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.

---

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.

     

On moduli of vector bundles on surfaces with negative Kodaira dimension

Summary Here we prove the following result. Fix integersq, τ,a’, b’, a’ _i, 1≤i≤τ,a’, b’, a’ _i, 1≤i≤τ; then there is an integerew such that for every integert≥w, for every algebraically closed fieldK for every smooth complete surfaceX with negative Kodaira dimension, irregularityq andK _X ² =8(1−q)−τ, the following condition holds; ifX→S is a sequence fo τ blowing-downs which gives a relatively minimal model with ruling ρ:S→C, take as basis of the Neron Severi groupNS(X) a smooth rational curve which is the total transform of a fiber ofC, the total transform of a minimal section of ρ and the total transformD _i, 1≤i≤τ, of the exceptional curver; then for everyH andL∈Pic (X) withH ample,H (resp.L) represented by the integersa’, b’, a’ _i, (resp.a’, b’, a’ _i), 1≤i≤τ, in the chosen basis ofNS(X) the moduli spaceM(ZX, 2,H, L, t) of rank 2H-stable vector bundles onX with determinantL andc ₂=t is generically smooth and the number, dimension and ?birational structure? of the irreducible components ofM(X, 2,H, L, t)_red do not depend on the choice ofK andX. Furthermore the birational structure of these irreducible components can be loosely described in terms of the birational structure of the components of suitableM(S, 2,H’, L’, t’)_red withS a relatively minimal model ofX.

---

Sunto SiaX una superficie algebrica liscia completa con dimensione di Kodaira negativa e definita su un campo algebricamente chiusoK; fissiamoH eL∈Pic (X),t∈Z; siaq l’irregolarità diX e τ≔8(1−q)−K _X ^Emphasis>2 ; siaM(X, 2,H, L, t) to schema dei moduli dei fibrati vettorialiH-stabili di rango 2 suX con determinateL ec ₂=t. Si dimostra che esiste una costantew che dipende solo daq, da τ e dalla classe numerica diH e diL (ma non da char (K) o dalla classe di isomorphismo diX) tale che per ognit≥w il numero, la dimensione e ?la struttura birazionale? delle componenti irriducibili diM(X, 2,H, L, t)_red non dipende dalla scelta di char (K),K eX ma solo daq, τ e dalle classi diH eL inNS(X). Inoltre la ?struttura birazionale? di queste componenti irriducibili può essere grossolanamente descritta in termini delle componenti di opportuni spazi di moduliM(S, 2,H’, L’, t’) (doveS è un modello minimale diX).

     

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 affine vector fields on weil bundles

We obtain necessary and sufficient conditions for a holomorphic vector field to be affine for a holomorphic linear connection defined on aWeil bundle. We also prove that the Lie algebra over R of holomorphic affine vector fields for the natural prolongation of a linear connection from the base to theWeil bundle is isomorphic to the tensor product of theWeil algebra by the Lie algebra of affine vector fields on the base.     

Holomorphic vector bundles on non-algebraic tori of dimension 2

We describe the Chern classes of holomorphic vector bundles on non-algebraic complex torus of dimension 2.     

Holomorphic vector bundles on open subsets of Stein manifolds

LetM be a connected two-dimensional Stein manifold withH ²(M,Z)=0 andS∈M a discrete subset withS≠ Ø. SetX:=M/S. Fix an integerr≥2. Then there exists a rankr vector bundleE onX such that there is no line bundleL onX with a non-zero mapL →E.     

Positive vector bundles on complex surfaces

A criterion for the positivity of a semi-stable vector bundle of rank 2 on a projective surface is proved. This is of particular interest for cotangent bundles of surfaces.Dedicated to Professor Karl Stein     

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.     

Rank-two vector bundles on Hirzebruch surfaces

We survey some parts of the vast literature on vector bundles on Hirzebruch surfaces, focusing on the rank-two case.     

Holomorphic Hermitian vector bundles over the Riemann sphere

We give a complete classification of isomorphism classes of all SU(2)-equivariant holomorphic Hermitian vector bundles on CP¹. We construct a canonical bijective correspondence between the isomorphism classes of SU(2)-equivariant holomorphic Hermitian vector bundles on CP¹ and the isomorphism classes of pairs ({Hn}_n∈Z,T), where each Hn is a finite dimensional Hilbert space with Hn=0 for all but finitely many n, and T is a linear operator on the direct sum _⊕n∈ZHn such that T(Hn)⊂Hn+2 for all n.     

A Note on vector bundles on Hirzebruch surfaces

In literature, two basic construction methods have been used to study vector bundles on a Hirzebruch surface. On the one hand, we have Serre?s method and elementary modifications, describing rank-2 bundles as extensions in a canonical way (Brînz?nescu and Stoia, 1984 [4], [5], Brînz?nescu, 1996 [6], Brosius, 1983 [7], Friedman, 1998 [9]), and on the other hand, we have a Beilinson-type spectral sequence (Buchdahl, 1987 [8]). Morally, the Beilinson spectral sequence indicates how to recover a bundle from the cohomology of its twists and from some sheaf morphisms (the differentials of the sequence). The aim of this Note is to show that the canonical extension of a rank-2 bundle can be deduced from the Beilinson spectral sequence of a suitable twist, called the normalization. In the final part we give a cohomological criterion for a topologically trivial vector bundle on a Hirzebruch surface to be trivial. To emphasize the relations and the differences between these two construction methods mentioned above, two different proofs are given.     

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)     

Rational surfaces and moduli spaces of vector bundles on rational surfaces

Let X be a smooth algebraic surface, L ? Pic(X) L \in \textrm{Pic}(X) and H an ample divisor on X. Set M_X,H(2; L, c₂) the moduli space of rank 2, H-stable vector bundles F on X with det(F) = L and c₂(F) = c₂. In this paper, we show that the geometry of X and of M_X,H(2; L, c₂) are closely related. More precisely, we prove that for any ample divisor H on X and any L ? Pic(X) L \in \textrm{Pic}(X) , there exists n₀ ? \mathbbZ n_0 \in \mathbb{Z} such that for all n₀ \leqq c₂ ? \mathbbZ n_0 \leqq c_2 \in \mathbb{Z} , M_X,H(2; L, c₂) is rational if and only if X is rational.     

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.

     

Holomorphic vector bundles on the Riemann sphere and the 21 st Hilbert problem

This report was presented at the first French-Russian colloquium on Geometry in Luminy in May 1992.