Workshop: Landau-Ginzburg models, mirror symmetry and Hodge theory
07.11. - 09.11.2012
Universität Mannheim, Germany
Aim
This workshop brings together people working on various aspects
of mirror symmetry and singularity theory. The talks will report on recent
and ongoing work around the following topics:
Quantum D-modules and their hypergeometric description
Mirror symmetry and Frobenius manifolds
Degeneration behavior at the large volume limit
Laurent polynomials and non-affine Landau-Ginzburg models
Construction of non-commutative Hodge structures in singularity theory
Rigid local systems and Hodge theory
Categorical aspects of Hodge theoretic invariants
Mirror symmetry for manifolds with positive Kodaira dimension
Program
Wednesday, 07.11.
14.00 - 15.00, Etienne Mann (Montpellier): Quantum D-module for toric nef hypersurfaces
Abstract:
On a smooth toric projective variety X, Iritani proves that its quantum D module is isomorphic to the GKZ system associated to X. We generalize this result to the pair (X,L) where L is an ample line bundle. More precisely, denote by Z the hypersurface defined by a generic section. We define the twisted quantum D-module which is a vector bundle with a flat connection, a flat pairing and a natural integrable structure. An appropriate quotient of it is isomorphic to the ambient part of the quantum D-module of Z.
As X is toric, these quantum D-modules are cyclic. The twisted quantum D-module can be presented via mirror symmetry by the GKZ system associated to the total space of the dual line bundle L*. A question is to know what is the system of equations that define the ambiant part of the
quantum D-module of Z. We construct this system as a quotient ideal of
the GKZ system. This work is done with Thierry Mignon.
16.00 - 17.00, Thomas Reichelt (Mannheim): Laurent polynomials, Radon transformation and hypergeometric systems (I)
Abstract:
I will explain a close relationship between Gauss-Manin systems of families of Laurent polynomials and the so called GKZ
systems of Gelfand, Kapranov and Zelevinsky. This endows the latter with the structure of a mixed Hodge module.
I will explain some ingredients of the proof like the Radon transform for D-modules as well as some results on
holonomic duality and the classification of GKZ systems.
Thursday, 08.11.
9.00 - 10.00, Thomas Reichelt (Mannheim): Laurent polynomials, Radon transformation and hypergeometric systems (II)
Abstract:
In this talk I explain how one can compute the Gauss-Manin system with respect to the intersection complex of the universal
family of hyperplane sections of a toric variety in terms of GKZ-systems. This is joint work with Christian Sevenheck.
11.00 - 12.00, Helge Ruddat (Mainz): Perverse varieties - how general type mirror duals appear inside Calabi-Yau manifolds (I)
Abstract: The Hodge structure of the mirror dual of a variety of general type, say of a genus two curve, is constructed using vanishing cycles. Very similar but yet different structures arise in the degeneration of Calabi-Yau threefolds. We want to study their relation and phrase some conjectures.
12.00 - 14.00, Lunch
14.00 - 15.00,
Dmytro Shklyarov (Freiburg) A "derived" viewpoint on classical invariants of isolated
singularities
Abstract:
A popular question in modern algebraic geometry is whether
or not a given topological or, say,
Hodge theoretic invariant of an algebraic variety is a
derived invariant, i.e. depends only on the
derived category of (quasi-)coherent sheaves on the
variety. Questions of this sort may, of course,
be considered in other geometric contexts, for example, in
singularity theory where the role of
the classical derived categories is played by the
categories of matrix factorizations.
The goal of the talk is to address the following question:
Is the Steenbrink-Varchenko spectrum of
an isolated singularity a derived invariant?
16.00 - 17.00, Claude Sabbah (Palaiseau): Hodge theory of the middle convolution (joint work with Michael Dettweiler, Bayreuth)
(notes of the talk)
Abstract:
We compute the behaviour of Hodge data by tensor product with a
unitary rank-one local system and middle convolution by a Kummer
unitary rank-one local system for an irreducible variation of
polarized complex Hodge structure on a punctured complex affine
line.
Friday, 09.11.
9.00 - 10.00 Helge Ruddat (Mainz): Perverse varieties - how general type mirror duals appear inside Calabi-Yau manifolds (II)
11.00 - 12.00, Christian Sevenheck (Mannheim): Landau-Ginzburg models for nef toric complete intersections
Abstract: I will report on joint work with Thomas Reichelt
concerning Landau-Ginzburg models for nef toric complete intersections. I will explain how to construct
non-commutative Hodge structures from them. For the case
of nef toric manifolds I will also describe a construction of Frobenius manifolds with logarithmic poles and how to identify them
with the A-model Frobenius structure.
12.00 - 14.00, Lunch
14.00 - Discussion
Registration
To register for the workshop, please send an email to Christian dot Sevenheck at uni-mannheim dot de
Participants (as registered on November 05, 2012)
- Christian Barz (Universität Mannheim)
- Michael Bogner (Universität Mainz)
- Helge Ruddat (Universität Mainz)
- Claus Hertling (Universität Mannheim)
- Thomas Reichelt (Universität Mannheim)
- Etienne Mann (Université de Montpellier)
- Thierry Mignon (Université de Montpellier)
- Claude Sabbah (Ecole Polytechnique, Palaiseau)
- Timo Schürg (Universät Bonn)
- Christian Sevenheck (Universität Mannheim)
- Dmytro Shklyarov (Universität Freiburg)
- Manuel Gonzalez Villa (Universität Heidelberg)