Here we give conditions and examples for the surjectivity or injectivity of the restriction map $H^0(X,F)\to H^0(Z,F|Z)$, where $X$ is a projective variety, $F$ is a vector bundle on $X$ and $Z$ is a “general” $0$-dimensional subscheme of $X$, $Z$ union of general “fat points”.

Soit $X$ une variété algébrique projective lisse irréductible. On appelle variété de modules fins de faisceaux sur $X$ une famille $ℱ$ de faisceaux cohérents sur $X$ paramétrée par une variété intègre $M$, possédant les propriétés suivantes : $ℱ$ est plate sur $M$; pour tous $x,y\in M$ distincts, les faisceaux ${ℱ}_x$ et ${ℱ}_y$ sur $X$ ne sont pas isomorphes et $ℱ$ est une déformation complète de ${ℱ}_x$; enfin $ℱ$ possède une propriété universelle locale évidente. On a aussi la notion de variété de modules fins définie localement, où $ℱ$ est...

We study various "generic" nefness and ampleness notions for holomorphic vector bundles on a projective manifold. We apply this in particular to the tangent bundle and investigate the relation to the geometry of the manifold.

In this article we establish an analogue of the Barth-Van de Ven-Tyurin-Sato theorem.We prove that a finite rank vector bundle on a complete intersection of finite codimension in a linear ind-Grassmannian is isomorphic to a direct sum of line bundles.

Here we study vector bundles E on the Hirzebruch surface F e such that their twists by a spanned, but not ample, line bundle M = $$𝒪$$ Fe(h + ef) have natural cohomology, i.e. h 0(F e, E(tM)) > 0 implies h 1(F e, E(tM)) = 0.

We study relatively semi-stable vector bundles and their moduli on non-Kähler principal elliptic bundles over compact complex manifolds of arbitrary dimension. The main technical tools used are the twisted Fourier-Mukai transform and a spectral cover construction. For the important example of such principal bundles, the numerical invariants of a 3-dimensional non-Kähler elliptic principal bundle over a primary Kodaira surface are computed.