With cedram.org English français Home Overview Search for an article Submit a paper Informations for the authors Subscription Limited access - RUCHE Table of contents for this issue | Previous article | Next article Kelly Jabbusch; Stefan KebekusPositive sheaves of differentials coming from coarse moduli spaces(Faisceaux positifs de différentielles provenant d’un espace de modules)Annales de l'institut Fourier, 61 no. 6 (2011), p. 2277-2290, doi: 10.5802/aif.2673 Article PDF | Reviews MR 2976311 | Zbl 1253.14009 Class. Math.: 14D07, 14D22Keywords: Moduli space, positivity of differentials Résumé - AbstractConsider 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. Bibliography[1] Frédéric Campana, “Orbifoldes spéciales et classification biméromorphe des variétés Kählériennes compactes”, preprint http://arxiv.org/abs/0705.0737v5, October 2008 [2] Hélène Esnault & Eckart Viehweg, Lectures on vanishing theorems, DMV Seminar 20, Birkhäuser Verlag, Basel, 1992  MR 1193913 |  Zbl 0779.14003[3] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977, Graduate Texts in Mathematics, No. 52  MR 463157 |  Zbl 0531.14001[4] Stefan Kebekus & Sándor J. Kovács, “Families of canonically polarized varieties over surfaces”, Invent. Math. 172 (2008) no. 3, p. 657-682 Article |  MR 2393082 |  Zbl 1140.14031[5] Stefan Kebekus & Sándor J. Kovács, “Families of varieties of general type over compact bases”, Adv. Math. 218 (2008) no. 3, p. 649-652 Article |  MR 2414316 |  Zbl 1137.14027[6] Stefan Kebekus & Sándor J. Kovács, “The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties”, Duke Math. J. 155 (2010) no. 1, p. 1-33 Article |  MR 2730371 |  Zbl 1208.14027[7] Stefan Kebekus & Luis Solá Conde, Existence of rational curves on algebraic varieties, minimal rational tangents, and applications, Global aspects of complex geometry, Springer, 2006, p. 359–416  MR 2264116 |  Zbl 1121.14012[8] János Kollár, “Projectivity of complete moduli”, J. Differential Geom. 32 (1990) no. 1, p. 235-268 Article |  MR 1064874 |  Zbl 0684.14002[9] Christian Okonek, Michael Schneider & Heinz Spindler, Vector bundles on complex projective spaces, Progress in Mathematics 3, Birkhäuser Boston, Mass., 1980  MR 561910 |  Zbl 0438.32016[10] Eckart Viehweg, 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 1368632 |  Zbl 0844.14004[11] Eckart Viehweg, 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 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001, p. 249–284  MR 1919460 |  Zbl 1092.14044[12] Eckart Viehweg & Kang Zuo, Base spaces of non-isotrivial families of smooth minimal models, Complex geometry (Göttingen, 2000), Springer, 2002, p. 279–328  MR 1922109 |  Zbl 1006.14004 © Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310