Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Tsuji's numerical trivial fibrations

Author: Thomas Eckl
Translated by:
Journal: J. Algebraic Geom. 13 (2004), 617-639
Published electronically: February 4, 2004
MathSciNet review: 2072764
Full-text PDF

Abstract | References | Additional Information

Abstract: This note grew out of an attempt to understand H. Tsuji's work on numerical trivial foliations. In this paper, the Reduction Map Theorem [H. Tsuji, Numerically trivial fibrations, Preprint, 2000] is corrected and proven. To this purpose, various definitions of Tsuji's new intersection numbers for pseudo-effective line bundles equipped with a positive singular hermitian metric are compared and their equivalence on sufficiently general smooth curves is shown. An important adjustment to the Reduction Map Theorem is to consider the fact that plurisubharmonic functions are singular on pluripolar sets. Then the author follows Tsuji's argument for the proof of the Reduction Map Theorem. Another important result of the paper is the characterization of numerically trivial varieties by a decomposition property of the curvature current.

References [Enhancements On Off] (What's this?)

  • [BCE$^+$00] Th. Bauer, F. Campana, Th. Eckl, St. Kebekus, Th. Peternell, S. Rams, T. Szemberg, and L. Wotzlaw. A reduction map for nef line bundles. In Analytic and Algebraic Methods in Complex Geometry, Konferenzbericht der Konferenz zu Ehren von Hans Grauert, Goettingen (April 2000).
  • [BM97] E. Bierstone and P. D. Milman. Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant. Invent. Math., 128:207-302, 1997.
  • [Dem00] J.-P. Demailly. Multiplier ideal sheaves and analytic methods in algebraic geometry. School on Vanishing theorems and effective results in Algebraic Geometry, ICTP in Lecture Notes 6, 1-148, 2000.
  • [Hoe94] L. Hoermander. Notions of convexity, Progress in Mathematics 127. Birkhaeuser, Boston, 1994.
  • [Laz00] R. Lazarsfeld. Multiplier ideals for algebraic geometers. preprint at http://, August 2000.
  • [ME00] H. Ben Messaoud and H. ElMir. Opérateur de Monge-Ampère et Tranchage des Courants Positifs Fermés. J. Geom. Analysis, 10(1):139-168, 2000.
  • [Siu74] Y.-T. Siu. Analyticity of sets associated to Lelong numbers and the extension of closed positive currents. Invent. Math., 27:53-156, 1974.
  • [Tsu00] H. Tsuji. Numerically trivial fibrations. Preprint, 2000.

Additional Information

Thomas Eckl
Affiliation: Institut für Mathematik, Universität Bayreuth, 95440 Bayreuth, Germany

Received by editor(s): April 8, 2002
Published electronically: February 4, 2004

American Mathematical Society