# Positive sheaves of differentials coming from coarse moduli spaces

Kelly Jabbusch^{[1]}; Stefan Kebekus^{[2]}

- [1] KTH Department of Mathematics 10044 Stockholm (Sweden)
- [2] Albert-Ludwigs-Universität Freiburg Mathematisches Institut Eckerstraße 1, 79104 Freiburg (Germany)

Annales de l’institut Fourier (2011)

- Volume: 61, Issue: 6, page 2277-2290
- ISSN: 0373-0956

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topJabbusch, 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 $m > 0$, 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 $m > 0$, 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 -

## References

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