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)     

Ample Vector Bundles with Small g − q

Let X be a smooth complex projective variety and let Z ? X be a smooth submanifold of dimension ≥ 2, which is the zero locus of a section of an ample vector bundle ? of rank dim X ? dim Z ≥ 2 on X. Let H be an ample line bundle on X, whose restriction H _Z to Z is generated by global sections. The structure of triplets (X,?,H) as above is described under the assumption that the curve genus of the corank-1 vector bundle ? ⊕ H ^{⊕ (dim Z?1)} is ≤ h ¹( _X ) + 2.     

On some Fano manifolds of large pseudoindex

In this paper we classify pairs (X,ℰ) with ℰ ample vector bundle of rank r on a smooth variety X of dimension n= 2r−1 such that K _X + det ℰ=? _x . Received: 7 April 2000     

Splitting of vector bundles on algebraic curves and elementary transformations

Let Ebe a rank nvector bundle on a smooth projective curve X. It is known that Emay be obtained from a splitted bundle +1≤i≤ L_i;, rank(L_i) = 1, by a finite number of elementary transformations. Here we give upper bounds for their minimal number. If n= 2 this is related to the order of stability of E.     

A generalization of Higgs bundles to higher dimensional varieties

Let X be a smooth n-dimensional projective variety defined over and let L be a line bundle on X. In this paper we shall construct a moduli space parametrizing -cohomology L-twisted Higgs pairs, i.e., pairs where E is a vector bundle on X and . If we take , the canonical line bundle on X, the variety is canonically identified with the cotangent bundle of the smooth locus of the moduli space of stable vector bundles on X and, as such, it has a canonical symplectic structure. We prove that, in the general case, in correspondence to the choice of a non-zero section , one can define, in a natural way, a Poisson structure on . We also analyze the relations between this Poisson structure on and the canonical symplectic structure of the cotangent bundle to the smooth locus of the moduli space of parabolic bundles over X, with parabolic structure over the divisor D defined by the section s. These results generalize to the higher dimensional case similar results proved in [Bo1] in the case of curves. Received November 4, 1997; in final form May 28, 1998     

A criterion for ample vector bundles over a curve in positive characteristic

Let X be a smooth projective curve defined over an algebraically closed field of positive characteristic. We give a necessary and sufficient condition for a vector bundle over X to be ample. This generalizes a criterion given by Lange in [Math. Ann. 238 (1978) 193-202] for a rank two vector bundle over X to be ample.     

ON CLIFFORD'S THEOREM FOR VECTOR BUNDLES ON ALGEBRAIC CURVES

Let X be a smooth projective curve of genus g ≥ 2 and E a rank r spanned vector bundle on X with deg(E)/rank(E) ≤ g ? 1. Here we give lower bounds for deg(E) refining the classical theorem of Clifford. Most results are for vector bundles with rank ≤ 5.     

Generalization of a criterion for semistable vector bundles

It is known that a vector bundle E on a smooth projective curve Y defined over an algebraically closed field is semistable if and only if there is a vector bundle F on Y such that both H⁰(X,EF) and H¹(X,EF) vanishes. We extend this criterion for semistability to vector bundles on curves defined over perfect fields. Let X be a geometrically irreducible smooth projective curve defined over a perfect field k, and let E be a vector bundle on X. We prove that E is semistable if and only if there is a vector bundle F on X such that Hⁱ(X,EF)=0 for all i. We also give an explicit bound for the rank of F.     

A theorem on the adjoint system for vector bundles

In this paper, we study the classification theory of uniruled varieties by means of the adjoint system for vector bundles on the varieties. We prove that ifE is an ample vector bundle on a smooth projective varietyX with rank(E)=dimX-2, thenK _X +C ₁ (E) is numerically effective except in a few cases. In all of the exceptional cases,X is a uniruled variety. As consequences, we generalized a result of Fujita [Fu3] and Ionescu [Io] and improve upon a theorem of Wiśniewski [Wi1].     

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 .      

Bogomolov restriction theorem for Higgs bundles

Let (E,θ) be a stable Higgs bundle of rank r on a smooth complex projective surface X equipped with a polarization H. Let C⊂X be a smooth complete curve with [C]=n⋅H. If where , then we prove that the restriction of (E,θ) to C is a stable Higgs bundle. This is a Higgs bundle analog of Bogomolov's restriction theorem for stable vector bundles.     

Generalization of Peternell, Le Potier and Schneider vanishing theorem

In this article we give a vanishing result for Dolbeault cohomology groups ${H^{p,q}(X, S^{\nu}E\otimes L)}$ , where ?? is a positive integer, E is a vector bundle generated by sections and L is an ample line bundle on a smooth projective variety X. We also give a condition for H ^p,q (X, S ^?? E) to vanish when E is s-ample and generated by sections. We also give an application related to a result of Barth-Lefschetz type. A general nonvanishing result under the same hypothesis is given to prove the optimality of the vanishing result for some parameter values.     

Unitary representations of the fundamental group of orbifolds

Let X be a smooth complex projective variety of dimension n and \(\mathcal {L}\) an ample line bundle on it. There is a well known bijective correspondence between the isomorphism classes of polystable vector bundles E on X with \(c_{1}(E) = 0 = c_{2} (E) \cdot c_{1} (\mathcal {L})^{n-2}\) and the equivalence classes of unitary representations of π₁(X). We show that this bijective correspondence extends to smooth orbifolds.     

An extension of Fujita’s non-extendability theorem for Grassmannians

In this paper we study smooth complex projective varieties X containing a Grassmannian of lines ${{\mathbb G}(1, r)}$ which appears as the zero locus of a section of a rank two nef vector bundle E. Among other things we prove that the bundle E cannot be ample.     

On Buchsbaum bundles on quadric hypersurfaces

Let E be an indecomposable rank two vector bundle on the projective space ℙ ⁿ , n ≥ 3, over an algebraically closed field of characteristic zero. It is well known that E is arithmetically Buchsbaum if and only if n = 3 and E is a null-correlation bundle. In the present paper we establish an analogous result for rank two indecomposable arithmetically Buchsbaum vector bundles on the smooth quadric hypersurface Q _n ⊂ ℙ ⁿ⁺¹, n ≥ 3. We give in fact a full classification and prove that n must be at most 5. As to k-Buchsbaum rank two vector bundles on Q ₃, k ≥ 2, we prove two boundedness results.     

Frobenius pull backs of vector bundles in higher dimensions

We prove that for a smooth projective variety X of arbitrary dimension and for a vector bundle E over X, the Harder?CNarasimhan filtration of a Frobenius pull back of E is a refinement of the Frobenius pull back of the Harder?CNarasimhan filtration of E, provided there is a lower bound on the characteristic p (in terms of rank of E and the slope of the destabilizing sheaf of the cotangent bundle of X). We also recall some examples, due to Raynaud and Monsky, to show that some lower bound on p is necessary. We also give a bound on the instability degree of the Frobenius pull back of E in terms of the instability degree of E and well defined invariants of X.     

Holomorphic Vector Bundle on Hopf Manifolds with Abelian Fundamental Groups

Let X be a Hopf manifolds with an Abelian fundamental group. E is a holomorphic vector bundle of rank r with trivial pull-back to W = ℂ ⁿ –{0}. We prove the existence of a non-vanishing section of L⊗E for some line bundle on X and study the vector bundles filtration structure of E. These generalize the results of D. Mall about structure theorem of such a vector bundle E. The research was supported by 973 Project Foundation of China and the Outstanding Youth Science Grant of NSFC (grant no. 19825105)     

Restriction of Stable Rank Two Vector Bundles in Arbitrary Characteristic

Let X be a smooth variety defined over an algebraically closed field of arbitrary characteristic and 𝒪 _X (H) be a very ample line bundle on X. We show that for a semistable X-bundle E of rank two, there exists an integer m depending only on Δ(E) · H ^dim(X)?2 and H ^dim(X) such that the restriction of E to a general divisor in |mH| is again semistable. As corollaries, we obtain boundedness results, and weak versions of Bogomolov's Theorem and Kodaira's vanishing theorem for surfaces in arbitrary characteristic.     

Cutting and Pasting of Elliptic Operators

If A : C∞E → C∞F is an elliptic operator between sections of vector bundles E, F over a closed smooth n-manifold X, Y a smooth (n – 1)-submanifold of X with trivial normal bundle, and Ψ = (ΨE, ΨF) a pair of automorphisms of E | Y and F | Y inducing a diffeomorphism f of Y and commuting with the principal symbol σ A of A over Y, then an elliptic operator A^Ψ is (uniquely up to operators of lower order) defined between sections of vector bundles E^Ψ, F^Ψ over a closed manifold X^f, all obtained by cutting and pasting the respective objects along Y. The difference index A^Ψ – index A is investigated and the relations with additivity properties of topological invariants and with classical transmission problems are explained.