Antoine Douai

Gauss-Manin systems, Brieskorn lattices and Frobenius structures (I)

This article explains a detailed example of the general result developed in the first part [3]. We were motivated by [1], where S. Barannikov describes a Frobenius structure attached to the Laurent polynomial f(u 0,..., u n ) = u 0 +...+ u n restricted to the torus U = {(u 0,..., u n ) ∈ ℂ n+1 | Π i u i = 1} and shows that it is isomorphic to the Frobenius structure attached to the quantum cohomology of the projective space ℙ n (ℂ) (as defined e.g., in [5]).

A canonical Frobenius structure

We show that it makes sense to speak of the Frobenius manifold attached to a convenient and nondegenerate Laurent polynomial.

The small quantum cohomology of a weighted projective space, a mirror D-module and their classical limits

We first describe a mirror partner (B-model) of the small quantum orbifold cohomology of weighted projective spaces (A-model) in the framework of differential equations: we attach to the A-model (resp. B-model) a quantum differential system (that is a trivial bundle equipped with a suitable flat meromorphic connection and a flat bilinear form) and we give an explicit isomorphism between these two quantum differential systems. On the A-side (resp. on the B-side), the quantum differential system alluded to is naturally produced by the small quantum cohomology (resp. a solution of the Birkhoff problem for the Brieskorn lattice of a Landau–Ginzburg model). Then we study the degenerations of these quantum differential systems and we apply our results to the construction of (classical, limit, logarithmic) Frobenius manifolds.

Sur le système de Gauss–Manin d'un polynôme modéré

Resume Given a cohomologically tame polynomial we compute the determinant of a period matrix and, when the Gauss–Manin connexion is described by a fuchsian system, we give a formula for the trace of the residue matrices.

Global spectra, polytopes and stacky invariants

Given a convex polytope, we define its geometric spectrum, a stacky version of Batyrev’s stringy E-functions, and we prove a stacky version of a formula of Libgober and Wood about the E-polynomial of a smooth projective variety. As an application, we get a closed formula for the variance of the geometric spectrum and a Noether’s formula for two dimensional Fano polytopes (polytopes whose vertices are primitive lattice points; a Fano polytope is not necessarily smooth). We also show that this geometric spectrum is equal to the algebraic spectrum (the spectrum at infinity of a tame Laurent polynomial whose Newton polytope is the polytope alluded to). This gives an explanation and some positive answers to Hertling’s conjecture about the variance of the spectrum of tame regular functions.