Positive sheaves of differentials coming from coarse moduli spaces
Annales de l'Institut Fourier, Volume 61 (2011) no. 6, pp. 2277-2290.

Consider a smooth projective family of canonically polarized complex manifolds over a smooth quasi-projective complex base Y , and suppose the family is non-isotrivial. If Y is a smooth compactification of Y , such that D:=YY 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 th symmetric power of this sheaf admits many sections. More precisely, the m 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 is a surface, we see that the non-isotriviality assumption implies that Y cannot be special in the sense of Campana.

On considère une famille projective lisse de variétés canoniquement polarisées sur une base quasi-projective lisse Y. Si la famille n’est pas iso-triviale, Viehweg et Zuo ont montré que toute bonne compactification de Y admet des formes pluricanoniques avec au plus des pôles logarithmiques le long du bord. Plus précisément leur résultat montre qu’une puissance symétrique suffisamment grande du faisceau des différentielles logarithmiques contient un sous-faisceau inversible dont la dimension de Kodaira-Iitaka est au moins égale à la variation de la famille. En suivant la construction de Viehweg-Zuo on montre que le faisceau inversible de Viehweg-Zuo provient, au moins génériquement, de l’espace de module “grossier” associé à la famille.

Comme corollaire immédiat on obtient que la base d’une famille non-isotriviale ne peut pas être spéciale au sens de Campana.

DOI: 10.5802/aif.2673
Classification: 14D07,  14D22
Keywords: Moduli space, positivity of differentials
Jabbusch, Kelly 1; Kebekus, Stefan 2

1 KTH Department of Mathematics 10044 Stockholm (Sweden)
2 Albert-Ludwigs-Universität Freiburg Mathematisches Institut Eckerstraße 1, 79104 Freiburg (Germany)
     author = {Jabbusch, Kelly and Kebekus, Stefan},
     title = {Positive sheaves of differentials coming from coarse moduli spaces},
     journal = {Annales de l'Institut Fourier},
     pages = {2277--2290},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {61},
     number = {6},
     year = {2011},
     doi = {10.5802/aif.2673},
     mrnumber = {2976311},
     zbl = {1253.14009},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2673/}
AU  - Jabbusch, Kelly
AU  - Kebekus, Stefan
TI  - Positive sheaves of differentials coming from coarse moduli spaces
JO  - Annales de l'Institut Fourier
PY  - 2011
DA  - 2011///
SP  - 2277
EP  - 2290
VL  - 61
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2673/
UR  - https://www.ams.org/mathscinet-getitem?mr=2976311
UR  - https://zbmath.org/?q=an%3A1253.14009
UR  - https://doi.org/10.5802/aif.2673
DO  - 10.5802/aif.2673
LA  - en
ID  - AIF_2011__61_6_2277_0
ER  - 
%0 Journal Article
%A Jabbusch, Kelly
%A Kebekus, Stefan
%T Positive sheaves of differentials coming from coarse moduli spaces
%J Annales de l'Institut Fourier
%D 2011
%P 2277-2290
%V 61
%N 6
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2673
%R 10.5802/aif.2673
%G en
%F AIF_2011__61_6_2277_0
Jabbusch, Kelly; Kebekus, Stefan. Positive sheaves of differentials coming from coarse moduli spaces. Annales de l'Institut Fourier, Volume 61 (2011) no. 6, pp. 2277-2290. doi : 10.5802/aif.2673. http://archive.numdam.org/articles/10.5802/aif.2673/

[1] Campana, Frédéric Orbifoldes spéciales et classification biméromorphe des variétés Kählériennes compactes (preprint http://arxiv.org/abs/0705.0737v5, October 2008)

[2] Esnault, Hélène; Viehweg, Eckart Lectures on vanishing theorems, DMV Seminar, 20, Birkhäuser Verlag, Basel, 1992 | MR | Zbl

[3] Hartshorne, Robin Algebraic geometry, Springer-Verlag, New York, 1977 (Graduate Texts in Mathematics, No. 52) | MR | Zbl

[4] Kebekus, Stefan; Kovács, Sándor J. Families of canonically polarized varieties over surfaces, Invent. Math., Volume 172 (2008) no. 3, pp. 657-682 | DOI | MR | Zbl

[5] Kebekus, Stefan; Kovács, Sándor J. Families of varieties of general type over compact bases, Adv. Math., Volume 218 (2008) no. 3, pp. 649-652 | DOI | MR | Zbl

[6] Kebekus, Stefan; Kovács, Sándor J. The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties, Duke Math. J., Volume 155 (2010) no. 1, pp. 1-33 | DOI | MR | Zbl

[7] Kebekus, Stefan; Solá Conde, Luis Existence of rational curves on algebraic varieties, minimal rational tangents, and applications, Global aspects of complex geometry, Springer, Berlin, 2006, pp. 359-416 | MR | Zbl

[8] Kollár, János Projectivity of complete moduli, J. Differential Geom., Volume 32 (1990) no. 1, pp. 235-268 | MR | Zbl

[9] Okonek, Christian; Schneider, Michael; Spindler, Heinz Vector bundles on complex projective spaces, Progress in Mathematics, 3, Birkhäuser Boston, Mass., 1980 | MR | Zbl

[10] Viehweg, Eckart Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 30, Springer-Verlag, Berlin, 1995 | MR | Zbl

[11] Viehweg, Eckart Positivity of direct image sheaves and applications to families of higher dimensional manifolds, School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000) (ICTP Lect. Notes), Volume 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001, pp. 249-284 | MR | Zbl

[12] Viehweg, Eckart; Zuo, Kang Base spaces of non-isotrivial families of smooth minimal models, Complex geometry (Göttingen, 2000), Springer, Berlin, 2002, pp. 279-328 | MR | Zbl

Cited by Sources: