Julien Grivaux

Chern classes in Deligne cohomology for coherent analytic sheaves

In this article, we construct Chern classes in rational Deligne cohomology for coherent sheaves on a smooth complex compact manifold. We prove that these classes satisfy the functoriality property under pullbacks, the Whitney formula and the Grothendieck–Riemann–Roch theorem for projective morphisms between smooth complex compact manifolds.

Loci in strata of meromorphic quadratic differentials with fully degenerate Lyapunov spectrum

We construct explicit closed GL(2, R)-invariant loci in strata of meromorphic differentials of arbitrary large dimension with fully degenerate Lyapunov spectrum. This answers a question of Forni-Matheus-Zorich.

The Hochschild–Kostant–Rosenberg Isomorphism for Quantized Analytic Cycles

In this article, we provide a detailed account of a construction sketched by Kashiwara in an unpublished manuscript concerning generalized HKR isomorphisms for smooth analytic cycles whose conormal exact sequence splits. It enables us, among other applications, to solve a problem raised recently by Arinkin and C\u{a}ld\u{a}raru about uniqueness of such HKR isomorphisms in the case of the diagonal injection. Using this construction, we also associate with any smooth analytic cycle endowed with an infinitesimal retraction a cycle class which is an obstruction for the cycle to be the vanishing locus of a transverse section of a holomorphic vector bundle.

Tian’s invariant of the Grassmann manifold

We prove that Tian’s invariant on the complex Grassmann manifold Gp,q(ℂ)is equal to 1/(p+ q).The method introduced here uses a Lie group of holomorphic isometries which operates transitively on the considered manifolds and a natural imbedding of (ℙ1 (ℂ))pin Gp,q (ℂ).RésuméOn prouve que l’invariant de Tian sur la grassmannienne Gp,q (ℂ)est 1/(p+ q).La méthode présentée dans cet article utilise un groupe de Lie d’isométries holomorphes qui opère transitivement sur les variétés considérées ainsi qu’un plongement naturel de (ℙ1(ℂ))p dans Gp,q (ℂ).

Variation of the holomorphic determinant bundle

In this paper, we prove that the Grothendieck-Riemann-Roch formula in Deligne cohomology computing the determinant of the cohomology of a holomorphic vector bundle on the fibers of a proper submersion between abstract complex manifolds is invariant by deformation of the bundle.

Topological properties of Hilbert schemes of almost-complex four-manifolds II

In this article, we study the rational cohomology rings of Voisin’s Hilbert schemes X anc associated with a symplectic compact four-manifold X . We prove that these rings can be universally constructed from H .X;Q/ and c1.X/, and that Ruan’s crepant resolution conjecture holds if c1.X/ is a torsion class. Next, we prove that for any almost-complex compact four-manifold X , the complex cobordism class of X anc depends only on the complex cobordism class of X . 32Q60; 14C05, 14J35

Topological properties of Hilbert schemes of almost-complex fourfolds (I)

In this article, we study topological properties of Voisin’s Hilbert schemes of an almost-complex four-manifold X. We compute in this setting their Betti numbers and construct Nakajima operators. We also define tautological bundles associated with any complex bundle on X, which are shown to be canonical in K–theory.