Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Vanishing theorems, boundedness and hyperbolicity over higher-dimensional bases

Author: Sándor J. Kovács
Journal: Proc. Amer. Math. Soc. 131 (2003), 3353-3364
MSC (2000): Primary 14J10
Published electronically: May 5, 2003
MathSciNet review: 1990623
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A vanishing theorem is proved for families over higher dimensional bases and applied to generalize some Shafarevich type statements to that setting.

References [Enhancements On Off] (What's this?)

  • [Abramovich-Karu00] D. Abramovich, K. Karu, Weak semistable reduction in characteristic $0$, Invent. Math. 139 (2000) 241-273. MR 2001f:14021
  • [Angehrn-Siu95] U. Angehrn, Y.T. Siu, Effective freeness and point separation for adjoint bundles, Invent. Math. 122 (1995) 291-308. MR 97b:32036
  • [Arakelov71] S. Arakelov, Families of algebraic curves with fixed degeneracies, Izv. A. N. SSSR 35 (1971) 1269-1293. MR 48:298
  • [Beauville81] A. Beauville, Le nombre minimum de fibres singulières d'une courbe stable sur $\mathbb P^{1}$, Astérisque 86 (1981) 97-108.
  • [Bedulev-Viehweg00] E. Bedulev, E. Viehweg, On the Shafarevich conjecture for surfaces of general type over function fields, Inv. Math. 139 (2000) 603-615. MR 2001f:14065
  • [Esnault-Viehweg90] H. Esnault, E. Viehweg, Effective bounds for semipositive sheaves and for the height of points on curves over complex function fields, Compositio Math. 76 (1990) 69-85. MR 91m:14038
  • [Esnault-Viehweg92] H. Esnault, E. Viehweg, Lectures on vanishing theorems, DMV Seminar 20, Birkhäuser (1992). MR 94a:14017
  • [Faltings83] G. Faltings, Arakelov's Theorem for abelian varieties, Inv. Math. 73 (1983) 337-348. MR 85m:14061
  • [Karu00] K. Karu, Minimal models and boundedness of stable varieties, J. Alg. Geom. 9 (2000) 93-109. MR 2001g:14059
  • [Kollár87] J. Kollár, Subadditivity of the Kodaira dimension: Fibers of general type, Advanced Studies in Pure Math. 10, Algebraic Geometry Sendai 85, Kinokuniya-North Holland (1987) 361-398. MR 89i:14029
  • [Kollár90] J. Kollár, Projectivity of Complete Moduli, J. Diff. Geom. 32 (1990) 235-268. MR 92e:14008
  • [Kovács96] S. J. Kovács, Smooth families over rational and elliptic curves, J. Alg. Geom. 5 (1996) 369-385. MR 97c:14035
  • [Kovács97a] S. J. Kovács, Families over a base with a birationally nef tangent bundle, Math. Ann. 308 (1997) 347-359. MR 98h:14039
  • [Kovács97b] S. J. Kovács, On the minimal number of singular fibres in a family of surfaces of general type, J. reine angew. Math. 487 (1997) 171-177. MR 98h:14038
  • [Kovács00a] S. J. Kovács, Algebraic hyperbolicity of fine moduli spaces, J. Alg. Geom. 9 (2000) 165-174. MR 2000i:14017
  • [Kovács00b] S. J. Kovács, Logarithmic vanishing theorems and Arakelov-Parshin boundedness for singular varieties, Comp. Math. 131 (2002) 291-317. MR 2003a:14025
  • [LePotier75] J. Le Potier, Annulation de la cohomologie à valeurs dans un fibré vectoriel holomorphe positif de rang quelconque, Math. Ann. 218 (1975) 35-53. MR 52:6044
  • [Migliorini95] L. Migliorini, A smooth family of minimal surfaces of general type over a curve of genus at most one is trivial, J. Alg. Geom. 4 (1995) 353-361. MR 95m:14023
  • [Parshin68] A. Parshin, Algebraic curves over function fields, Izv. A. N. SSSR 32 (1968) 1145-1170. MR 41:1740
  • [Shiffman-Sommese85] B. Shiffman, A. J. Sommese, Vanishing theorems on complex manifolds, Progress. in Math. 56, Birkhäuser (1985). MR 86h:32048
  • [Viehweg83a] E. Viehweg, Weak positivity and the additivity of Kodaira dimension for certain fibre spaces, Advanced Studies in Pure Math. 1, Algebraic Varieties and Analytic Varieties (1983) 329-353. MR 85b:14041
  • [Viehweg83b] E. Viehweg, Weak positivity and the additivity of the Kodaira dimension II, Classification of algebraic and analytic manifolds, Birkhäuser (1983) 567-590. MR 85i:14020
  • [Viehweg89] E. Viehweg, Weak positivity and the stability of certain Hilbert points, Inv. Math. 96 (1989) 639-667; II 101 (1990) 191-223; III 101 (1990) 521-544. MR 90i:14037; MR 91f:14032; MR 91f:14032
  • [Viehweg95] E. Viehweg, Quasi-Projective Moduli of Polarized Manifolds, Springer (1995). MR 97j:14001
  • [Viehweg00] E. Viehweg, Positivity of direct image sheaves and applications to families of higher dimensional manifolds, Lecture notes, ``School on Vanishing Theorems and Effective Results in Algebraic Geometry", ICTP, Trieste, (2000).
  • [Viehweg-Zuo01a] E. Viehweg, K. Zuo, On the isotriviality of families of projective manifolds over curves, J. Alg. Geom. 10 (2001) 781-799. MR 2002g:14012
  • [Viehweg-Zuo01b] E. Viehweg, K. Zuo, Base spaces of non-isotrivial families of smooth minimal models, preprint (2001).
  • [Viehweg-Zuo01c] E. Viehweg, K. Zuo, On the Brody hyperbolicity of moduli spaces for canonically polarized manifolds, preprint (2001).

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 14J10

Retrieve articles in all journals with MSC (2000): 14J10

Additional Information

Sándor J. Kovács
Affiliation: Department of Mathematics, University of Washington, 354350, Seattle, Washington 98195

Received by editor(s): April 4, 2001
Received by editor(s) in revised form: June 10, 2002
Published electronically: May 5, 2003
Additional Notes: This work was supported in part by NSF Grants DMS-0196072, DMS-0092165, and a Sloan Research Fellowship.
Communicated by: Michael Stillman
Article copyright: © Copyright 2003 American Mathematical Society