolds, namely SL(n,H)-manifolds admitting an HKT metric. This is joint work with M. Verbitsky.

  • Jarosław Wiśniewski, Warsaw University.
    Title: Contractions of hyperkähler 4-folds

Colloquium (cancelled)

During the research group M. Verbitsky will deliver a colloquium talk adressed to a wide mathematical audience (on Friday, April 13th at 15:00).
Title: Algebraic geometry over quaternions

Abstract: A hyperkähler manifold is a manifold equipped with an action of quaternions I, J, K, and a quaternionic-Hermitian metric, making I, J, K Kähler. By Calabi-Yau theorem, this structure naturally occurs on any compact holomorphically symplectic Kähler manifold. A surprising number of algebro-geometric structures, such as moduli spaces and a desingularization, makes sense in hyperkähler situation, producing hyperkähler varieties as moduli spaces of stable bundles and subvarieties. Also, a desingularization of a singular hyperkähler variety is defined, and it is smooth and hyperkähler.

Location and accomodation

The research group takes place at the Banach Center in the Institute of Mathematics of the Polish Academy of Sciences. Here is a description of the access.

The guest rooms of the Institute can accommodate most of the participants. Support of accommodation expenses for a limited number of participants is possible. Unfortunately, we cannot offer support of travel expenses.

Registration

People interested in participation should contact the organisers before 29.Feb.2012. Participants requesting a financial support of accommodation expenses or an invitation letter should provide a brief letter of recommendation from a senior mathematician.

Schedule

Arrival: Apr 11 afternoon and Apr 12 morning.

Apr 12

10:20-10:30 Welcome
10:30-11:30 D. Greb
11:30-12:00 Coffee and tea
12:00-13:00 A. Soldatenkov
lunch break
15:00-16:00 G. Kapustka
16:00-16:30 Coffee and tea
16:30-17:30 P. Pragacz

Apr 13

09:30-10:30 D. Greb
10:30-11:00 Coffee and tea
11:00-12:00 J. Wiśniewski
lunch break
15:00-16:00 D. Greb
16:00-16:30 Coffee and tea
16:30-17:30 Ch. Lehn

Apr 14

09:30-10:30 D. Greb
10:30-11:00 Coffee and tea
11:00-12:00 M. Kapustka

Lectures take place in room 403. Coffee, tea and cookies will be served in the common room 409. Both rooms 403 and 409 are on the fourth floor (counting from 0, i.e. ground floor). You can read about lunch possibilities here (information collected by Piotr Pragacz).

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Acknowledgements

We would like to thank Marysia Donten-Bury for preparing the web page of the research group.