Searcharxiv⌕ Search

arXiv subjects

O. Debarre

Publications and source records attributed to O. Debarre.

11 recordsLinked to original sources
arXiv2013

Pseudo-effective classes and pushforwards

Given a morphism between complex projective varieties, we make several conjectures on the relations between the set of pseudo-effective (co)homology classes which are annihilated by pushforward and the set of classes of varieties contracted by the morphism. We prove these conjectures for classes of curves or divisors. We also prove that one of these conjectures implies Grothendieck's generalized Hodge conjecture for varieties with Hodge coniveau at least 1.

math.AG↗
arXiv2011

Varieties With Ample Cotangent Bundle

We study smooth projective complex varieties with ample cotangent bundle. Our main result is that in an abelian variety of dimension n, a complete intersection of at least n/2 general hypersurfaces of sufficiently high degrees has ample cotangent bundle. We discuss the conjecture that the analogous statement should hold in the projective space. Finally, we present a construction due to Bogomolov of varieties with ample cotangent bundle as linear sections of a product of varieties with big cotangent bundle.

math.AG↗
arXiv2011

Varieties with vanishing holomorphic Euler characteristic

We study smooth complex projective varieties $X$ of maximal Albanese dimension and of general type satisfying with vanishing holomorphic Euler characteristic. We prove that the Albanese variety of $X$ has at least three simple factors. Examples were constructed by Ein and Lazarsfeld, and we prove that in dimension 3, these examples are (up to abelian étale covers) the only ones. By results of Ueno, another source of examples is provided by varieties $X$ of maximal Albanese dimension and of general type with $p_g(X)=1$. Examples were constructed by Chen and Hacon, and again, we prove that in dimension 3, these examples are (up to abelian étale covers) the only ones. We also formulate a conjecture on the general structure of these varieties in all dimensions.

math.AG↗
arXiv2008

On the period map for prime Fano threefolds of degree 10

We prove that the deformations of a smooth complex Fano threefold X with Picard number 1, index 1, and degree 10, are unobstructed. The differential of the period map has two-dimensional kernel. We construct two two-dimensional components of the fiber of the period map through X: one is isomorphic to the variety of conics in X, modulo an involution, another is birationally isomorphic to a moduli space of semistable rank-2 torsion-free sheaves on X, modulo an involution. The threefolds corresponding to points of these components are obtained from X via conic and line (birational) transformations.

math.AG↗
arXiv2002

Sur une conjecture de Mukai

Generalizing a question of Mukai, we conjecture that a Fano manifold $X$ with Picard number $ρ_X$ and pseudo-index $ι_X$ satisfies $ρ_X (ι_X-1) \le \dim(X)$. We prove this inequality in several situations: $X$ is a Fano manifold of dimension $\le 4$, $X$ is a toric Fano manifold of dimension $\le 7$ or $X$ is a toric Fano manifold of arbitrary dimension with $ι_X \ge \dim(X)/3+1$. Finally, we offer a new approach to the general case.

math.AG↗
arXiv1996

Schémas de Fano

Let X be a subvariety of $P^n$ defined by equations of degrees $ d =(d_1,...,d_s)$, over an algebraically closed field k of any characteristic. We study properties of the Fano scheme $F_r(X)$ that parametrizes linear subspaces of dimension r contained in X. We prove that $F_r(X)$ is connected and smooth of the expected dimension for n big enough (this was previously known in characteristic 0 or for r=1). Using Bott's theorem, we prove a vanishing theorem for certain bundles on the Grassmannian and use it to calculate the cohomology groups of $F_r(X)$ in degree $\le \dim X-2r$, and to prove that $F_r(X)$ is projectively normal in the Grassmannian. Finally, we prove that for n big enough, the rational Chow group $A_1(F_r(X))$ is of rank 1, and $F_r(X)$ is unirational. All bounds on n are effective.

alg-geom↗
arXiv1993

Very ample linear systems on abelian varieties

Let $(X,L)$ be a polarized complex abelian variety of dimension $g$ where $L$ is a polarization of type $(1,...,1,d)$. For $(X,L)$ genberic we prove the following: (1) If $d \ge g+2$, then $ϕ_L\colon X \to {\bf P}^{d-1}$ defines a birational morphism onto its image. (2) If $d > 2^g$, then $L$ is very ample. We show the latter by checking it on a suitable rank-$(g-1)$-degeneration.

alg-geom↗