Workshop on Integral Stokes Structures and Applications

Giordano Cotti: t.b.a.

Nan-Kuo Ho: Lie theoretic approach to the geometry of the tt*-Toda equations.

Abstract: We propose a Lie-theoretic definition of the tt*-Toda equations for any complex simple Lie algebra, based on the concept of topological-antitopological fusion introduced by Cecotti and Vafa. Our first result concerns the Stokes data of a certain meromorphic connection, whose isomonodromic deformations are controlled by these equations. By exploiting a framework introduced by Boalch, we show that this data can be described using Kostant’s theory of Cartan subalgebras in apposition and Steinberg’s theory of conjugacy classes of regular elements, and it can be visualized on the Coxeter Plane. Moreover, when the Lie algebra is sl_{n+1}(C), we look at the whole monodromy data corresponding to these connection 1-forms, and show that it is a (real) symplectic groupoid over the Stokes data of these meromorphic connections. This is a joint work with Martin Guest.

Yichen Qin: Irregular Hodge numbers of nondegenerate functions.

Abstract: Given a smooth quasi-projective complex variety U equipped with a regular function f, there are several ways to producing Hodge numbers, including the irregular Hodge numbers and those constructed by Katzarkov-Kontsevich-Pantev (KKP). Esnault-Sabbah-Yu have shown that one type of KKP numbers coincide with the irregular Hodge numbers. In this talk, we discuss the relation of another kind of KKP numbers with the irregular Hodge numbers in the case of non-degenerate functions. This is a joint work with Dingxin Zhang.

Christian Sevenheck: Hodge theory of tautological systems.

Abstract: In this talk I will discuss a certain class of differential systems that are naturally attached to group actions on algebraic varieties, the so-called tautological systems. They appear in particular as periods of hyperplane sections of homogeneous spaces, and they can naturally be seen as generalizations of hypergeometric systems. I will discuss how to construct such systems in a functorial way, how to understand their Hodge theory. From an algebraic point of view, these systems are related to a generalization of the Chevalley- Eilenberg complex from Lie algebra theory. If time permits, I will explain this complex, along with a few applications to the duality theory of tautological systems.

Takuro Mochizuki: Algebraic integrable connections with bounded irregularity.

Abstract: According to the higher dimensional non-abelian Hodge theory, any algebraic vector bundle with an integrable connection on a quasi-projective complex variety underlies a wild harmonic bundle. In particular, an algebraic Lagrangian cover is attached to an integrable connection. It is useful to understand the irregularity of the integrable connection. In this talk, we shall discuss an application to the study of boundedness of the family of algebraic integrable connections with bounded irregularity.

Andrea D'Agnolo: On a topological approach to the Stokes phenomenon.

Abstract: A few years ago, Masaki Kashiwara and I proposed a Riemann-Hilbert correspondence in the possibly irregular case. This allows for a topological approach to the Stokes phenomenon. In this talk, I will recall some simple examples that we obtained using this method.

Marco Hien: t.b.a.

Andreas Hohl: Rationality conditions for sheaves and D-modules.

Abstract: Classically, monodromy data and Stokes structures associated to differential systems are a priori defined over the field of complex numbers. In many applications (such as in Hodge theory) it is, however, important to dispose of a structure over certain subfields (like the rational numbers) or even the integers. In this talk, we explain the concept of Galois descent in the context of the irregular Riemann-Hilbert correspondence, and we will show how one can use it to obtain explicit sufficient conditions for the rationality of Stokes phenomena, for instance in the case of hypergeometric systems.

Davide Guzzetti: A survey of analytic results on Stokes matrices and Dubrovin-Frobenius manifolds over the past few years.

Abstract: I will present results, from several viewpoints, on the Stokes phenomenon and monodromy of  an n x n linear system with two isolated singularities of respectively  Poincare' rank 0 and 1, when the eigenvalues of the leading matrix at rank 1 coalesce.  I will realize this in the geometry of a semisimple Dubrovin-Frobenius manifold, where the coalescence may correspond to two opposite cases: a semisimple point or a point of the caustic (non semisimple locus).

Claudio Meneses: Variations of stability and hyper-Kähler structures for moduli of parabolic Higgs bundles on the Riemann sphere. 

Abstract: An interesting feature of moduli spaces of stable parabolic Higgs bundles on compact Riemann surfaces is that they determine an infinite series of families of hyper-Kähler metrics, in part coming from the dependence of the former on choices of stability parameters. Although the wall-crossing phenomena associated with such dependence is well understood as a problem in birational geometry, the analogous differential-geometric problem for the hyper-Kähler structure is still outstanding.
In this talk I will discuss results on the variation problem in the special case of genus 0 and rank 2, obtained in collaboration with Lynn Heller and Sebastian Heller. Concretely, I will explain how a suitable renormalised limit of the Hitchin hyper-Kähler metrics converge to the hyperpolygon-space hyper-Kähler metrics as the stability parameters approach 0.

Thomas Krämer: A Tannakian view on monodromy.

Abstract: I will discuss a conjectural relation between Tannaka groups of perverse sheaves and the monodromy of local systems on irregular varieties, and illustrate it by a recent example of local systems whose algebraic monodromy is exceptional of type E6 (work in progress with Daniel Litt and Marco Maculan).

Yudai Hateruma: Experiments with the braid group action on Stokes data for the tt*-Toda equations.

Abstract: Cecotti-Vafa introduced the topological-antitopological fusion (tt*) equations in the 1990's and investigated relations between their (conjectural) solutions and deformations of quantum field theories. They focused on theories of Landau-Ginzburg type, where physics agreed with and (conjecturally) extended some well known results in Picard-Lefschetz theory. Examples with “cyclic symmetry” were prominent in their work; in this case, the tt* equations are a special case of the Toda equations, and these “tt*-Toda equations” were solved by Guest-Its-Lin in a series of articles during the past 15 years. We shall explain our work (in progress) on the role of the Stokes data in the tt*-Toda case. This data is a subset of the Stokes data of the associated Dubrovin connection, restricted by the nature of (global) solutions to the tt*-Toda equations.

Luisa Fiorot: Relative regular Riemann-Hilbert correspondence.

Abstract: I will present the relative regular Riemann-Hilbert correspondence which is a joint work with Teresa Monteiro Fernandes and Claude Sabbah. Deligne (1970) proved the equivalence between meromorphic bundles with flat connections having regular singularities and local systems of complex vector spaces. Later in 1984, Kashiwara on the one hand, and Mebkhout on the other hand, generalized this correspondence in the regular Riemann-Hilbert correspondence, by establishing an equivalence between the triangulated category of regular holonomic D-modules and that of constructible complexes of sheaves. Holomorphic families of such objects occur in various contexts. Extending the regular Riemann-Hilbert correspondence to this 'relative' setting is the purpose of our work. This program started in 2013, when Teresa Monteiro Fernandes and Claude Sabbah provided the framework for this relative Riemann-Hilbert correspondence, in particular by making clear the notion of holomorphic family of constructible complexes. They proved this correspondence in the context of regular mixed twistor D-modules, as introduced by T. Mochizuki. The general case was achieved in our work, where we provide a complete proof of this relative correspondence.

Claude Sabbah: Some applications of Stokes structures in higher dimensions.

Abstract: I will explain the proof of an equality between global Euler characteristics of de Rham complexes of algebraic holonomic D-modules by using the Riemann-Hilbert correspondence between good meromorphic flat bundles and good Stokes-filtered local systems. I will take this opportunity to explain a few results on Stokes structures in higher dimensions.