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We review some classical and modern aspects of hypergeometric differential equations, including A-hypergeometric systems of Gel$$'$$ ′ fand, Graev, Kapranov and Zelevinsky. Some recent advances in this theory, such as Euler–Koszul homology, rank jump phenomena, irregularity questions and Hodge theoretic aspects are discussed with more details. We also give some applications of the theory of hypergeometric systems to toric mirror symmetry.

We consider mixed Hodge module structures on GKZ-hypergeometric differential systems. We show that the Hodge filtration on these D-modules is given by the order filtration, up to suitable shift. As an application, we prove a conjecture on the existence of non-commutative Hodge structures on the reduced quantum D-module of a nef complete intersection inside a toric variety.

We show that certain one-dimensional hypergeometric differential systems underlie objects of the category of irregular mixed Hodge modules, which was recently introduced by Sabbah, and compute the irregular Hodge filtration for them. We also provide a comparison theorem between two different types of Fourier–Laplace transformation for algebraic integrable twistor D -modules.

We introduce tautological systems defined by prehomogeneous actions of reductive algebraic groups. If the complement of the open orbit is a linear free divisor satisfying a certain finiteness condition, we show that these systems underly mixed Hodge modules. A dimensional reduction is considered and gives rise to one-dimensional differential systems generalizing the quantum differential equation of projective spaces.

We study Betti structures in the solution complexes of confluent hypergeometric equations. We use the framework of enhanced ind-sheaves and the irregular Riemann–Hilbert correspondence of D’Agnolo–Kashiwara. The main result is a group theoretic criterion that ensures that enhanced solutions of such systems are defined over certain subfields of $\mathds{ C}$. The proof uses a description of the hypergeometric systems as exponentially twisted Gauß–Manin systems of certain Laurent polynomials.

For any integer $d\times (n+1)$ matrix $A$ and parameter $\beta\in\CC^d$ let $M_A(\beta)$ be the associated $A$-hypergeometric (or GKZ) system in the variables $x_0,\ldots,x_n$. We describe bounds for the (roots of the) $b$-functions of both $M_A(\beta)$ and its Fourier transform along the hyperplanes $(x_j=0)$. We also give an estimate for the $b$-function for restricting $M_A(\beta)$ to a generic point.

We construct Landau-Ginzburg models for numerically effective complete intersections in toric manifolds as partial compactifications of families of Laurent polynomials. We show a mirror statement saying that the quantum D-module of the ambient part of the cohomology of the submanifold is isomorphic to an intersection cohomology D-module defined from this partial compactification and we deduce Hodge properties of these differential systems.

We describe mirror symmetry for weak Fano toric manifolds as an equivalence of filtered D \mathcal {D} -modules. We discuss in particular the logarithmic degeneration behavior at the large radius limit point and express the mirror correspondence as an isomorphism of Frobenius manifolds with logarithmic poles. The main tool is an identification of the Gauß-Manin system of the mirror Landau-Ginzburg model with a hypergeometric D \mathcal {D} -module, and a detailed study of a natural filtration defined on this differential system. We obtain a solution of the Birkhoff problem for lattices defined by this filtration and show the existence of a primitive form, which yields the construction of Frobenius structures with logarithmic poles associated to the mirror Laurent polynomial. As a final application, we show the existence of a pure polarized non-commutative Hodge structure on a Zariski open subset of the complexified Kähler moduli space of the variety.

We determine the irregular Hodge filtration, as introduced by Sabbah, for the purely irregular hypergeometric $\mathcal{D}$-modules. We obtain in particular a formula for the irregular Hodge numbers of these systems. We use the reduction of hypergeometric systems from GKZ-systems as well as comparison results to Gauss-Manin systems of Laurent polynomials via Fourier-Laplace and Radon transformations., Comment: 32 pages, final version

We investigate differential systems occurring in the study of particular non-isolated singularities, the so-called linear free divisors. We obtain a duality theorem for these \({\mathcal{D}}\) -modules taking into account filtrations, and deduce degeneration properties of certain Frobenius manifolds associated to linear sections of the Milnor fibres of the divisor.

We review a version of the mirror correspondence for smooth toric varieties with a numerically effective anticanonical bundle. We give a precise description of the so-called B-model, which involves the Gaus-Manin system of a family of Laurent polynomials. We show how to derive from these data a variation of non-commutative Hodge structures and describe general results on period maps and classifying spaces for these generalized Hodge structures. Finally, we explain a version of mirror symmetry as an isomorphism of Frobenius manifolds.

We investigate variations of Brieskorn lattices over non-compact parameter spaces, and discuss the corresponding limit objects on the boundary divisor. We study the associated variation of twistors and the corresponding limit mixed twistor structures. We construct a compact classifying space for regular singular Brieskorn lattices and prove that its pure polarized part carries a natural hermitian structure and that the induced distance makes it into a complete metric space., 55 pages

We study tt*-geometry on the classifying space for regular singular TERP-structures, e.g., Fourier-Laplace transformations of Brieskorn lattices of isolated hypersurface singularities. We show that (a part of) this classifying space can be canonically equipped with a hermitian structure. We derive an estimate for the holomorphic sectional curvature of this hermitian metric, which is the analogue of a similar result for classifying spaces of pure polarized Hodge structures., Comment: 25 pages

We prove that deformations of a Lagrangian singularity are unobstructed if the usual (flat) deformations are unobstructed and if a cohomological vanishing condition is satisfied. This gives another application to deformation theory of the Lagrangian de Rham complex introduced in Sevenheck and van Straten (Math. Ann. 327 (1) (2003) 79–102). To prove our theorem, we use the T 1 -lifting criterion due to Ran, Kawamata and others. To cite this article: C. Sevenheck, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

We discuss Bernstein polynomials of reductive linear free divisors. We define suitable Brieskorn lattices for these non-isolated singularities, and show the analogue of Malgrange's result relating the roots of the Bernstein polynomial to the residue eigenvalues on the saturation of these Brieskorn lattices., Comment: 14 pages

We study linear functions on fibrations whose central fibre is a linear free divisor. We analyse the Gauss-Manin system associated to these functions, and prove the existence of a primitive and homogenous form. As a consequence, we show that the base space of the semi-universal unfolding of such a function carries a Frobenius manifold structure., 37 pages

We study a generalization of Hodge structures which first appeared in the work of Cecotti and Vafa. It consists of twistors, that is, holomorphic vector bundles on P^1, with additional structure, a flat connection on C^*, a real subbundle and a pairing. We call these objects TERP-structures. We generalize to TERP-structures a correspondence of Cattani, Kaplan and Schmid between nilpotent orbits of Hodge structures and polarized mixed Hodge structures. The proofs use work of Simpson and Mochizuki on variations of twistor structures and a control of the Stokes structures of the poles at zero and infinity. The results are applied to TERP-structures which arise via oscillating integrals from holomorphic functions with isolated singularities., 43 pages, some very minor modifications, misprints corrected

In this article we prove a rigidity theorem for lagrangian singularities by studying the local cohomology of the lagrangian de Rham complex that was introduced in math.AG/0002083. The result can be applied to show the rigidity of all open swallowtails of dimension greater or equal than two. In the case of lagrangian complete intersection singularities the lagrangian de Rham complex turns out to be perverse. We also show that lagrangian complete intersection in dimension greater than two cannot be regular in codimension one., 11 pages