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.     

On rationality of moduli spaces of vector bundles on real Hirzebruch surfaces

Let X be a real form of a Hirzebruch surface. Let M _H (r,c ₁, c ₂) be the moduli space of vector bundles on X. Under some numerical conditions on r, c ₁ and c ₂, we identify those M _H (r,c ₁,c ₂) that are rational.     

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.     

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     

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.

---

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.

     

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.     

Brill-Noether theory on Hirzebruch surfaces

In this paper, we investigate higher rank Brill-Noether problems for stable vector bundles on Hirzebruch surfaces. Using suitable non-splitting extensions, we deal with the non-emptiness. Results concerning the emptiness follow as a consequence of a generalization of Clifford’s theorem for line bundles on curves to vector bundles on surfaces.     

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.      

Analytic torsion of Hirzebruch surfaces

Using different forms of the arithmetic Riemann-Roch theorem and the computations of Bott-Chern secondary classes, we compute the analytic torsion and the height of Hirzebruch surfaces.     

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.

     

Special rank two vector bundles over Enriques surfaces

Partially supported by the European Science project Geometry of Algebraic Varieties, Contract SCJ-0398-C(A)     

On vector bundles over surfaces and Hilbert schemes

Let X be an irreducible smooth projective surface over ${{\mathbb{C}}}$ and Hilb ^d (X) the Hilbert scheme parametrizing the zero-dimensional subschemes of X of length d. Given a vector bundle E on X, there is a naturally associated vector bundle ${{\mathcal{F}}_d(E)}$ over Hilb ^d (X). If E and V are semistable vector bundles on X such that ${{\mathcal{F}}_d(E)}$ and ${{\mathcal{F}}_d(V)}$ are isomorphic, we prove that E is isomorphic to V. A key input in the proof is provided by Biswas and Nagaraj (see [1]).