Etienne Mann

Smooth toric Deligne-Mumford stacks

Abstract We give a geometric definition of smooth toric Deligne-Mumford stacks using the action of a “torus”. We show that our definition is equivalent to the one of Borisov, Chen and Smith in terms of stacky fans. In particular, we give a geometric interpretation of the combinatorial data contained in a stacky fan. We also give a bottom up classification in terms of simplicial toric varieties and fiber products of root stacks.

ORBIFOLD QUANTUM COHOMOLOGY OF WEIGHTED PROJECTIVE SPACES

This article is a revised, short and english version of my PhD thesis. First, we show a mirror theorem : the Frobenius manifold associated to the orbifold quantum cohomology of weighted projective space is isomorphic to the one attached to a specific Laurent polynomial. Secondly, we show a reconstruction theorem, that is, we can reconstruct in an algorithmic way the full genus 0 Gromov-Witten potential from the 3-point invariants.

The cohomological crepant resolution conjecture for P(1,3,4,4)

We prove the cohomological crepant resolution conjecture of Ruan for the weighted projective space ℙ(1,3,4,4). To compute the quantum corrected cohomology ring, we combine the results of Coates–Corti–Iritani–Tseng on ℙ(1,1,1,3) and our previous results.

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.

QUANTUM D-MODULES FOR TORIC NEF COMPLETE INTERSECTIONS

Let X be a smooth projective toric variety with k ample line bundles. Let Z be the zero locus of k generic sections. It is well known that the ambient quantum 𝒟-module of Z is cyclic i.e. is defined by an ideal of differential operators. In this paper, we give an explicit construction of this ideal as a quotient ideal of a GKZ system associated to the toric data of X and the line bundles. This description can be seen as a “left cancellation procedure”. We consider some examples where this description enables us to compute generators of this ideal, and thus to give a presentation of the ambient quantum 𝒟-module.

A model for the orbifold Chow ring of weighted projective spaces

We construct an isomorphism of graded Frobenius algebras between the orbifold Chow ring of weighted projective spaces and graded algebras of groups of roots of the unity.

Quantum Serre Theorem as a Duality Between Quantum D-Modules

We give an interpretation of quantum Serre theorem of Coates and Givental as a duality of twisted quantum D-modules. This interpretation admits a non-equivariant limit, and we obtain a precise relationship among (1) the quantum D-module of X twisted by a convex vector bundle E and the Euler class, (2) the quantum D-module of the total space of the dual bundle E∨ → X, and (3) the quantum D-module of a submanifold Z ⊂ X cut out by a regular section of E . When E is the anticanonical line bundle K−1 X , we identify these twisted quantum D-modules with second structure connections with different parameters, which arise as Fourier–Laplace transforms of the quantum D-module of X. In this case, we show that the duality pairing is identified with Dubrovin’s second metric (intersection form).

Computing certain Gromov-Witten invariants of the crepant resolution of ℙ(1, 3, 4, 4)

We prove a formula computing the Gromov-Witten invariants of genus zero with three marked points of the resolution of the transversal A 3 -singularity of the weighted projective space ℙ(1,3,4,4) using the theory of deformations of surfaces with A n -singularities. We use this result to check Ruan’s conjecture for the stack ℙ(1,3,4,4).