相关论文: (Semi)simple exercises in quantum cohomology

We describe genus g>1 potentials of semisimple Frobenius structures. Our formula can be considered as a definition in the axiomatic context of Frobenius manifolds. In Gromov-Witten theory, it becomes a conjecture expressing higher genus…

We formulate some conjectures that relates semisimple Frobenius manifolds, their spectral curves and integrable hierarchies.

Let G be a reductive algebraic group over a field of prime characteristic. One can associate to G (or subgroups thereof) its Lie algebra, its Frobenius kernels, and the finite Chevalley group of points over a finite field. The…

We review and streamline our previous results and the results of Y.Ostrover on the existence of Calabi quasi-morphisms and symplectic quasi-states on symplectic manifolds with semi-simple quantum homology. As an illustration, we discuss the…

We introduce a "limiting Frobenius structure" attached to any degeneration of projective varieties over a finite field of characteristic p which satisfies a p-adic lifting assumption. Our limiting Frobenius structure is shown to be…

The quantum cohomology algebra of a projective manifold X is the cohomology H(X,Q) endowed with a different algebra structure, which takes into account the geometry of rational curves in X. We show that this algebra takes a remarkably…

We study the quantum cohomology of quasi-minuscule and quasi-cominuscule homogeneous spaces. The product of any two Schubert cells does not involve powers of the quantum parameter higher than 2. With the help of the quantum to classical…

In the simplicial theory of hypercoverings, we replace the indexing category $\Delta$ by the \emph{symmetric simplicial category} $\Delta S$ and study (a class of) $\Delta S$-hypercoverings, which we call \emph{spaces admitting symmetric…

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…

We obtain algebraic Frobenius manifolds from classical $W$-algebras associated to subregular nilpotent elements in simple Lie algebras of type $D_r$ where $r$ is even and $E_r$. The resulting Frobenius manifolds are certain hypersurfaces in…

Certain semigroups are known to admit a `strong semilattice decomposition' into simpler pieces. We introduce a class of Banach algebras that generalise the $\ell^1$-convolution algebras of such semigroups, and obtain a disintegration…

Following T.-J. Li, W. Zhang [Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom.], we continue to study the link between the cohomology of an almost-complex manifold…

Assuming a certain "purity" conjecture, we derive a formula for the (complex) cohomology groups of the affine Springer fiber corresponding to any unramified regular semi-simple element. We use this calculation to present a complex analog of…

In the first part of this paper, we give a new analytical proof of a theorem of C. Sabbah on integrable deformations of meromorphic connections on $\mathbb P^1$ with coalescing irregular singularities of Poincar\'e rank 1, and generalizing…

We establish simplicial triviality of the convolution algebra $\ell^1(S)$, where $S$ is a band semigroup. This generalizes results of the first author [Glasgow Math. J. 2005, Houston J. Math. 2010]. To do so, we show that the cyclic…

We prove some extension theorems for analytic objects, in particular sections of a coherent sheaf, defined in semi q-coronae of a complex space. Semi q-coronae are domains whose boundary is the union of a Levi flat part, a q-pseudoconvex…

We study Frobenius-Schur indicators of the regular representations of finite-dimensional semisimple Hopf algebras, especially group-theoretical ones. Those of various Hopf algebras are computed explicitly. In view of our computational…

Let $p,q$ be prime numbers with $p>q^3$, and $k$ an algebraically closed field of characteristic 0. In this paper, we obtain the structure theorems for semisimple Hopf algebras of dimension $pq^3$.

In this note we prove a generalization of the Frobenius-Schur theorem for finite groups for the case of semisimple Hopf algebra over an algebraically closed field of characteristic 0. A similar result holds in characteristic $p > 2$ if the…

In this article formulas for the quantum product of a rational surface are given, and used to give an algebro-geometric proof of the associativity of the quantum product for strict Del Pezzo surfaces, those for which $-K$ is very ample. An…