Extension of holomorphic vector bundles across a boundary point

Let U ? C ⁿ , n ≥ 3, be a domain and P ∈ ?U such that U is 2-concave at P. Here we prove the existence of a holomorphic vector bundle on U which does not extend across P, but it extends across every Q ∈ ?U with Q ≠ P. We also prove a similar result taking a Stein space X instead of C ⁿ .     

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)     

Rank 2 totally arithmetically Cohen–Macaulay vector bundles on Hirzebruch surfaces

Here we study the totally arithmetically Cohen–Macaulay rank 2 vector bundles on any Hirzebruch surface F _e . E. Ballico was partially supported by MIUR and GNSAGA of INdAM (Italy).     

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.     

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.

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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.

     

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)     

Holomorphic vector bundles on compact complex non-algebraic surfaces

LetX be a smooth complex compact surface without non-constant meromorphic functions. Here we prove the existence of rank holomorphic vector bundles onX containing exactly one rank one saturated subsheaf.

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Sunto SiaX una superficie complessa compatta non singolare senza funzioni meromorfe non costanti. In questo lavoro si domstra cheX possiede molti fibrati olomorfi di rango 2 contenenti un unico fibrato in rette.

     

Some results on special stable vector bundles of rank 3 on algebraic curves

The authors discuss the existence and classification of stable vector bundles of rank 3, with 2 3 or 4 linearly independent holomorphic sections. The sets of all such bundles are denoted by ω3^2,d and w3 respectively. Our argument leads to sufficient and necessary conditions for the existence of both kinds of bundles. The conclusion is very interesting because of its contradiction to the conjectured dimension formula of stable bundles. Finally, we give a preliminary classification of ω3^2,4 and a complete discussion on the structure of ω3^3,2/3g＋2.     

Holomorphic triples of genus 0

Here we study the relationship between the stability of coherent systems and the stability of holomorphic triples over a curve of arbitrary genus. Moreover we apply these results to study some properties and give some examples of holomorphic triples on the projective line.      

Holomorphic functions and embedded real surfaces

The paper is devoted to the study of necessary and sufficient topological conditions for an embedded real surface to lie in a strictly pseudoconvex domain on a complex surface. These results are used to construct Stein domains on algebraic manifolds and to describe envelopes of holomorphy of real surfaces in P ² and in some other complex surfaces.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 599–606, April, 1998.The author is grateful to A. Vitushkin for the statement of the problem and for constant encouragement.This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-01002 and by the International Science Foundation under grant No. a97-1850.     

Vector bundles on Hirzebruch surfaces whose twists by a non-ample line bundle have natural cohomology

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 = _Fe (h + ef) have natural cohomology, i.e. h ⁰(F _e , E(tM)) > 0 implies h ¹(F _e , E(tM)) = 0.      

Vector bundles as direct images of line bundles

LetX be a smooth irreducible projective variety over an algebraically closed fieldK andE a vector bundle onX. We prove that, if dimX ≥ 1, there exist a smooth irreducible projective varietyZ overK, a surjective separable morphismf:Z →X which is finite outside an algebraic subset of codimension ≥ 3 inX and a line bundleL onX such that the direct image ofL byf is isomorphic toE. WhenX is a curve, we show thatZ, f, L can be so chosen thatf is finite and the canonical mapH ¹(Z, O) →H ¹(X, EndE) is surjective. Dedicated to the memory of Professor K G Ramanathan     

Regulous vector bundles

Among recently introduced new notions in real algebraic geometry is that of regulous functions. Such functions form a foundation for the development of regulous geometry. Several interesting results on regulous varieties and regulous sheaves are already available. In this paper, we define and investigate regulous vector bundles. We establish algebraic and geometric properties of such vector bundles, and identify them with stratified‐algebraic vector bundles. Furthermore, using new results on curve‐rational functions, we characterize regulous vector bundles among families of vector spaces parametrized by an affine regulous variety. We also study relationships between regulous and topological vector bundles.     

Some remarks about 1-convex manifolds on which all holomorphic line bundles are trivial

We give an abridged proof of an example already considered in [M. Col?oiu, On 1-convex manifolds with 1-dimensional exceptional set, Rev. Roumaine Math. Pures et Appl. 43 (1998) 97-104] of a 1-convex manifold X of dimension 3 such that all holomorphic line bundles on X are trivial. We also point out several mistakes of [Vo Van Tan, On the quasiprojectivity of compactifiable strongly pseudoconvex manifolds, Bull. Sci. Math. 129 (2005) 501-522] concerning this topic.     

Manifolds polarized by vector bundles

LetX be a complex projective manifold of dimension n and let ε be an ample vector bundle of rank r. Let also τ = τ (X,ε) = min {t ∈ ℝ : K_X + t det ε is nef} be the nef value of the pair (X, ε). In this paper we classify the pairs (X, ε) such that{ Mathematics Subject Classification (2000)14J60; 14J40; 14E30     

Ample and spanned vector bundles of top Chern number two on smooth projective varieties

The purpose of this paper is to classify ample and spanned vector bundles of top Chern number two on smooth projective varieties of arbitrary dimension defined over an algebraically closed field of characteristic zero.

     

Vector bundles on a product of real cubic curves

In this paper we give a characterization of then-tuples (C ₁,...,C _n ) of nonsingular projective real cubic curves such that every topological complex vector bundle onC ₁×...×C _n admits an algebraic structure. The results are very explicit and can be expressed in an especially simple form for cubies defined over the rationals.The second author was supported by an NSF grant.     

On a characterization of finite vector bundles as vector bundles admitting a flat connection with finite monodromy group

We prove that a holomorphic vector bundle over a compact connected Kähler manifold admits a flat connection, with a finite group as its monodromy, if and only if there are two distinct polynomials and , with nonnegative integral coefficients, such that the vector bundle is isomorphic to . An analogous result is proved for vector bundles over connected smooth quasi-projective varieties, of arbitrary dimension, admitting a flat connection with finite monodromy group.

When the base space is a connected projective variety, or a connected smooth quasi-projective curve, the above characterization of vector bundles admitting a flat connection with finite monodromy group was established by M. V. Nori.

     

Unitary Equivalence, Similarity and Calculation of K0-Group

In this paper, we are concerned with the classification of operators on complex separable Hilbert spaces, in the unitary equivalence sense and the similarity sense, respectively. We show that two strongly irreducible operators A and B are unitary equivalent if and only if W＊（A＋B）′≈M2（C）, and two operators A and B in B1（Ω） are similar if and only if A′（AGB）/J≈M2（C）. Moreover, we obtain V（H^∞（Ω,μ）≈N and Ko（H^∞（Ω,μ）≈Z by the technique of complex geometry, where Ω is a bounded connected open set in C, and μ is a completely non-reducing measure on Ω.