Restriction properties for the Krull–Schmidt decomposition of vector bundles

We obtain decomposability criteria for vector bundles on smooth projective varieties [Formula: see text] by comparing the Krull–Schmidt decomposition on [Formula: see text], on one hand, and along the vanishing locus of a section in an ample vector bundle over [Formula: see text], on the other hand. We determine effective bounds for the amplitude of and also genericity conditions for its sections which ensure that the irreducible components of and those of its restriction correspond bijectively. Moreover, we get a simple splitting criterion for vector bundles on partial flag varieties.

The threefold containing the Bordiga surface of degree ten as a hyperplane section

Let be a very ample vector bundle of rank 2 on $\Bbb P^2$ with c1() = 4 and c2() = 6. Then it is proved that is the cokernel of a bundle monomorphism $\mathcal O_{\Bbb P^2}(1)^{\oplus 2}\to T_{\Bbb P^2}^{\oplus 2}$, where $T_{\Bbb P^2}$ is the tangent bundle of $\Bbb P^2$. This gives a new example of a threefold containing a Bordiga surface as a hyperplane section.

Projective manifolds of sectional genus three as zero loci of sections of ample vector bundles

Let ϵ 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 ϵ whose zero locus Z is a smooth subvariety of dimension n-r ≥ 2 of X. Let H be an ample line bundle on X such that the restriction HZ of H to Z is very ample. Triplets (X, ϵ, H) with g(Z, HZ) = 3 are classified, where g(Z, HZ) is the sectional genus of (Z, HZ).

A Vanishing Theorem

The main result is a general vanishing theorem for the Dolbeault cohomology of an ample vector bundle obtained as a tensor product of exterior powers of some vector bundles. It is also shown that the conditions for the vanishing given by this theorem are optimal for some parameter values.

Some Adjunction Properties of Ample Vector Bundles

Let ε be an ample vector bundle of rank r on a projective variety X with only log-terminal singularities. We consider the nefness of adjoint divisors KX +(t−r) det ε when t ≥ dim X and t > r. As an application, we classify pairs (X, ε) with cr-sectional genus zero.

Special varieties in adjunction theory and ample vector bundles

In this paper varieties are always assumed to be defined over the field [Copf ] of complex numbers.Given a smooth projective variety Z, the classification of smooth projective varieties X containing Z as an ample divisor occupies an extremely important position in the theory of polarized varieties and it is well-known that the structure of Z imposes severe restrictions on that of X. Inspired by this philosophy, we set up the following condition ([midast ]) in [LM1] in order to generalize several results on ample divisors to ample vector bundles:([midast ]) [Escr ] is an ample vector bundle of rank r [ges ] 2 on a smooth projective variety X of dimension n such that there exists a global section s ∈ Γ([Escr ]) whose zero locus Z = (s)0 is a smooth subvariety of X of dimension n − r [ges ] 1.

AMPLE VECTOR BUNDLES WITH SECTIONS VANISHING ON SPECIAL VARIETIES

Let ε be an ample vector bundle of rank r on a complex projective manifold X such that there exists a section s ∈ Γ(ε) whose zero locus Z = (s = 0) is a smooth submanifold of the expected dimension dim X-r:= n - r. Assume that Z is not minimal; we investigate the hypothesis under which the extremal contractions of Z can be lifted to X. Finally we study in detail the cases in which Z is a scroll, a quadric bundle or a del Pezzo fibration.

Ample Vector Bundles of Curve Genus One

We investigate the pairs (X, ε) consisting of a smooth complex projective variety X of dimension n and an ample vector bundle ε of rank n − 1 on X such that ε has a section whose zero locus is a smooth elliptic curve.