Multigraded regularity: syzygies and fat points.
Nef value of homogeneous line bundles and related vanishing theorems.
On a Grauert-Riemenschneider vanishing theorem for Frobenius split varieties in characteristic p.
On abelian subvarieties generated by symmetric correspondences.
On connected divisors.
On the Castelnuovo-Mumford regularity of products of ideal sheaves.
On the k-regularity of some proyective manifolds.
The conjecture on the (degree-codimension + 1) - regularity of projective varieties is proved for smooth linearly normal polarized varieties (X,L) with L very ample, for low values of Delta(X,L) = degree-codimension-1. Results concerning the projective normality of some classes of special varieties including scrolls over curves of genus 2 and quadric fibrations over elliptic curves, are proved.
Positivity on subvarieties and vanishing of higher cohomology
We study the relationship between positivity of restriction of line bundles to general complete intersections and vanishing of their higher cohomology. As a result, we extend classical vanishing theorems of Kawamata–Viehweg and Fujita to possibly non-nef divisors.
Pre-Tango structures and uniruled varieties
The pre-Tango structure is an ample invertible sheaf of locally exact differentials on a variety of positive characteristic. It is well known that pre-Tango structures on curves often induce pathological uniruled surfaces. We show that almost all pre-Tango structures on varieties induce higher-dimensional pathological uniruled varieties, and that each of these uniruled varieties also has a pre-Tango structure. For this purpose, we first consider the p-closed rational vector field induced...
Regularity bounds by minimal generators and Hilbert function.
Rings of global sections in two-dimensional schemes.
Some consequences of perversity of vanishing cycles
For a holomorphic function on a complex manifold, we show that the vanishing cohomology of lower degree at a point is determined by that for the points near it, using the perversity of the vanishing cycle complex. We calculate this order of vanishing explicitly in the case the hypersurface has simple normal crossings outside the point. We also give some applications to the size of Jordan blocks for monodromy.
Some remarks on multiplier ideals and vector bundles.
Some vanishing for the cohomology of stable rank two vector bundles on P3.
Sur la cohomologie des fibrés associés au fibré quotient universel sur la grassmannienne
Tensor products of modules, rigidity and local cohomology.
Théorèmes d’annulation pour des fibrés munis d’une forme non dégénérée
Je démontre des théorèmes d’annulation de la cohomologie de Dolbeault de fibrés vectoriels amples sur une variété projective lisse, munis d’une forme symplectique ou d’une forme quadratique non-dégénérée à valeurs dans un fibré en droites. L’hypothèse d’existence d’une telle forme permet d’améliorer les résultats similaires précédents. Je fais aussi des remarques sur la cohomologie des fibrés en droites sur les grassmanniennes isotropes.
Théorèmes d'annulation sur certaines variétés projectives.
Tight closure and the Kodaira Vanishing Theorem.
T-symmetrical tensor forms on complete intersections.