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In 1970, Kobayashi conjectured that general hypersurfaces of sufficiently large degree in are hyperbolic. In this paper we prove that a general sufficiently ample hypersurface in a smooth projective variety is hyperbolic. To prove this statement, we construct hypersurfaces satisfying a property which is Zariski open and which implies hyperbolicity. These hypersurfaces are chosen such that the geometry of their higher order jet spaces can be related to the geometry of a universal family of complete intersections. To do so, we introduce a Wronskian construction which associates a (twisted) jet differential to every finite family of global sections of a line bundle.
@article{PMIHES_2017__126__1_0,
author = {Brotbek, Damian},
title = {On the hyperbolicity of general hypersurfaces},
journal = {Publications Math\'ematiques de l'IH\'ES},
pages = {1--34},
publisher = {Springer Berlin Heidelberg},
address = {Berlin/Heidelberg},
volume = {126},
year = {2017},
doi = {10.1007/s10240-017-0090-3},
mrnumber = {3735863},
zbl = {1458.32022},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1007/s10240-017-0090-3/}
}
TY - JOUR AU - Brotbek, Damian TI - On the hyperbolicity of general hypersurfaces JO - Publications Mathématiques de l'IHÉS PY - 2017 SP - 1 EP - 34 VL - 126 PB - Springer Berlin Heidelberg PP - Berlin/Heidelberg UR - http://geodesic.mathdoc.fr/articles/10.1007/s10240-017-0090-3/ DO - 10.1007/s10240-017-0090-3 LA - en ID - PMIHES_2017__126__1_0 ER -
%0 Journal Article %A Brotbek, Damian %T On the hyperbolicity of general hypersurfaces %J Publications Mathématiques de l'IHÉS %D 2017 %P 1-34 %V 126 %I Springer Berlin Heidelberg %C Berlin/Heidelberg %U http://geodesic.mathdoc.fr/articles/10.1007/s10240-017-0090-3/ %R 10.1007/s10240-017-0090-3 %G en %F PMIHES_2017__126__1_0
Brotbek, Damian. On the hyperbolicity of general hypersurfaces. Publications Mathématiques de l'IHÉS, Tome 126 (2017), pp. 1-34. doi: 10.1007/s10240-017-0090-3
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