Duke Mathematical Journal

The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties

Stefan Kebekus and Sándor J. Kovács

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Abstract

Generalizing the well-known Shafarevich hyperbolicity conjecture, it has been conjectured by Viehweg that a quasi-projective manifold that admits a generically finite morphism to the moduli stack of canonically polarized varieties is necessarily of log general type. Given a quasi-projective threefold $Y^\circ$ that admits a nonconstant map to the moduli stack, we employ extension properties of logarithmic pluriforms to establish a strong relationship between the moduli map and the minimal model program of $Y^\circ$: in all relevant cases the minimal model program leads to a fiber space whose fibration factors the moduli map. A much-refined affirmative answer to Viehweg's conjecture for families over threefolds follows as a corollary. For families over surfaces, the moduli map can often be described quite explicitly. Slightly weaker results are obtained for families of varieties with trivial or more generally semiample canonical bundle.

Article information

Source
Duke Math. J. Volume 155, Number 1 (2010), 1-33.

Dates
First available in Project Euclid: 23 September 2010

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1285247217

Digital Object Identifier
doi:10.1215/00127094-2010-049

Mathematical Reviews number (MathSciNet)
MR2730371

Zentralblatt MATH identifier
1208.14027

Subjects
Primary: 14J10: Families, moduli, classification: algebraic theory
Secondary: 14D22: Fine and coarse moduli spaces

Citation

Kebekus, Stefan; Kovács, Sándor J. The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties. Duke Math. J. 155 (2010), no. 1, 1--33. doi:10.1215/00127094-2010-049. https://projecteuclid.org/euclid.dmj/1285247217.


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References

  • M. C. Beltrametti and A. J. Sommese, The Adjunction Theory of Complex Projective Varieties, de Gruyter Exp. Math. 16, Walter de Gruyter, Berlin, 1995.
  • C. Birkar, P. Cascini, C. D. Hacon, and J. Mckernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), 405--468.
  • S. Boucksom, J.-P. Demailly, M. Păun, and T. Peternell, The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, preprint.
  • L. Caporaso, J. Harris, and B. Mazur, Uniformity of rational points, J. Amer. Math. Soc. 10 (1997), 1--35.
  • A. Corti, ed., Flips for 3-Folds and 4-Folds, Oxford Lecture Ser. Math. Appl. 35, Oxford Univ. Press, Oxford, 2007.
  • M. Demazure and A. Grothendieck, eds., Schémas en groupes, I: Propriétés générales des schémas en groupes, Séminaire de Géométrie Algébrique du Bois Marie 1962--1964 (SGA 3), Lecture Notes in Math. 151, Springer, Berlin, 1970.
  • H. Esnault and E. Viehweg, Lectures on Vanishing Theorems, DMV Seminar 20, Birkhäuser, Basel, 1992.
  • D. Greb, S. Kebekus, and S. J. Kovács, Extension theorems for differential forms and Bogomolov-Sommese vanishing on log canonical varieties, Composito Math. 146 (2010), 193--219.
  • D. Greb, S. Kebekus, S. J. Kovács, and T. Peternell, Differential forms on log canonical spaces, preprint.
  • R. Hartshorne, Algebraic Geometry, Grad. Texts in Math. 52, Springer, New York, 1977.
  • D. Huybrechts and M. Lehn, The Geometry of Moduli Spaces of Sheaves, Aspects Math. E31, Friedr. Vieweg, Braunschweig, 1997.
  • S. Iitaka, Algebraic Geometry, Grad. Texts in Math. 76, Springer, New York, 1982
  • S. Kebekus and S. J. Kovács, Families of canonically polarized varieties over surfaces, Invent. Math. 172 (2008), 657--682.
  • S. Kebekus and L. Solá Conde, ``Existence of rational curves on algebraic varieties, minimal rational tangents, and applications,'' in Global Aspects of Complex Geometry, Springer, Berlin, 2006, 359--416.
  • S. Kebekus, L. Solá Conde, and M. Toma, Rationally connected foliations after Bogomolov and McQuillan, J. Algebraic Geom. 16 (2007), 65--81.
  • S. Keel, K. Matsuki, and J. Mckernan, Log abundance theorem for threefolds, Duke Math. J. 75 (1994), 99--119.; Correction, Duke Math. J. 122 (2004), 625--630. ${\!}$;
  • S. Keel and J. Mckernan, Rational curves on quasi-projective surfaces, Mem. Amer. Math. Soc. 140 (1999), no. 669.
  • J. Kollár, ed., Flips and abundance for algebraic threefolds, Astérisque 211, Soc. Math. France, Montrouge, 1992.
  • —, Lectures on resolution of singularities, Ann. of Math. Stud. 166, Princeton Univ. Press, Princeton, 2007.
  • J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge Tracts in Math. 134, Cambridge Univ. Press, Cambridge, 1998.
  • S. J. Kovács, Algebraic hyperbolicity of fine moduli spaces, J. Algebraic Geom. 9 (2000), 165--174.
  • R. Lazarsfeld, Positivity in Algebraic Geometry, II, Ergeb. Math. Grenzgeb. 3, Folge, A Series of Modern Surveys in Mathematics 49, Springer, Berlin, 2004.
  • Y. Miyaoka, Deformations of a morphism along a foliation and applications, Proc. Sympos. Pure Math. 46, Amer. Math. Soc., Providence, 1987, 245--268.
  • D. Mumford, ``Picard groups of moduli problems'' in Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), Harper & Row, New York, 1965, 33--81.
  • M. Reid, ``Young person's guide to canonical singularities'' in Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math. 46, Amer. Math. Soc., Providence, 1987, 345--414.
  • I. R. Shafarevich, ``Algebraic number fields,'' in Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963; English translation in Amer. Math. Soc. Transl. Ser. 2 31 (1963), 25--39., 163--176.
  • —, Basic Algebraic Geometry, 1, 2nd ed., Springer, Berlin, 1994
  • E. Viehweg, ``Positivity of direct image sheaves and applications to families of higher dimensional manifolds'' in School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), ICTP Lect. Notes, 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001, 249--284.
  • E. Viehweg and K. Zuo, ``Base spaces of non-isotrivial families of smooth minimal models'' in Complex Geometry (Göttingen, 2000), Springer, Berlin, 2002, 279--328.
  • —, On the isotriviality of families of projective manifolds over curves, J. Algebraic Geom. 10 (2001), 781--799.