Union of linear spaces in infinite-dimensional projective spaces and holomorphic vector bundles

Let V be a localizing Banach space with an unconditional countable basis, X an equicodimensional transversal union of finite-codimensional linear subspaces of P(V) and E a holomorphic vector bundle of finite rank on X. Here we prove that Hⁱ(X,E) = 0 for every i > 0 and that E is isomorphic to a direct sum of line bundles ^OX(t).     

Topologically trivial holomorphic vector bundles on infinite-dimensional projective varieties

Let V be an infinite-dimensional locally convex complex space, X a closed subset of P(V) defined by finitely many continuos homogeneous equations and E a holomorphic vector bundle on X with finite rank. Here we show that E is holomorphically trivial if it is topologically trivial and spanned by its global sections and in a few other cases.     

Vector Bundles on Certain Non-algebraic Surfaces

It is well-known that every holomorphic vector bundle is filtrable on a projective algebraic     

Höhere Sekantenvarietäten und Vektorbündel auf Kurven

If E denotes a vector bundle of rank 2 an a smooth projective curve X, an upper bound for the number m(E) of sublinebundles of maximal degree of E is given in terms of the genus of X and the invariant s(E). The proof is an application of an enumerative result for higher secant varieties of curves in projective space.     

On the twisted endomorphism algebra of a vector bundle on a curve

Let X be a smooth projective curve, L∈Pic (X) and E a vector bundle on X. Here we study the bigraded algebra , mainly when E is a general stable bundle on X. We give informations on the degrees of the minimal generators and study a generalization of the notion of projective normality. Entrata in Redazione il 19 gennaio 2000.     

Ein einfacher Beweis des Verschwindungssatzes für positive holomorphe Vektorraumbündel

The object of this note is to give a simple proof of the following vanishing theorem of le Potier [8]: Let X be a compact complex manifold of dimension n and E a weakly positive holomorphic vector bundle of rank r on X. Then $$H^q (X,\Omega ^p (E)) = 0forp + q \geqslant n + r.$$      

HORIZONTAL LAPLACE OPERATOR IN REAL FINSLER VECTOR BUNDLES

A vector bundle F over the tangent bundle TM of a manifold M is said to be a Finsler vector bundle if it is isomorphic to the pull-back π^＊E of a vector bundle E over M（[1]）. In this article the authors study the h-Laplace operator in Finsler vector bundles. An h-Laplace operator is defined, first for functions and then for horizontal Finsler forms on E. Using the h-Laplace operator, the authors define the h-harmonic function and ho harmonic horizontal Finsler vector fields, and furthermore prove some integral formulas for the h-Laplace operator, horizontal Finsler vector fields, and scalar fields on E.     

BRILL－NOETHER MATRIX FOR RANK TWO VECTOR BUNDLES

Lei X be an arbitrary smooth irreducible complex projective curve, E (?) X a rank two vector bundle generated by its sections. The author first represents E as a triple {D1,D2,f}, where D1 , D2 are two effective divisors with d = deg(D1) + deg(D2), and f ∈ H0(X, [D1] |D2) is a collection of polynomials. E is the extension of [D2] by [D1] which is determined by f. By using f and the Brill-Noether matrix of D1 + D2, the author constructs a 2g X d matrix WE whose zero space gives Im{H0(X,[D1]) (?) H0(X, [D1] |D1)}(?)Im{H0(X, E) (?) H0(X,[D2]) (?) H0(X,[D2] |D2)}. From this and H0(X,E) = H0(X, [D1]) (?) Im{H0(X, E) (?) H0(X, [D2])}, it is got in particular that dimH0(X, E) = deg(E) - rank(WE) + 2.     

Generalized Adjunction of k-Very Ample Vector Bundles on Algebraic Surfaces

We study the k-very ampleness of the adjoint bundle K _S+det E associated to a k-very ample vector bundle E on an algebraic surface S. We extend the results of Beltrametti and Sommese to the vector bundle case and give the classification of the pairs (S, E) such that the preservation of k-very ampleness fails.     

Parallel connections over symmetric spaces

Let M be a simply connected Riemannian symmetric space, with at most one flat direction. We show that every Riemannian (or unitary) vector bundle with parallel curvature over M is an associated vector bundle of a canonical principal bundle, with the connection inherited from the principal bundle. The problem of finding Riemannian (or unitary) vector bundles with parallel curvature then reduces to finding representations of the structure group of the canonical principal bundle.     

Algebraic morphisms into grassmannians:differential geometry and frobenius

Let Xbe a projective variety (over Spec(K)) and f:X→G(r,v) a morphism to a Grassmannian, i.e. a pair (E,V) where E is a rank r vector bundle on V?H^O(X,E) is a subspace spanning E with dim(V) = v. Here we study the differential properties of f and their relations to a sequence of quotient bundles E→E₁→E₂→of E called the derived bundles of (E,V). In the first 5 sections we study the case X a smooth curve, char(K) >0 (the case char(K) = 0, being due to D. Perkinson). Then we give a general duality theorem for the derived bundles when Xis any normal variety.     

Some extensions of horrocks criterion to vector bundles on Grassmannians and quadrics

Summary In this paper we prove that a vector bundle E on a grassmannian (resp. on a quadric) splits as a direct sum of line bundles if and only if certain cohomology groups involving E and the quotient bundle (resp. the spinor bundle) are zero. When rank E=2 a better criterion is obtained considering only finitely many suitably chosen cohomology groups.This paper has been written while the author was enrolled in the Research Doctorate of the University of Florence. Partially supported by MPI 40% funds.     

Pseudo-effective cone of Grassmann bundles over a curve

Let \(E\) be a vector bundle over a smooth projective curve \(X\) defined over an algebraically closed field \(k\) . For any integer \(1\,\le \, r\, <\, \mathrm{rank}(E)\) , let \(\mathrm{Gr}_r(E)\,\longrightarrow \, X\) be a Grassmann bundle parametrizing all \(r\) dimensional quotients of the fibers of \(E\) . We compute the pseudo-effective cone in the real Néron–Severi group \(\mathrm{NS}(\mathrm{Gr}_r(E))_\mathbb{R }\) . We prove that this cone coincides with the nef cone in \(\mathrm{NS}(\mathrm{Gr}_r(E))_\mathbb{R }\) if and only if the vector bundle \(E\) is semistable (respectively, strongly semistable) when the characteristic of \(k\) is zero (respectively, positive). Examples are given to show that this characterization of (strong) semistability is not true for vector bundles on higher dimensional projective varieties.     

A general vanishing theorem

Let E be a vector bundle and L be a line bundle over a smooth projective variety X. In this article, we give a condition for the vanishing of Dolbeault cohomology groups of the form $H^{p,q}(X,{\mathcal S}^{\alpha}E\otimes \wedge^{\beta} E\otimes L)$ when S ^α?+?β E???L is ample. This condition is shown to be invariant under the interchange of p and q. The optimality of this condition is discussed for some parameter values.     

Nonorientable Manifolds, Complex Structures, and Holomorphic Vector Bundles

A generalization of the notion of almost complex structure is defined on a nonorientable smooth manifold M of even dimension. It is defined by giving an isomorphism J from the tangent bundle TM to the tensor product of the tangent bundle with the orientation bundle such that JJ=–Id _TM . The composition JJ is realized as an automorphism of TM using the fact that the orientation bundle is of order two. A notion of integrability of this almost complex structure is defined; also the Kähler condition has been extended. The usual notion of a complex vector bundle is generalized to the nonorientable context. It is a real vector bundle of even rank such that the almost complex structure of a fiber is given up to the sign. Such bundles have generalized Chern classes. These classes take value in the cohomology of the tensor power of the local system defined by the orientation bundle. The notion of a holomorphic vector bundle is extended to the context under consideration. Stable vector bundles and Einstein–Hermitian connections are also generalized. It is shown that a generalized holomorphic vector bundle on a compact nonorientable Kähler manifold admits an Einstein–Hermitian connection if and only if it is polystable.     

A Universal Construction for Moduli Spaces of Decorated Vector Bundles over Curves

Let $X$ be a smooth projective curve over the field of complex numbers, and fix a homogeneous representation $\rho\colon \mathop{\rm GL}(r)\rightarrow \mathop{\rm GL}(V)$. Then one can associate to every vector bundle $E$ of rank $r$ over $X$ a vector bundle $E_\rho$ with fibre $V$. We would like to study triples $(E,L,\phi)$ where $E$ is a vector bundle of rank $r$ over $X$, $L$ is a line bundle over $X$, and $\phi\colon E_\rho\rightarrow L$ is a nontrivial homomorphism. This setup comprises well known objects such as framed vector bundles, Higgs bundles, and conic bundles. In this paper, we will formulate a general (parameter dependent) semistability concept for such triples, which generalizes the classical Hilbert--Mumford criterion, and we establish the existence of moduli spaces for the semistable objects. In the examples which have been studied so far, our semistability concept reproduces the known ones. Therefore, our results give in particular a unified construction for many moduli spaces considered in the literature.     

Kähler Metrics on the Projective Bundle of a Holomorphic Finsler Vector Bundle

Let (M, g) be a compact Kähler manifold and (E, F) be a holomorphic Finsler vector bundle of rank r ≥ 2 over M. In this paper, we prove that there exists a Kähler metric φ defined on the projective bundle P (E) of E, which comes naturally from g and F. Moreover, a necessary and sufficient condition for φ having positive scalar curvature is obtained, and a sufficient condition for φ having positive Ricci curvature is established.     

Determinant line bundle on moduli space of parabolic bundles

In Biswas and Raghavendra (Proc Indian Acad Sci (Math Sci) 103:41–71, 1993; Asian J Math 2:303–324, 1998), a parabolic determinant line bundle on a moduli space of stable parabolic bundles was constructed, along with a Hermitian structure on it. The construction of the Hermitian structure was indirect: The parabolic determinant line bundle was identified with the pullback of the determinant line bundle on a moduli space of usual vector bundles over a covering curve. The Hermitian structure on the parabolic determinant bundle was taken to be the pullback of the Quillen metric on the determinant line bundle on the moduli space of usual vector bundles. Here a direct construction of the Hermitian structure is given. For that we need to establish a version of the correspondence between the stable parabolic bundles and the Hermitian–Einstein connections in the context of conical metrics. Also, a recently obtained parabolic analog of Faltings’ criterion of semistability plays a crucial role.     

Microlocal analysis on wonderful varieties. Regularized traces and global characters

Let be a connected reductive complex algebraic group with split real form . Consider a strict wonderful ‐variety X equipped with its σ‐equivariant real structure, and let X be the corresponding real locus. Further, let E be a real differentiable G‐vector bundle over X . In this paper, we introduce a distribution character for the regular representation of G on the space of smooth sections of E given in terms of the spherical roots of , and show that on a certain open subset of G of transversal elements it is locally integrable and given by a sum over fixed points.