Positive sheaves of differentials coming from coarse moduli spaces

Jabbusch, Kelly, and Kebekus, Stefan. "Positive sheaves of differentials coming from coarse moduli spaces." Annales de l’institut Fourier 61.6 (2011): 2277-2290. <http://eudml.org/doc/219813>.

@article{Jabbusch2011,
abstract = {Consider a smooth projective family of canonically polarized complex manifolds over a smooth quasi-projective complex base $Y^\circ $, and suppose the family is non-isotrivial. If $Y$ is a smooth compactification of $Y^\circ $, such that $D:=Y \setminus Y^\circ $ is a simple normal crossing divisor, then we can consider the sheaf of differentials with logarithmic poles along $D$. Viehweg and Zuo have shown that for some $m &gt; 0$, the $m^\{\rm th\}$ symmetric power of this sheaf admits many sections. More precisely, the $m^\{\rm th\}$ symmetric power contains an invertible sheaf whose Kodaira-Iitaka dimension is at least the variation of the family. We refine this result and show that this “Viehweg-Zuo sheaf” comes from the coarse moduli space associated to the given family, at least generically.As an immediate corollary, if $Y^\circ $ is a surface, we see that the non-isotriviality assumption implies that $Y^\circ $ cannot be special in the sense of Campana.},
affiliation = {KTH Department of Mathematics 10044 Stockholm (Sweden); Albert-Ludwigs-Universität Freiburg Mathematisches Institut Eckerstraße 1, 79104 Freiburg (Germany)},
author = {Jabbusch, Kelly, Kebekus, Stefan},
journal = {Annales de l’institut Fourier},
keywords = {Moduli space; positivity of differentials; moduli space; canonically polarized manifold; special varieties; geometric orbifold},
language = {eng},
number = {6},
pages = {2277-2290},
publisher = {Association des Annales de l’institut Fourier},
title = {Positive sheaves of differentials coming from coarse moduli spaces},
url = {http://eudml.org/doc/219813},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Jabbusch, Kelly
AU - Kebekus, Stefan
TI - Positive sheaves of differentials coming from coarse moduli spaces
JO - Annales de l’institut Fourier
PY - 2011
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 6
SP - 2277
EP - 2290
AB - Consider a smooth projective family of canonically polarized complex manifolds over a smooth quasi-projective complex base $Y^\circ $, and suppose the family is non-isotrivial. If $Y$ is a smooth compactification of $Y^\circ $, such that $D:=Y \setminus Y^\circ $ is a simple normal crossing divisor, then we can consider the sheaf of differentials with logarithmic poles along $D$. Viehweg and Zuo have shown that for some $m &gt; 0$, the $m^{\rm th}$ symmetric power of this sheaf admits many sections. More precisely, the $m^{\rm th}$ symmetric power contains an invertible sheaf whose Kodaira-Iitaka dimension is at least the variation of the family. We refine this result and show that this “Viehweg-Zuo sheaf” comes from the coarse moduli space associated to the given family, at least generically.As an immediate corollary, if $Y^\circ $ is a surface, we see that the non-isotriviality assumption implies that $Y^\circ $ cannot be special in the sense of Campana.
LA - eng
KW - Moduli space; positivity of differentials; moduli space; canonically polarized manifold; special varieties; geometric orbifold
UR - http://eudml.org/doc/219813
ER -