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     

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 log terminal varieties

Let be an -dimensional variety and an ample vector bundle on of rank . We give a complete classification of pairs , with log terminal and such that is not ample. The results we obtain were conjectured by Fujita, and recently by Zhang.

     

Twisted holomorphic forms on generalized flag varieties

In this paper we prove some vanishing theorems for the twisted Dolbeault cohomology of the complete flag varieties associated to a simple, simply connected algebraic group.     

Grassmannians of 2-Sided Vector Spaces

Let k ? K be an extension of fields, and let A ? M _n (K) be a k-algebra. We study parameter spaces of m-dimensional subspaces of K ⁿ which are invariant under A. The space  _A (m, n), whose R-rational points are A-invariant, free rank m summands of R ⁿ , is well known. We construct a distinct parameter space,  _A (m, n), which is a fiber product of a Grassmannian and the projectivization of a vector space. We then study the intersection  _A (m, n) ∩  _A (m, n), which we denote by  _A (m, n). Under suitable hypotheses on A, we construct affine open subschemes of  _A (m, n) and  _A (m, n) which cover their K-rational points. We conclude by using  _A (m, n),  _A (m, n), and  _A (m, n) to construct parameter spaces of 2-sided subspaces of 2-sided vector spaces.     

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.     

Bioriented flags and resolutions of Schubert varieties

We use incidence relations running in two directions in order to construct a Kempf–Laksov type resolution for any Schubert variety of the complete flag manifold but also an embedded resolution for any Schubert variety in the Grassmannian. These constructions are alternatives to the celebrated Bott–Samelson resolutions. The second process led to the introduction of W-flag varieties, algebro-geometric objects that interpolate between the standard flag manifolds and products of Grassmannians, but which are singular in general. The surprising simple desingularization of a particular such type of variety produces an embedded resolution of the Schubert variety within the Grassmannian.     

Vector bundles over real algebraic varieties

The object of this paper is to study continuous vector bundles, over real algebraic varieties, admitting an algebraic structure. For large classes of real varieties, we obtain explicit information concerning the Grothendieck group of algebraic vector bundles. We show that in many cases this group is small compared to the corresponding group of continuous vector bundles. These results are used elsewhere to study the geometry of real algebraic varieties.Dedicated to Professor Alexander Grothendieck on the occasion of his 60th birthdaySupported by the NSF Grant DMS-8602672.     

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)     

Real double flag varieties for the symplectic group

In this paper we study a key example of a Hermitian symmetric space and a natural associated double flag variety, namely for the real symplectic group G and the symmetric subgroup L, the Levi part of the Siegel parabolic $P_S$. We give a detailed treatment of the case of the maximal parabolic subgroups Q of L corresponding to Grassmannians and the product variety of $G/P_S$ and $L/Q$; in particular we classify the L-orbits here, and find natural explicit integral transforms between degenerate principal series of L and G.     

Cohomology of line bundles on Schubert varieties: The rank two case

In this paper we describe vanishing and non-vanishing of cohomology of “most” line bundles over Schubert subvarieties of flag varieties for rank 2 semisimple algebraic groups.     

Vector bundles on toric varieties

Following Sam Payne?s work, we study the existence problem of nontrivial vector bundles on toric varieties. The first result we prove is that every complete fan admits a nontrivial conewise linear multivalued function. Such functions could potentially be the Chern classes of toric vector bundles. Then we use the results of Cortiñas, Haesemeyer, Walker and Weibel to show that the (non-equivariant) Grothendieck group of the toric 3-fold studied by Payne is large, so the variety has a nontrivial vector bundle. Using the same computation, we show that every toric 3-fold X either has a nontrivial line bundle, or there is a finite surjective toric morphism from Y to X, such that Y has a large Grothendieck group.     

Stability of Frobenius pull-backs of tangent bundles and generic injectivity of Gauss maps in positive characteristic

For a smooth projective variety X of dimension n in a projective space defined over an algebraically closed field k, the Gauss mapis a morphism from X to the Grassmannian of n-plans in sending to the embedded tangent space .The purpose of this paper is to prove the generic injectivity of Gauss mapsin positive characteristic for two cases; (1) weighted complete intersectionsof dimension of general type; (2) surfaces or 3-folds with -semistable tangent bundles; based on a criterion of Kaji by looking atthe stability of Frobenius pull-backs of their tangent bundles. The first result implies that a conjecture of Kleiman--Piene is true in case X is of general type of dimension . The second result is a generalization of the injectivity for curves.     

Desingularizations of Schubert varieties in double Grassmannians

Let X = Gr(k, V) × Gr(l, V) be the direct product of two Grassmann varieties of k-and l-planes in a finite-dimensional vector space V, and let B ? GL(V) be the isotropy group of a complete flag in V. We consider B-orbits in X, which are an analog of Schubert cells in Grassmannians. We describe this set of orbits combinatorially and construct desingularizations for the closures of these orbits, similar to the Bott-Samelson desingularizations for Schubert varieties.     

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 on real affine varieties

Studiamo i fibrati fortemente algebrici sulle superficie reali aperte e i moduli proiettivi su certi anelli di funzioni razionali reali in tre variabili.     

Characterization and structure of Finsler spaces with constant flag curvature

The geometric characterization and structure of Finsler manifolds with constant flag curvature (CFC) are studied. It is proved that a Finsler space has constant flag curvature 1 (resp. 0) if and only if the Ricci curvature along the Hilbert form on the projective sphere bundle attains identically its maximum (resp. Ricci scalar). The horizontal distributionH of this bundle is integrable if and only ifM has zero flag curvature. When a Finsler space has CFC, Hilbert form’s orthogonal complement in the horizontal distribution is also integrable. Moreover, the minimality of its foliations is equivalent to given Finsler space being Riemannian, and its first normal space is vertical Project supported by Wang KC Fundation of Hong Kong and the National Natural Science Foundation of China (Grant No. 19571005).     

Joins of projective varieties and multisecant spaces

Let , , be integral varieties. For any integers 0$">, , and set and . Let be the set of all linear -spaces contained in a linear -space spanned by points of , points of , ..., points of . Here we study some cases where has the expected dimension. The case was recently considered by Chiantini and Coppens and we follow their ideas. The two main results of the paper consider cases where each is a surface, more particularly:

or

     

Realizing representations on generalized flag manifolds

Let G be a complex reductive linear algebraic group and a real form. Suppose P is a parabolic subgroup of G and assume that P has a Levi factor L such that is a real form of L.Using the minimal globalization V _min of a finite length admissible representation for L ₀, one can define a homogeneous analytic vector bundle on the G ₀ orbit S of P in the generalized flag manifold . Let denote the corresponding sheaf of polarized sections. In this article we analyze the G ₀ representations obtained on the compactly supported sheaf cohomology groups .