Theta functions on the Kodaira-Thurston manifold

We construct an analog of the classical theta function on an abelian variety for the Kodaira-Thurston nilmanifold, which is defined as a (nonholomorphic) section of a special complex line bundle over the Kodaira-Thurston manifold. The theta functions we introduce are used for a canonical symplectic embedding of the Kodaira-Thurston manifold into a complex projective space (an analog of the Lefschetz theorem).     

Some Koszul rings from geometry

We give examples of Koszul rings that arise naturally in algebraic geometry. In the first part, we prove a general result on Koszul property associated to an ample line bundle on a projective variety. Specifically, we show how Koszul property of multiples of a base point free ample line bundle depends on its Castelnuovo–Mumford regularity. In the second part, we give examples of Koszul rings that come from adjoint line bundles on minimal irregular surfaces of general type.     

Generating infinitesimal almost projective transformations in the tangent bundle of a general space of paths by projective transformations on the base

We obtain conditions under which an almost projective infinitesimal transformation on the tangent bundle of a general space of paths is a Yano-Okubo-Kagan complete lift of an infinitesimal projective transformation of a base manifold.     

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     

Base loci of linear series are numerically determined

We introduce a numerical invariant, called a moving Seshadri constant, which measures the local positivity of a big line bundle at a point. We then show how moving Seshadri constants determine the stable base locus of a big line bundle.

     

Un théorème d'annulation “à la Kawamata-Viehweg”

The vanishing theorem of Kawamata and Viehweg is an important extension of Kodaira's theorem to nef and big line bundles. We extend it to nef vector bundles of arbitrary rank over smooth projective varieties: the hypothesis of a positive self intersection becomes a positivity condition for caracteristic numbers defined by certain Schur polynomials. We derive this condition from an expression of the self-intersection of a line bundle on a relative flag manifold, which provides some insight into the corresponding Gysin morphism. This expression is itself a byproduct of some expansions of the Chern character of symmetric powers, that should be of independant interest.      

A geometric characterization of arithmetic varieties

A result of Belyi can be stated as follows. Every curve defined over a number field can be expressed as a cover of the projective line with branch locus contained in a rigid divisor. We define the notion of geometrically rigid divisors in surfaces and then show that every surface defined over a number field can be expressed as a cover of the projective plane with branch locus contained in a geometrically rigid divisor in the plane. The main result is the characterization of arithmetically defined divisors in the plane as geometrically rigid divisors in the plane.     

On cyclic covers of the projective line

We construct configuration spaces for cyclic covers of the projective line that admit extra automorphisms and we describe the locus of curves with given automorphism group. As an application we provide examples of arbitrary high genus that are defined over their field of moduli and are not hyperelliptic.     

Rationality in families of threefolds

We prove that in a family of projective threefolds defined over an algebraically closed field, the locus of rational fibers is a countable union of closed subsets of the locus of separably rationally connected fibers. When the ground field has characteristic zero, this implies that the locus of rational fibers in a smooth family of projective threefolds is the union of at most countably many closed subfamilies.     

Vector Bundles on Certain Non-algebraic Surfaces

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

Szegö kernel, regular quantizations and spherical CR-structures

We compute the Szegö kernel of the unit circle bundle of a negative line bundle dual to a regular quantum line bundle over a compact Kähler manifold. As a corollary we provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds (disk bundles over homogeneous Hodge manifolds) for which the log-terms in the Fefferman expansion of the Szegö kernel vanish and which are not locally CR-equivalent to the sphere. We also give a proof of the fact that, for homogeneous Hodge manifolds, the existence of a locally spherical CR-structure on the unit circle bundle alone implies that the manifold is biholomorphic to a projective space. Our results generalize those obtained by Engli? (Math Z 264(4):901–912, 2010) for Hermitian symmetric spaces of compact type.     

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.     

When K+(n-4)L fails to be nef

Let X be a smooth complex projective variety of dimension n and let L be an ample line bundle on X. We study polarized pairs (X,L) for which K+(n−3)L is nef but K+(n−4)L fails to be nef. Supported by MURST funds     

On 4-manifolds homotopy equivalent to surface bundles over surfaces

We show that a closed 4-manifold is homotopy equivalent to the total space of a surface bundle over a surface if the obviously necessary conditions on the fundamental group and Euler characteristic hold. When the base is the 2-sphere we need also conditions on the characteristic classes of the manifold. (Our results are incomplete when the base is the projective plane.) In most cases we can show the manifold is s-cobordant to the total space of the bundle.     

Lines on projective varieties and applications

The first part of this note contains a review of basic properties of the variety of lines contained in an embedded projective variety and passing through a general point. In particular we provide a detailed proof that for varieties defined by quadratic equations the base locus of the projective second fundamental form at a general point coincides, as a scheme, with the variety of lines. The second part concerns the problem of extending embedded projective manifolds, using the geometry of the variety of lines. Some applications to the case of homogeneous manifolds are included.     

Mumford's example and a general construction

We construct a line bundle on a complex projective manifold (a general ruled variety over a curve) which is not ample, but whose restriction to every proper subvariety is ample. This example is of interest in connection with ampleness questions of vector bundles on varieties of dimension greater than one. The method of construction shows that a stable bundle of positive degree on a curve is ample. The example can be used to show that there is no restriction theorem for Bogomolov stability.     

On the structure of infinitesimal almost projective transformations in the tangent bundle of a general space of paths

We establish necessary conditions for a vector field on the tangent bundle of a general space of paths to be an infinitesimal almost projective transformation in the case when the tensor fields determining the complexes of autoparallel curves are Yano-Okubo-Kagan complete lifts of tensor fields from the base manifold.     

Seshadri constants at very general points

We study the local positivity of an ample line bundle at a very general point of a smooth projective variety. We obtain a slight improvement of the result of Ein, Küchle, and Lazarsfeld.

     

On some examples of obstructed irregular surfaces

We determine the base space of the Kuranishi family of some complete intersections in the product of an abelian variety and a projective space.As a consequence,we obtain new examples of obstructed irregular surfaces with ample canonical bundle and maximal Albanese dimension.