arXiv subjects

Philippe Eyssidieux

Publications and source records attributed to Philippe Eyssidieux.

23 records · Page 2Linked to original sources

arXiv2009

Variations of Mixed Hodge Structure attached to the deformation theory of a Complex Variation of Hodge Structures

We construct a Mixed Hodge Structure on the local complete ring of the representation scheme at the holonomy of a VHS on a compact Kähler manifold and prove that the corresponding tautological representation is the holonomy of a VMHS. In order to carry out this construction we revisit the well-known work of Goldman and Millson on the deformation theory of representations of Kähler groups.

math.AG↗

arXiv2008

Singular Kahler-Einstein metrics

We study degenerate complex Monge-Ampère equations of the form $(ω+dd^c φ)^n = e^{t φ} μ$ where $ω$ is a big semi-positive form on a compact Kähler manifold $X$ of dimension $n$, $t \in \R^+$, and $μ=fω^n$ is a positive measure with density $f\in L^p(X,ω^n)$, $p>1$. We prove the existence and unicity of bounded $ω$-plurisubharmonic solutions. We also prove that the solution is continuous under a further technical condition. In case $X$ is projective and $ω=ψ^*ω'$, where $ψ:X\to V$ is a proper birational morphism to a normal projective variety, $[ω']\in NS_{\R} (V)$ is an ample class and $μ$ has only algebraic singularities, we prove that the solution is smooth in the regular locus of the equation. We use these results to construct singular Kähler-Einstein metrics of non-positive curvature on projective klt pairs, in particular on canonical models of algebraic varieties of general type.

math.AG↗

arXiv2004

On the Validity or Failure of Gap Rigidity for Certain pairs of Bounded Symmetric Domains

In our previous work "Characterization of certain homorphic geodesic cycles on Hermitian locally symmetric manifolds of the noncompact type" in "Modern methods in Complex Analysis" Annals of Math. Studies 138 (1995) 85-118, we formulated a conjecture: the so called "gap phenomenon". The purpose of the article is two-fold. We give a counterexample to the gap phenomenon in the most general situation. We give new examples of situations in which the gap phenomenon holds and a unified conceptual, hopefully definitive, presentation of these situations. In the last section of the article, we survey some open problems connected to the gap phenomenon.

math.AG↗

arXiv1998

Un théoréme de Nakai-Moishezon pour certaines classes de type (1,1)

Let $X$ be a smooth compact projective variety over $\mathbb C$. Let $H^2(π_1(X),\mathbb R)^{1,1}$ be the intersection of $H^{1,1}(X,{\mathbb R})$ with the image of the map $H^2(π_1(X),{\mathbb R})\to H^2(X)$ induced by the classifying map $X\to Bπ_1(X)$. Let $NS(X)$ be the Néron-Severi group of $X$. Let $[ω]\in H^2(π_1(X),\mathbb R)^{1,1}+ NS(X)\otimes {\mathbb R}$. In this note, we prove that $[ω]$ is the cohomology class of a Kähler metric if and only if for every $d$-dimensional reduced closed algebraic subvariety $Z\subset X$, $[ω]^d.Z>0$.

math.AG↗

arXiv1998

Invariants de Von Neumann des faisceaux coherents

Inspired by some recent work of M. Farber, W. Lück and M. Shubin on L2 homotopy invariants of infinite Galois coverings of simplicial complexes (L2 Betti numbers and Novikov-Shubin invariants), this article extends Atiyah's L2 index theory to coherent analytic sheaves on complex analytic spaces. Let $X$ be a complex analytic space with a proper cocompact biholomorphic action of a discrete group $G$. Let $F$ be a $G$-equivariant coherent analytic sheaf on $X$. We give a meaningful notion of a L2 section of $F$ on $X$. We also construct L2 cohomology groups. We prove that these L2 cohomology groups belong to an abelian category of topological $G$-modules introduced by M. Farber. On this category there are two kinds of invariants: Von Neumann dimension and Novikov-Shubin invariants. The alternating sum of the Von Neumann dimensions of the L2 cohomology groups of $F$ can be computed by an analogue of Atiyah's L2 index theorem. Novikov-Shubin invariants show up when the L2 cohomology groups are non-Hausdorff and, like in algebraic topology, are still very intriguing (and not very well understood).

math.AG↗