Frédéric Campana

A Reduction Map for Nef Line Bundles

In [Ts00], H. Tsuji stated several very interesting assertions on the structure of pseudo-effective line bundles L on a projective manifold X. In particular he postulated the existence of a meromorphic “reduction map”, which essentially says that through the general point of X there is a maximal irreducible L-flat subvariety. Moreover the reduction map should be almost holomorphic, i.e. has compact fibers which do not meet the indeterminacy locus of the reduction map. The proofs of [Ts00], however, are extremely difficult to follow.

The algebraic dimension of compact complex threefolds with vanishing second Betti number

This note investigates compact complex manifolds X of dimension 3 with second Betti number b2(X) = 0. If X admits a non-constant meromorphic function, then we prove that either b1(X) = 1 and b3(X) = 0 or that b1(X) = 0 and b3(X) = 2. The main idea is to show that c3(X) = 0 by means of a vanishing theorem for generic line bundles on X. As a consequence a compact complex threefold homeomorphic to the 6-sphere S6 cannot admit a non-constant meromorphic function. Furthermore we investigate the structure of threefolds with b2(X) = 0 and algebraic dimension 1, in the case when the algebraic reduction X → P1 is holomorphic.

4-folds with numerically effective tangent bundles and second Betti numbers greater than one

In this paper we investigate projective 4-dimensional manifolds X whose tangent bundles TX are numerically effective and give an almost complete classification. An important technical tool is the “Mori theory” of projective manifolds X whose canonical bundles KX are not numerically effective.

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

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

Représentations linéaires des groupes kählériens : factorisations et conjecture de Shafarevich linéaire

We extend to compact Kahler manifolds some classical results on linear representation of fundamental groups of complex projective manifolds. Our approach, based on an interversion lemma for fibrations with tori versus general type manifolds as fibers, gives a refinement of the classical work of Zuo. We extend to the Kahler case some general results on holomorphic convexity of coverings such as the linear Shafarevich conjecture.

Special orbifolds and birational classification: a survey

In this paper, we discuss a proof of existence of log minimal models or Mori fibre spaces for klt pairs

Positivity properties of the bundle of logarithmic tensors on compact Kähler manifolds

(X/Z,B)

Variétés faiblement spéciales à courbes entières dégénérées

with

Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields

B

Geometric stability of the cotangent bundle and the universal cover of a projective manifold

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