arXiv subjects
Publications and source records attributed to F. Laytimi.
On the flag variety $ \mathcal{F}l_s(E)$ associated to a vector bundle $E,$ , a sequence $s$ and a partition $a,$ there is a line bundle $\it Q^a_s$ on $ \mathcal{F}l_s(E).$ The aim of this paper is to prove the following conjecture: $Q^a_s $ on $ \mathcal{F}l_s(E)$ is big if only if $\pi_*(Q^a_s)=S_a(E)$ on X is big. The "if" part is proven here, the "only if" part is proven under the V-bigness hypothesis.
Hartshorne in "Ample vector bundles" proved that $E$ is ample if and only if $\OOO_{P(E)}(1)$ is ample. Here we generalize this result to flag manifolds associated to a vector bundle $E$ on a complex manifold $X$: For a partition $a$ we show that the line bundle $\it Q_a^s$ on the corresponding flag manifold $\mathcal{F}l_s(E)$ is ample if and only if $ \SSS_aE $ is ample. In particular $\det Q$ on $\it{G}_r(E)$ is ample if and only if $\wedge ^rE$ is ample.\\ We give also a proof of the Ampleness Dominance theorem that does not depend on the saturation property of the Littlewood-Richardson semigroup.
In this article we prove a general result on a nef vector bundle $E$ on a projective manifold $X$ of dimension $n$ depending on the vector space $H^{n,n} (X, E). $ It is also shown that $H^{n,n} (X, E)=0$ for an indecomposable nef rank 2 vector bundles $E$ on some specific type of $n$ dimensional projective manifold $X.$ The same vanishing shown to hold for indecomposable nef and big rank 2 vector bundles on any variety with trivial canonical bundle.
We show that tensor products of semiample vector bundles are semiample. For k-ampleness in the sens of Sommese, we show that over compact complex manifolds tensor products of semiample and k-ample vector bundles are k-ample, and the sum of k-ample vector bundles is k-ample. In particular results of Sommese on k-ampleness are strenghtened.
In this note we describe the image of $\PP^2$ in $ Gr(2, \CC^{4})$ under a morphism given by a rank two vector bundle on $\PP^2$ with Chern classes $(2,2).$
We prove a new vanishing theorem generalizing that of Le Potier for Schur functors of a vector bundle.
For ample vector bundles $E$ over compact complex varieties $X$ and a Schur functor $S_I$ corresponding to an arbitrary partition $I$ of the integer $|I|$, one would like to know the optimal vanishing theorem for the cohomology groups $H^{p,q}(X, S_I(E))$, depending on the rank of $E$ and the dimension $n$ of $X$. Three years ago (Nov. 1995), in an unpublished paper one of us (W.N.) proved a vanishing theorem for the situation where the partition $I$ is a hook. Here we give a simpler proof of this theorem. We also treat the same problem under weaker positivity assumptions, in particular under the hypothesis of ample $Λ^m E$ with $m\in \N^*$. In this case we also need some bound on the weight $|I|$ of the partition. Moreover, we prove that the same vanishing condition applies for $H^{q,p}(X, S_I(E))$, with $p,q$ interchanged.