On a question of Sean Keel

Let X be a smooth projective surface defined over , and let L be a line bundle over X such that for every complete curve Y contained in X. A question of Keel asks whether L is ample. If X is a P¹-bundle over a curve, we prove that this question has an affirmative answer.     

On the Picard bundle

Fix a holomorphic line bundle ξ over a compact connected Riemann surface X of genus g, with g?2, and also fix an integer r such that degree(ξ)>r(2g−1). Let Mξ(r) denote the moduli space of stable vector bundles over X of rank r and determinant ξ. The Fourier-Mukai transform, with respect to a Poincaré line bundle on X×J(X), of any F∈Mξ(r) is a stable vector bundle on J(X). This gives an injective map of Mξ(r) in a moduli space associated to J(X). If g=2, then Mξ(r) becomes a Lagrangian subscheme.     

On the F-fundamental group scheme

Let X be a smooth projective variety defined over a perfect field k of positive characteristic, and let FX be the absolute Frobenius morphism of X. For any vector bundle E→X, and any polynomial g with non-negative integer coefficients, define the vector bundle using the powers of FX and the direct sum operation. We construct a neutral Tannakian category using the vector bundles with the property that there are two distinct polynomials f and g with non-negative integer coefficients such that . We also investigate the group scheme defined by this neutral Tannakian category.     

Direct image of parabolic line bundles

Given a vector bundle E, on an irreducible projective variety X, we give a necessary and sufficient criterion for E to be a direct image of a line bundle under a surjective étale morphism. The criterion in question is the existence of a Cartan subalgebra bundle of the endomorphism bundle $\text{End}(E)$. As a corollary, a criterion is obtained for E to be the direct image of the structure sheaf under an étale morphism. The direct image of a parabolic line bundle under any ramified covering map has a natural parabolic structure. Given a parabolic vector bundle, we give a similar criterion for it to be the direct image of a parabolic line bundle under a ramified covering map.     

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     

On the positivity of high-degree Schur classes of an ample vector bundle

Let X be a smooth projective variety of dimension n,and let E be an ample vector bundle over X.We show that any Schur class of E,lying in the cohomology group of bidegree(n-1,n-1),has a representative which is strictly positive in the sense of smooth forms.This conforms the prediction of Griffiths conjecture on the positive polynomials of Chern classes/forms of an ample vector bundle on the form level,and thus strengthens the celebrated positivity results of Fulton and Lazarsfeld(1983)for certain degrees.     

Discriminant loci of ample and spanned line bundles

Let (X,L,V) be a triplet where X is an irreducible smooth complex projective variety, L is an ample and spanned line bundle on X and V⊆H⁰(X,L) spans L. The discriminant locus D(X,V)⊂|V| is the algebraic subset of singular elements of |V|. We study the components of D(X,V) in connection with the jumping sets of (X,V), generalizing the classical biduality theorem. We also deal with the degree of the discriminant (codegree of (X,L,V)) giving some bounds on it and classifying curves and surfaces of codegree 2 and 3. We exclude the possibility for the codegree to be 1. Significant examples are provided.     

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     

Normally generated line bundles on general curves

Let L be a very ample line bundle of degree d on a general curve X of genus g≥2. Here we prove that if then L is globally generated, i.e. L embeds X as a projectively normal curve in PH⁰(L).     

Syzygies of projective bundles

In this paper we study defining equations and syzygies among them of projective bundles. We prove that for a given p≥0, if a vector bundle on a smooth complex projective variety is sufficiently ample, then the embedding given by the tautological line bundle satisfies property N_p.     

Simple immersions of wonderful varieties

Let G be a semisimple connected linear algebraic group over , and X a wonderful G-variety. We study the possibility of realizing X as a closed subvariety of the projective space of a simple G-module. We describe the wonderful varieties having this property as well as the linear systems giving rise to such immersions. We also prove that any ample line bundle on a wonderful variety is very ample. Research supported by European Research Training Network LIEGRITS (MRTN-CT 2003- 505078), in contract with CNRS DR17, No 2.     

Deformations of the generalised Picard bundle

Let X be a nonsingular algebraic curve of genus g?3, and let M_ξ denote the moduli space of stable vector bundles of rank n?2 and degree d with fixed determinant ξ over X such that n and d are coprime. We assume that if g=3 then n?4 and if g=4 then n?3, and suppose further that n₀, d₀ are integers such that n₀?1 and nd₀+n₀d>nn₀(2g-2). Let E be a semistable vector bundle over X of rank n₀ and degree d₀. The generalised Picard bundle W_ξ(E) is by definition the vector bundle over M_ξ defined by the direct image where U_ξ is a universal vector bundle over X×M_ξ. We obtain an inversion formula allowing us to recover E from W_ξ(E) and show that the space of infinitesimal deformations of W_ξ(E) is isomorphic to H¹(X,End(E)). This construction gives a locally complete family of vector bundles over M_ξ parametrised by the moduli space M(n₀,d₀) of stable bundles of rank n₀ and degree d₀ over X. If (n₀,d₀)=1 and W_ξ(E) is stable for all E∈M(n₀,d₀), the construction determines an isomorphism from M(n₀,d₀) to a connected component M⁰ of a moduli space of stable sheaves over M_ξ. This applies in particular when n₀=1, in which case M⁰ is isomorphic to the Jacobian J of X as a polarised variety. The paper as a whole is a generalisation of results of Kempf and Mukai on Picard bundles over J, and is also related to a paper of Tyurin on the geometry of moduli of vector bundles.     

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.     

Rational curves and ampleness properties¶of the tangent bundle of algebraic varieties

Let X be a projective manifold, a locally free ample subsheaf of the tangent bundle T _X . If and or n, we prove that . Furthermore we investigate ampleness properties of T _X on large families of curves and the relation to rational connectedness. Received: 2 July 1996     

Kähler structure on moduli spaces of principal bundles

Let M be a moduli space of stable principal G-bundles over a compact Kähler manifold (X,ωX), where G is a reductive linear algebraic group defined over C. Using the existence and uniqueness of a Hermite-Einstein connection on any stable G-bundle P over X, we have a Hermitian form on the harmonic representatives of H¹(X,ad(P)), where ad(P) is the adjoint vector bundle. Using this Hermitian form a Hermitian structure on M is constructed; we call this the Petersson-Weil form. The Petersson-Weil form is a Kähler form, a fact which is a consequence of a fiber integral formula that we prove here. The curvature of the Petersson-Weil Kähler form is computed. Some further properties of this Kähler form are investigated.     

Ample vector bundles with zero loci of sectional genus two

Let X be a smooth complex projective variety and let be a smooth submanifold of dimension , which is the zero locus of a section of an ample vector bundle of rank on X. Let H be an ample line bundle on X, whose restriction H_Z to Z is generated by global sections. Triplets as above are classified under the assumption that is a polarized manifold of sectional genus 2. This can be regarded as a step towards the classification of ample vector bundles of corank one and curve genus two. Received: 6 June 2003     

Absolutely split real algebraic vector bundles over a real form of projective space

Let M be a projective variety, defined over the field of real numbers, with the property that the base change MR_×C is isomorphic to CPN for some N. A real algebraic vector bundle E over M will be called absolutely split if the vector bundle ER_⊗C over MR_×C splits into a direct sum of line bundles. We classify the isomorphism classes of absolutely split vector bundles over M.     

Galois closure of essentially finite morphisms

Let X be a reduced connected k-scheme pointed at a rational point x∈X(k). By using tannakian techniques we construct the Galois closure of an essentially finite k-morphism f:Y→X satisfying the condition H⁰(Y,O_Y)=k; this Galois closure is a torsor dominating f by an X-morphism and universal for this property. Moreover, we show that is a torsor under some finite group scheme we describe. Furthermore we prove that the direct image of an essentially finite vector bundle over Y is still an essentially finite vector bundle over X. We develop for torsors and essentially finite morphisms a Galois correspondence similar to the usual one. As an application we show that for any pointed torsor f:Y→X under a finite group scheme satisfying the condition H⁰(Y,O_Y)=k, Y has a fundamental group scheme π₁(Y,y) fitting in a short exact sequence with π₁(X,x).     

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.     

Opers and theta functions

We construct natural maps (the Klein and Wirtinger maps) from moduli spaces of semistable vector bundles over an algebraic curve X to affine spaces, as quotients of the nonabelian theta linear series. We prove a finiteness result for these maps over generalized Kummer varieties (moduli space of torus bundles), leading us to conjecture that the maps are finite in general. The conjecture provides canonical explicit coordinates on the moduli space. The finiteness results give low-dimensional parametrizations of Jacobians (in for generic curves), described by 2Θ functions or second logarithmic derivatives of theta.We interpret the Klein and Wirtinger maps in terms of opers on X. Opers are generalizations of projective structures, and can be considered as differential operators, kernel functions or special bundles with connection. The matrix opers (analogues of opers for matrix differential operators) combine the structures of flat vector bundle and projective connection, and map to opers via generalized Hitchin maps. For vector bundles off the theta divisor, the Szegö kernel gives a natural construction of matrix oper. The Wirtinger map from bundles off the theta divisor to the affine space of opers is then defined as the determinant of the Szegö kernel. This generalizes the Wirtinger projective connections associated to theta characteristics, and the associated Klein bidifferentials.