We investigate the role of Hertling-Manin condition on the structure constants of an associative commutative algebra in the theory of integrable systems of hydrodynamic type. In such a framework we introduce the notion of $F$-manifold with compatible connection generalizing a structure introduced by Manin.

O’Grady showed that certain special sextics in ${\mathbb{P}}^5$ called EPW sextics admit smooth double covers with a holomorphic symplectic structure. We propose another perspective on these symplectic manifolds, by showing that they can be constructed from the Hilbert schemes of conics on Fano fourfolds of degree ten. As applications, we construct families of Lagrangian surfaces in these symplectic fourfolds, and related integrable systems whose fibers are intermediate Jacobians.

Soit V une variété close de dimension 3. Dans cet article, on montre que les classes dhomotopie de champs de plans sur V qui contiennent des structures de contact tendues sont en nombre fini et que, si V est atoroïdale, les classes disotopie des structures de contact tendues sur V sont elles aussi en nombre fini.

In this paper we study topological properties of stable Hamiltonian structures. In particular, we prove the following results in dimension three: The space of stable Hamiltonian structures modulo homotopy is discrete; stable Hamiltonian structures are generically Morse-Bott (i.e. all closed orbits are Bott nondegenerate) but not Morse; the standard contact structure on $S^3$ is homotopic to a stable Hamiltonian structure which cannot be embedded in ${ℝ}^4$. Moreover, we derive a structure theorem for stable...

In this note, we study formal deformations of derived representations of the principal series representations of $\mathrm{SL}(2,ℝ)$. In particular, we recover all the representations of the derived principal series by deforming one of them. Similar results are also obtained for $\mathrm{SL}(2,ℂ)$.

Let $K$ be a compact Lie group acting in a Hamiltonian way on a symplectic manifold $(M,\Omega )$ which is pre-quantized by a Kostant-Souriau line bundle. We suppose here that the moment map $\Phi$ is proper so that the reduced space $M_{\mu }:={\Phi }^{-1}(K·\mu )/K$ is compact for all $\mu$. Then, we can define the “formal geometric quantization” of $M$ as$${𝒬}_K^{-\infty }\left(M\right):=\sum _{\mu \in \widehat{K}}𝒬\left(M_{\mu }\right)V_{\mu }^K.$$The aim of this article is to study the functorial properties of the assignment $(M,K)\to {𝒬}_K^{-\infty }\left(M\right)$.

We review topological properties of Kähler and symplectic manifolds, and of their odd-dimensional counterparts, coKähler and cosymplectic manifolds. We focus on formality, Lefschetz property and parity of Betti numbers, also distinguishing the simply-connected case (in the Kähler/symplectic situation) and the b1 = 1 case (in the coKähler/cosymplectic situation).

Let $M$ be a differential manifold. Let $\Phi$ be a Drinfeld associator. In this paper we explain how to construct a global formality morphism starting from $\Phi$. More precisely, following Tamarkin’s proof, we construct a Lie homomorphism “up to homotopy" between the Lie algebra of Hochschild cochains on $C^{\infty }\left(M\right)$ and its cohomology $(\Gamma (M,\Lambda TM),{[-,-]}_S$). This paper is an extended version of a course given 8 - 12 March 2004 on Tamarkin’s works. The reader will find explicit examples, recollections on $G_{\infty }$-structures, explanation of the...

2000 Mathematics Subject Classification: 37J55, 53D10, 53D17, 53D35.In this paper, we study contact forms on a 3-manifold having a common Reeb vector field R. The main result is that when the contact forms induce the same orientation, they are diffeomorphic.

We study the parametrized Hamiltonian action functional for finite-dimensional families of Hamiltonians. We show that the linearized operator for the $L^2$-gradient lines is Fredholm and surjective, for a generic choice of Hamiltonian and almost complex structure. We also establish the Fredholm property and transversality for generic $S^1$-invariant families of Hamiltonians and almost complex structures, parametrized by odd-dimensional spheres. This is a foundational result used to define $S^1$-equivariant...