Mirror Symmetry and Irregular Singularities coming from Physics – SISYPH

(a) Moduli spaces are at the core of most questions considered in this project. There is a variety of objects they parametrize (level curves, stable maps, singularities, linear differential equations), of enumerative/classification questions they intend to answer (counting curves, classifying singularities, finding algebraic solutions to non-linear differential equations like Painleve´'s) and of techniques (birational geometry, various old-and-new aspects of Hodge theory, Poisson geometry). Understanding their geometry is one of the main objectives of this project. Advances in one topic should be profitable to the other ones.
(b) Irregular singularities of linear differential equations in higher dimensions (holonomic D-modules) is a subject which has experienced a rapid development in the recent years. The techniques and results obtained in this domain should quickly have consequences in various questions mirror symmetry is concerned with. Developing Hodge theory in this context and applying this theory to get more structures on moduli spaces is another objective of this project.
(c) Higher dimensional hypergeometric differential systems (GKZ systems) will be a central object to deal with, as they provide at the same time a framework for certain quantum D-modules and a practical laboratory for highly non-trivial problems concerned with irregular singularities. They will serve as experimental data for testing the interplay between mirror symmetry and irregular singularities.

Analysis of fundamental examples of wild character varieties.
Introduction of hybrid Landau-Ginzburg models.
Explicit formulas for the virtual fundamental cycle in some theories in genus zero. Application to the equivalence of some hierarchies of integrable systems.
Detailed description of the local monodromy and the determinant of Katz' middle convolution functor in the l-adic setting.
Study of some non-tame Gauss-Manin systems related to the small quantum cohomology of hypersurfaces in weighted projective spaces.
Introduction of a new conceptual language for the geometrical objects underlying Painlevé equations, and new results on real solutions of a Painlevé III equation.
Interpretation of the quantum Serre theorem as a duality of twisted quantum D-modules.
Determination of the Hodge filtration of some hypergeometric differential systems (GKZ) and applications to the existence of non-commutative Hodge structures on some quantum D-modules.
Computation of the mirror map for some canonical formal families of Calabi-Yau varieties.
New structure of Hodge type on the cohomology associated to some Landau-Ginzburg potentials. Proof of some Hodge-type properties for these structures.
Definition of the topological Laplace transformation for a local system equipped with a Stokes structure on the complex affine line, and computation in the case of an elementary differential system.
Confirmation of a conjecture on quasi-homogeneous isolated complete intersection singularities with small embedding dimension, and counter-example in big embedding dimension.
Study of quasi-homogeneous free divisors.

Possible computations of Gromov-Witten invariants, mainly for Calabi-Yau varieties, but also other algebraic varieties (generic hypersurfaces in weighted projective spaces, Fano varieties, varieties of general type). We will focus on the geometry of the relevant moduli spaces. Construction of various kinds of mirror symmetries (homological or in terms of Frobenius manifolds), e.g. for varieties of general type or for toric degenerations, will be considered. We will aim at a better understanding of the relations between these various mirror symmetries, in particular by looking at the quantum D-module approach.
We also aim at
- constructing a classifying space of irregular non-commutative Hodge structures and studying period maps for variations of them,
- obtaining a good theory of irregular Hodge D-modules, which can be applied at various places in Mirror Symmetry,
- obtaining a mirror correspondence for moduli spaces of irregular singular linear differential equations, within the framework of nonabelian Hodge theory.
The Stokes phenomenon will be regarded as an organizing principle in various ways. Making precise, along lines envisioned by Cecotti-Vafa, the relation between Stokes matrices and spectral numbers of singularities (which can be regarded as the wild Hodge numbers) is an exciting perspective. This is an occurrence of the relation between the Betti and the Hodge aspects in wild Hodge theory. The question of which Stokes matrices turn up in the case of singularities and within mirror symmetry shall be treated. The role of the Stokes phenomenon in wild character varieties also needs to be further understood.
Transversally to these three topics, the GKZ systems are of central interest. Making precise the irregularity invariants (slopes along specific subvarieties, irregularity sheaf, Stokes phenomenon) is the perspective for GKZ systems.

1. M. Granger & M. Schulze, Normal crossing properties of complex hypersurfaces via logarithmic residues, Compos. Math. 150 (2014), no. 9, p. 1607-1622.
2. M. Granger & M. Schulze, Quasihomogeneity of curves and the Jacobian endomorphism ring, Commun. Algebra 43 (2015), no. 2, p. 861-870.
3. M. Hien & C. Sabbah, The local Laplace transform of an elementary irregular meromorphic connection, Rend. Sem. Mat. Univ. Padova, to appear.
4. Th. Reichelt & Ch. Sevenheck, Logarithmic Frobenius manifolds, hypergeometric systems and quantum D-modules, J. Algebr. Geom. 24 (2015), no. 2, p. 201-281.
5. Résumés des exposés de la conférence d’Oberwolfach, à paraître dans les « Oberwolfach Reports » (Birkhäuser).
6. Ph. Boalch, Geometry and braiding of Stokes data; fission and wild character varieties, Ann. Math. (2) 179 (2014), no. 1, p. 301-365.
7. Ph. Boalch, Poisson varieties from Riemann surfaces, Indag. Math., New Ser. 25 (2014), no. 5, p. 872-900.
8. A. Douai, Quantum differential systems and construction of rational structures, Manuscr. Math. 145 (2014), No. 3-4, 285-317.
9. Hiroshi Iritani, Étienne Mann & Thierry Mignon, Quantum Serre theorem as a duality between quantum D-modules, IMRN, to appear.
10. H. Ruddat, N. Sibilla, D. Treumann, E. Zaslow: Skeleta of Affine Hypersurfaces, Geometry & Topology 18 (2014), 41p.
11. H. Ruddat, Perverse Curves and Mirror Symmetry, J. Algebraic Geom., to appear.
12. C. Sabbah & Morihiko Saito, Kontsevich’s conjecture on an algebraic formula for vanishing cycles of local systems, Algebr. Geom. 1 (2014), no. 1, 107-130.
13. Hélène Esnault, C. Sabbah & Jeng-Daw Yu, E1-degeneration of the irregular Hodge filtration (with an appendix by Morihiko Saito), J. reine angew. Math. (2015).
15. C. Sabbah & Jeng-Daw Yu, On the irregular Hodge filtration of exponentially twisted mixed Hodge modules, Forum Math. Sigma 3 (2015), Article ID e9, 71 p.
15. C. Sabbah, Differential systems of pure Gaussian type, Izv. Math., to appear.

The overall aim of this project is to make important progress on the interplay between mirror symmetry and singularities of linear differential equations in any dimension. The general objective and the originality of the project consists in profiting from the expertise of the participants in various diverse fields in order to advance each of the research topics. The French (ANR-funded) and German (DFG-funded) partners have complementary expertise in all topics and a long standing experience of collaboration. The following topics will be considered both separately and in connection to each other:

- Mirror symmetry as an effective tool for the computation of Gromov-Witten invariants of various kinds of smooth algebraic varieties or orbifolds,

- Irregular singularities of linear differential systems in any dimension, either from the point of view of holonomic D-modules or from that of isomonodromy deformations,

- Hodge theoretic aspects of such differential systems.

One of the original aspects of the project consists in obtaining results in each topic by exhibiting the interplay between these topics through the use of various tools and methods (algebraic geometry, non commutative Hodge theory, singularity theory and D-modules, symplectic geometry) with, in the background, motivations and conjectures formulated by physicists.

A central object of interest will be the generalized hypergeometric systems of linear differential equations (GKZ systems) as models for the quantum D-module of toric manifolds or orbifolds. These GKZ systems also provide a large class of examples of holonomic D-modules with irregular singularities, where conjectures and preliminary results can be tested.

The understanding of the geometry of different types of moduli spaces like those for isolated hypersurface singularities, for curves, or more generally for stable mappings (entering in the very definition of Gromov-Witten invariants), and for meromorphic connections on vector bundles, is one of the most important motivations of the whole project. Although the first ones are known to be essential for mirror symmetry, a basic question will be to make sense/fully understand the notion of mirror symmetry for the moduli spaces of irregular singular connections on Riemann surfaces.

The Stokes phenomenon, which is a fundamental property of irregular singularities of differential equations, is a basic object to be understood in the context of either Gromov-Witten theory or Landau-Ginzburg models and their extensions to singularity theory. Its relationship with Hodge-theoretic properties (in particular their non-commutative aspects) will allow the analysis of moduli spaces of singularities.

This project would allow the French and German teams of researchers the full means to realize their scientific goals and collaborate more effectively. The structure of the project motivates that part of the funding we apply for is devoted to financing travel and invitation costs of members of the project and also of some other recognized scholars. However, most part of the funding will be devoted to post-doctoral financial support, in order to fully develop specific directions.