Ample Cotangent Bundle Research Articles

Hyperbolicity and Uniformity of Varieties of Log General type

AbstractProjective varieties with ample cotangent bundle satisfy many notions of hyperbolicity, and one goal of this paper is to discuss generalizations to quasi-projective varieties. A major hurdle is that the naive generalization is false—the log cotangent bundle is never ample. Instead, we define a notion called almost ample that roughly asks that it is as positive as possible. We show that all subvarieties of a quasi-projective variety with almost ample log cotangent bundle are of log general type. In addition, if one assumes globally generated then we obtain that such varieties contain finitely many integral points. In another direction, we show that the Lang–Vojta conjecture implies the number of stably integral points on curves of log general type, and surfaces of log general type with almost ample log cotangent sheaf are uniformly bounded.

Effective bounds on ampleness of cotangent bundles

We prove that a general complete intersection of dimension $n$, codimension $c$ and type $d_1, \dots, d_c$ in $\mathbb{P}^N$ has ample cotangent bundle if $c \geq 2n-2$ and the $d_i$'s are all greater than a bound that is $O(1)$ in $N$ and quadratic in $n$. This degree bound substantially improves the currently best-known super-exponential bound in $N$ by Deng, although our result does not address the case $n \leq c < 2n-2$.

An Ampleness Criterion for Rank 2 Vector Bundles on Surfaces

We give an ampleness criterion for globally generated rank 2 vector bundles on certain surfaces. This applies to the Lazarsfeld–Mukai bundles, to congruences of lines in $\mathbb {P}^{3}$, and possibly to the construction of surfaces with ample cotangent bundle.

Rational points of varieties with ample cotangent bundle over function fields

Let K be the function field of a smooth curve over an algebraically closed field k. Let X be a scheme, which is smooth and projective over K. Suppose that the cotangent bundle Omega _{X/K} is ample. Let R:=mathrm{Zar}(X(K)cap X) be the Zariski closure of the set of all K-rational points of X, endowed with its reduced induced structure. We prove that for each irreducible component {mathfrak R} of R, there is a projective variety {mathfrak R}'_0 over k and a finite and surjective K^mathrm{sep}-morphism {mathfrak R}'_{0,K^mathrm{sep}}rightarrow {mathfrak R}_{K^mathrm{sep}}, which is birational when mathrm{char}(K)=0. This improves on results of Noguchi and Martin-Deschamps in characteristic 0. In positive characteristic, our result can be used to give the first examples of varieties, which are not embeddable in abelian varieties and satisfy an analog of the Bombieri–Lang conjecture.

Examples of geometric Bombieri–Lang–Noguchi outside Mordell–Lang: Nonrigid varieties with ample but not globally generated cotangent bundle

In this note we provide examples of nonisotrivial varieties over a function fields with ample cotangent bundle and zero Albanese. This provides explicit examples of varieties which are not embeddable into Abelian varieties for which the geometric Bombieri–Lang–Noguchi conjecture is known to be true.

Symmetric differential forms on complete intersection varieties and applications

In this paper we study the cohomology of the symmetric powers of the cotangent bundle of complete intersection varieties in projective space. We provide an explicit description of some of those cohomology groups in terms of the equations defining the complete intersection. As a first application, we give a new example illustrating the fact that the dimension of the space of holomorphic symmetric differential forms is not deformation invariant. Then, as our main application, we construct varieties with ample cotangent bundle, providing new results towards a conjecture of Debarre.

Hyperbolicity related problems for complete intersection varieties

AbstractIn this paper we examine different problems regarding complete intersection varieties of high multidegree in a smooth complex projective variety. First we prove an existence theorem for jet differential equations that generalizes a theorem of Diverio. Then we show how one can deduce hyperbolicity for generic complete intersections of high multidegree and high codimension from the known results on hypersurfaces. Finally, motivated by a conjecture of Debarre, we focus on the positivity of the cotangent bundle of complete intersections, and prove some results towards this conjecture; among other things, we prove that a generic complete intersection surface of high multidegree in a projective space of dimension at least four has an ample cotangent bundle.

The generic rank of the Baum–Bott map for foliations of the projective plane

Our main result says that the generic rank of the Baum–Bott map for foliations of degree is not the pull-back of a foliation of smaller degree. In Appendix A we show that the monodromy of the singular set of the universal foliation with very ample cotangent bundle is the full symmetric group.

Varieties with ample cotangent bundle

The aim of this article is to provide methods for constructing smooth projective complex varieties with ample cotangent bundle. We prove that the intersection of at least n/2 sufficiently ample general hypersurfaces in a complex abelian variety of dimension n has ample cotangent bundle. We also discuss analogous questions for complete intersections in the projective space. Finally, we present an unpublished result of Bogomolov which states that a general linear section of small dimension of a product of sufficiently many smooth projective varieties with big cotangent bundle has ample cotangent bundle.

Branched coverings of surfaces with ample cotangent bundle

Let /: X —> Y be a branched covering of compact complex surfaces, where the ramification set in X consists of smooth curves meeting with at most normal crossings and Y has ample cotangent bundle. We further assume that / is locally of form (u,υ) —> (un ,vm). We characterize ampleness of T*X . A class of examples of such X, which are branched covers of degree two, is provided. 1. Introduction. An interesting problem in surface theory is the construction and characterization of surfaces with ample cotangent bundle. They are necessarily algebraic surfaces of general type. Natural examples occur among the complete intersection surfaces of abelian varieties. More subtle examples are those constructed by Hirzebruch [6] using line-arrangements in the plane. The characterization of those of Hirzebruch's line-arrangement surfaces with ample cotangent bundle is due to Sommese [8]. In this article, we will give a characterization of ampleness of the cotangent bundle of a class of surfaces which branch cover another surface with ample cotangent bundle. We will also construct certain branched coverings of explicit line-arrangement surfaces; these constructions will again have ample cotangent bundle. For any vector bundle E over a base manifold M, the projectivization P(E) is a fiber bundle over M, with fiber Pq(E) over q e M given by T?g(E) « (2?*\0)/C*. There is a tautological linebundle ξβ over P(E) satisfying (i) ζE\F ^ « 0(1)P ^ V<? e M, and (ii) the projection pE: P(E) -» M gives /?^ (ξβ) ~ E. In the case that E = T*X we will denote p£ = Pτ*x simply by p. DEFINITION. The vector bundle E is ample if ζβ over ~P(E) is