Intrinsic pseudo-volume forms for logarithmic pairs

Thomas Dedieu

Bulletin de la Société Mathématique de France (2010)

  • Volume: 138, Issue: 4, page 543-582
  • ISSN: 0037-9484

Abstract

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We study an adaptation to the logarithmic case of the Kobayashi-Eisenman pseudo-volume form, or rather an adaptation of its variant defined by Claire Voisin, for which she replaces holomorphic maps by holomorphic K -correspondences. We define an intrinsic logarithmic pseudo-volume form Φ X , D for every pair ( X , D ) consisting of a complex manifold X and a normal crossing Weil divisor D on X , the positive part of which is reduced. We then prove that Φ X , D is generically non-degenerate when X is projective and K X + D is ample. This result is analogous to the classical Kobayashi-Ochiai theorem. We also show the vanishing of Φ X , D for a large class of log- K -trivial pairs, which is an important step in the direction of the Kobayashi conjecture about infinitesimal measure hyperbolicity in the logarithmic case.

How to cite

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Dedieu, Thomas. "Intrinsic pseudo-volume forms for logarithmic pairs." Bulletin de la Société Mathématique de France 138.4 (2010): 543-582. <http://eudml.org/doc/272461>.

@article{Dedieu2010,
abstract = {We study an adaptation to the logarithmic case of the Kobayashi-Eisenman pseudo-volume form, or rather an adaptation of its variant defined by Claire Voisin, for which she replaces holomorphic maps by holomorphic $K$-correspondences. We define an intrinsic logarithmic pseudo-volume form $\Phi _\{X,D\}$ for every pair $(X,D)$ consisting of a complex manifold $X$ and a normal crossing Weil divisor $D$ on $X$, the positive part of which is reduced. We then prove that $\Phi _\{X,D\}$ is generically non-degenerate when $X$ is projective and $K_X+D$ is ample. This result is analogous to the classical Kobayashi-Ochiai theorem. We also show the vanishing of $\Phi _\{X,D\}$ for a large class of log-$K$-trivial pairs, which is an important step in the direction of the Kobayashi conjecture about infinitesimal measure hyperbolicity in the logarithmic case.},
author = {Dedieu, Thomas},
journal = {Bulletin de la Société Mathématique de France},
keywords = {log-$K$-correspondence; Kobayashi-Eisenman pseudo-volume form; logarithmic pair},
language = {eng},
number = {4},
pages = {543-582},
publisher = {Société mathématique de France},
title = {Intrinsic pseudo-volume forms for logarithmic pairs},
url = {http://eudml.org/doc/272461},
volume = {138},
year = {2010},
}

TY - JOUR
AU - Dedieu, Thomas
TI - Intrinsic pseudo-volume forms for logarithmic pairs
JO - Bulletin de la Société Mathématique de France
PY - 2010
PB - Société mathématique de France
VL - 138
IS - 4
SP - 543
EP - 582
AB - We study an adaptation to the logarithmic case of the Kobayashi-Eisenman pseudo-volume form, or rather an adaptation of its variant defined by Claire Voisin, for which she replaces holomorphic maps by holomorphic $K$-correspondences. We define an intrinsic logarithmic pseudo-volume form $\Phi _{X,D}$ for every pair $(X,D)$ consisting of a complex manifold $X$ and a normal crossing Weil divisor $D$ on $X$, the positive part of which is reduced. We then prove that $\Phi _{X,D}$ is generically non-degenerate when $X$ is projective and $K_X+D$ is ample. This result is analogous to the classical Kobayashi-Ochiai theorem. We also show the vanishing of $\Phi _{X,D}$ for a large class of log-$K$-trivial pairs, which is an important step in the direction of the Kobayashi conjecture about infinitesimal measure hyperbolicity in the logarithmic case.
LA - eng
KW - log-$K$-correspondence; Kobayashi-Eisenman pseudo-volume form; logarithmic pair
UR - http://eudml.org/doc/272461
ER -

References

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