Related papers: Quantum Cohomology of Hilb^2(P^1xP^1) and Enumerat…

We compute quantum cohomology ring of elliptic $\mathbb{P}^1$ orbifolds via orbi-curve counting. The main technique is the classification theorem which relates holomorphic orbi-curves with certain orbifold coverings. The countings of…

Let $g$ be an even positive integer, and $p$ be a prime number. We compute the cohomological invariants with coefficients in $\mathbb{Z}/p\mathbb{Z}$ of the stacks of hyperelliptic curves $\mathscr{H}_g$ over an algebraically closed field…

We describe a method for counting maps of curves of given genus (and variable moduli) to $\Bbb P^2$, essentially by splitting the $\Bbb P^2$ in two; then specialising to the case of genus 0 we show that the method of quantum cohomology may…

We compute the Gromov-Witten invariants of the projective plane blown up in r general points. These are determined by associativity from r+1 intial values. Applications are given to the enumeration of rational plane curves with prescribed…

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…

The $L^p$-cohomology in degree 1 of Riemannian homogeneous spaces is computed. It turns out that reduced cohomology does not vanish exactly for spaces quasiisometric to negatively curved homogeneous spaces.

This paper studies the (small) quantum homology and cohomology of fibrations $p: P\to S^2$ whose structural group is the group of Hamiltonian symplectomorphisms of the fiber $(M,\om)$. It gives a proof that the rational cohomology splits…

We determine the two-point invariants of the equivariant quantum cohomology of the Hilbert scheme of points of surface resolutions associated to type A_n singularities. The operators encoding these invariants are expressed in terms of the…

Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $G$ be a finite $p$-group. The results of Harbater, Katz and Gabber associate a $G$-cover of the projective line ramified only over $\infty$ to every $k$-linear…

We compute the cohomological invariants with coefficients in $\mathbb{Z}/p\mathbb{Z}$ of the stack $\mathscr{H}_3$ of hyperelliptic curves of genus $3$ over an algebraically closed field.

Let V be a symplectic vector space and LG be the Lagrangian Grassmannian which parametrizes maximal isotropic subspaces in V. We give a presentation for the (small) quantum cohomology ring QH^*(LG) and show that its multiplicative structure…

For any hyperelliptic curve X, we give an explicit basis of the first de-Rham cohomology of X in terms of \v{C}ech cohomology. We use this to produce a family of curves in characteristic p>2 for which the Hodge-de-Rham short exact sequence…

Within the framework of deformation quantization, a first step towards the study of star-products is the calculation of Hochschild cohomology. The aim of this article is precisely to determine the Hochschild homology and cohomology in two…

We give elementary geometric proofs of the main theorems about the (small) quantum cohomology of partial flag varieties SL(n)/P, including the quantum Pieri and quantum Giambelli formulas and the presentation.

We determine the ring structure of the equivariant quantum cohomology of the Hilbert scheme of points in the complex plane. The operator of quantum multiplication by the divisor class is a nonstationary deformation of the quantum…

We investigate space curves with large cohomology. To this end we introduce curves of subextremal type. This class includes all subextremal curves. Based on geometric and numerical characterizations of curves of subextremal type, we show…

We study possible real structures in the space of solutions to the quantum differential equation. We show that, under mild conditions, a real structure in orbifold quantum cohomology yields a pure and polarized tt^*-geometry near the large…

The (small) quantum cohomology ring of a flag manifold F encodes enumerative geometry of rational curves on F. We give a proof of the presentation of the ring and of a quantum Giambelli formula, which is more direct and geometric than the…

When the genus $g$ is even, we extend the computation of mod 2 cohomological invariants of $\mathcal{H}_g$ to non algebraically closed fields, we give an explicit functorial description of the invariants and we completely describe their…

We apply classical invariant theory of binary forms to explicitly characterize isomorphism classes of hyperelliptic curves of small genus and, conversely, propose algorithms for reconstructing hyperelliptic models from given invariants. We…