Ample vector bundles and Bordiga surfaces

Let X be a smooth complex projective variety and let Z ? X be a smooth surface, which is the zero locus of a section of an ample vector bundle ? of rank dimX – 2 ≥ 2 on X. Let H be an ample line bundle on X, whose restriction H _Z to Z is a very ample line bundle and assume that (Z, H _Z ) is a Bordiga surface, i.e., a rational surface having (?², ?? (4)) as its minimal adjunction theoretic reduction. Triplets (X, ?, H) as above are discussed and classified. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the adjunction mapping of very ample vector bundles of corank one

Let be a very ample vector bundle of rank over a smooth complex projective variety of dimension . The structure of being known when , we investigate the structure of the adjunction mapping when .

     

Chern numbers of ample vector bundles on toric surfaces

This article shows a number of strong inequalities that hold for the Chern numbers , of any ample vector bundle of rank on a smooth toric projective surface, , whose topological Euler characteristic is . One general lower bound for proven in this article has leading term . Using Bogomolov instability, strong lower bounds for are also given. Using the new inequalities, the exceptions to the lower bounds 4e(S)$"> and e(S)$"> are classified.

     

Ample and spanned vector bundles of sectional genera three

Work partially supported by the M.P.I. of the Italian Government     

Vector bundles on flag varieties

We study vector bundles on flag varieties over an algebraically closed field k. In the first part, we suppose $G=G_k(d,n)$ $(2\le d\le n-d)$ to be the Grassmannian parameterizing linear subspaces of dimension d in $k^n$, where k is an algebraically closed field of characteristic $p>0$. Let E be a uniform vector bundle over G of rank $r\le d$. We show that E is either a direct sum of line bundles or a twist of the pullback of the universal subbundle $H_d$ or its dual $H_d^{\vee }$ by a series of absolute Frobenius maps. In the second part, splitting properties of vector bundles on general flag varieties $F(d_1,\text{…},d_s)$ in characteristic zero are considered. We prove a structure theorem for bundles over flag varieties which are uniform with respect to the ith component of the manifold of lines in $F(d_1,\text{…},d_s)$. Furthermore, we generalize the Grauert–M$\ddot{\mathrm{u}}$lich–Barth theorem to flag varieties. As a corollary, we show that any strongly uniform i-semistable $(1\le i\le n-1)$ bundle over the complete flag variety splits as a direct sum of special line bundles.     

On the Cayley-Bacharach property and the construction of vector bundles

We study the Cayley-Bacharach property on complex projective smooth varieties of dimension n≥2 for zero dimensional subscheme defined by the zero set of the wedge of r-n + 1 global sections of a rank r≥n vector bundle,and give a construction of high rank reflexive sheaves and vector bundles from codimension 2 subschemes.     

Exceptional vector bundles on Enriques surfaces

The main purpose of this paper is to study exceptional vector bundles on Enriques surfaces. Translated fromMatematicheskie Zametki, Vol. 61, No. 6, pp. 825–834, June, 1997.     

A lower bound for the number of components of the moduli schemes of stable rank 2 vector bundles on projective 3-folds

Fix a smooth projective 3-fold , , with ample, and . Assume the existence of integers with such that is numerically equivalent to . Let be the moduli scheme of -stable rank 2 vector bundles, , on with and . Let be the number ofits irreducible components. Then .

     

ACM vector bundles on scrolls

Here we show that certain low rank ACM vector bundles on scrolls over smooth curves are iterated extensions of line bundles. Partially supported by MIUR and GNSAGA of INDAM (Italy)     

Ample vector bundles with zero loci having a bielliptic curve section of low degree

Let be an ample vector bundle of rank r ≥ 2 on a smooth complex projective variety X of dimension n such that there exists a global section of whose zero locus Z is a smooth subvariety of dimension n − r ≥ 3 of X. Let H be an ample line bundle on X such that its restriction H _Z to Z is very ample. Triplets are classified under the assumption that (Z,H _Z ) has a smooth bielliptic curve section of genus ≥ 3 with .      

Rank two vector bundles with canonical determinant

Denote by B^k_2,K the locus of vector bundles of rank two that have canonical determinant and at least k sections. We show that for a generic curve of genus g,B^k_2,K is non‐empty and has a component of the expected dimension if g is sufficiently large. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Classification and syzygies of smooth projective varieties with 2-regular structure sheaf

The geometric and algebraic properties of smooth projective varieties with 1-regular structure sheaf are well understood, and the complete classification of these varieties is a classical result. The aim of this paper is to study the next case: smooth projective varieties with 2-regular structure sheaf. First, we give a classification of such varieties using adjunction mappings. Next, under suitable conditions, we study the syzygies of section rings of those varieties to understand the structure of the Betti tables, and show a sharp bound for Castelnuovo–Mumford regularity.     

A note on tensor products of ample line bundles on abelian varieties

Let be a polarized abelian variety defined over the complex number field. Then we classify with such that is not -jet ample nor -very ample.

     

Holomorphic and algebraic vector bundles on 0-convex algebraic surfaces

LetX be a smooth complex algebraic surface such that there is a proper birational morphism/:X → Y withY an affine variety. Let X_hol be the 2-dimensional complex manifold associated toX. Here we give conditions onX which imply that every holomorphic vector bundle onX is algebraizable and it is an extension of line bundles. We also give an approximation theorem of holomorphic vector bundles on X_hol (X normal algebraic surface) by algebraic vector bundles.     

Holomorphic vector bundles on a neighborhood of non-Stein compact subsets of Stein spaces

LetK be a compact subset of a complex spaceX. Here we give conditions onX andK assuring the existence of a fundamental systemU of open neighborhoods, ofK such that for everyU∈U there is a holomorphic vector bundleE onU which is not holomorphically trivial.

---

Sunto SiaX uno spazio complesso eK∩X un compatto. In questo lavoro diamo condizioni suX eK che garantiscono l'esistenza di un sistema fondamentale di intorni apertiU diK inX tali che per ogniU∈U esiste un fibrato olomorfo non-triviale suU.

     

Nefness of adjoint bundles for ample vector bundles of corank 3

Let be an ample vector bundle of rank on a smooth complex projective variety X of dimension n. The aim of this paper is to describe the structure of pairs as above whose adjoint bundles are not nef for . Furthermore, we give some immediate consequences of this result in adjunction theory.     

Finite morphisms of projective and K?hler manifolds

In this paper we study finite morphisms of projective and compact K?hler manifolds, in particular, positivity properties of the associated vector bundle, deformation theory and ramified endomorphisms. Dedicated to Professor LU QiKeng on the occasion of his 80th birthday