Thomas Peternell

Bio: Thomas Peternell is an academic researcher from University of Bayreuth. The author has contributed to research in topics: Line bundle & Kodaira dimension. The author has an hindex of 32, co-authored 196 publications receiving 4911 citations. Previous affiliations of Thomas Peternell include University of Münster.

The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension

Abstract: We prove that a holomorphic line bundle on a projective manifold is pseudo-effective if and only if its degree on any member of a covering family of curves is non-negative. This is a consequence of a duality statement between the cone of pseudo-effective divisors and the cone of " movable curves " , which is obtained from a general theory of movable intersections and approximate Zariski decomposition for closed positive (1, 1)-currents. As a corollary, a projective manifold has a pseudo-effective canonical bundle if and only if it is not uniruled. We also prove that a 4-fold with a canonical bundle which is pseudo-effective and of numerical class zero in restriction to curves of a good covering family, has non-negative Kodaira dimension.

Compact complex manifolds with numerically effective tangent bundles

Abstract: 1. Basic properties of nef line bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 5 1.A. Nef line bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 5 1.B. Nef vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p. 11

The pseudo-effective cone of a compact K\"ahler manifold and varieties of negative Kodaira dimension

Abstract: We prove that a holomorphic line bundle on a projective manifold is pseudo-effective if and only if its degree on any member of a covering family of curves is non-negative. This is a consequence of a duality statement between the cone of pseudo-effective divisors and the cone of ``movable curves'', which is obtained from a general theory of movable intersections and approximate Zariski decomposition for closed positive (1,1)-currents. As a corollary, a projective manifold has a pseudo-effective canonical bundle if and only if it is is not uniruled. We also prove that a 4-fold with a canonical bundle which is pseudo-effective and of numerical class zero in restriction to curves of a covering family, has non negative Kodaira dimension.

Projective manifolds whose tangent bundles are numerically effective.

Abstract: In 1979, Mori [Mo] proved the so-called Hartshorne-Frankel conjecture: Every projective n-dimensional manifold with ample tangent bundle is isomorphic to the complex projective space P,. A differential-geometric analogon assuming the existence of a K/ihler metric on X with positive holomorphic bisectional curvature is independently due to Siu-Yau [SY]. Thus it seems natural to classify projective manifolds X whose tangent bundle Tx satisfy a degenerate condition of ampleness: numerical effectivity (abbreviated by "nef'). This means that the tautological quotient line bundle d~(1) on F(Tx) is numerically effective, i.e. C_>_0

Differential Forms on Log Canonical Spaces

Abstract: The present paper is concerned with differential forms on log canonical varieties It is shown that any p-form defined on the smooth locus of a variety with canonical or klt singularities extends regularly to any resolution of singularities In fact, a much more general theorem for log canonical pairs is established The proof relies on vanishing theorems for log canonical varieties and on methods of the minimal model program In addition, a theory of differential forms on dlt pairs is developed It is shown that many of the fundamental theorems and techniques known for sheaves of logarithmic differentials on smooth varieties also hold in the dlt setting Immediate applications include the existence of a pull-back map for reflexive differentials, generalisations of Bogomolov-Sommese type vanishing results, and a positive answer to the Lipman-Zariski conjecture for klt spaces

Existence of minimal models for varieties of log general type

Abstract: Assuming finite generation in dimension n − 1, we prove that pl-flips exist in dimension n.

Singularities of pairs

Abstract: These are the revised notes of my lectures at the 1995 Santa Cruz summer institute. Changes from the Jan.26,1996 version: Major: 7.1--2; Medium: 3.7, 3.11, 5.3.3, 7.9.2, 9.8.

Convex bodies associated to linear series

Abstract: Dans son travail sur la log-concavite des multiplicites, Okounkov montre au passage que l'on peut associer un corps convexe a un systeme lineaire sur une variete projective, puis utiliser la geometrie convexe pour etudier ces systemes lineaires. Bien qu'Okounkov travaille essentiellement dans le cadre classique des fibres en droites amples, il se trouve que sa construction s'etend au cas d'un grand diviseur arbitraire. De plus, ce point de vue permet de rendre transparentes de nombreuses proprietes de base des invariants asymptotiques des systemes lineaires, et ouvre la porte a de nombreuses extensions. Le but de cet article est d'initier un developpement systematique de la theorie et de donner quelques applications et exemples.