The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties. (English) Zbl 1208.14027

Kebekus, Stefan; Kovács, Sándor J.

Duke Math. J. 155, No. 1, 1-33 (2010).

Let \(f^\circ : X^\circ \rightarrow Y^\circ\) be a family of canonically polarised projective manifolds over a smooth base, i.e. a smooth morphism between quasi-projective manifolds such that the fibres are projective manifolds with ample canonical bundle. The family induces a map from the base \(Y^\circ\) to some moduli stack (of canonically polarised manifolds with fixed Hilbert polynomial) and we define the variation of the family \(f\) as the dimension of the image of this map. If the variation is maximal, i.e. equal to the dimension of \(Y^\circ\), a conjecture of Viehweg claims that the manifold \(Y^\circ\) is of log-general type. This conjecture is classically known to be true if the fibres are curves S. Ju. Arakelov [Math. USSR, Izv. 5(1971), 1277–1302 (1972; Zbl 0248.14004)] or the base \(Y\) is a curve E. Viehweg and K. Zuo [J. Algebr. Geom. 10, No. 4, 781–799 (2001; Zbl 1079.14503)]. More recently the authors dealt with the case where \(Y\) is a surface [Invent. Math. 172, No. 3, 657–682 (2008; Zbl 1140.14031)].
In the paper under review the authors prove a generalised version of Viehweg’s conjecture in the threefold case, i.e. if \(f^\circ : X^\circ \rightarrow Y^\circ\) is a smooth projective family of varieties with semiample canonical bundle and the dimension of the base is at most three, then \(f^\circ\) has maximal variation only if \(Y^\circ\) has log-general type. If the canonical bundle is ample, it is possible to describe the map to the moduli stack: if \((Y, D)\) is a compactification of \(Y^\circ\), any minimal model program for the pair \((Y, D)\) will terminate in a Kodaira or Mori fibre space whose fibration factors the moduli map birationally. Note also that while the authors’ proof of the surface case relied on an explicit discussion of curve arrangements on surfaces and a difficult result by S. Keel and J. McKernan [Rational curves on quasi-projective surfaces. Mem. Am. Math. Soc. 669 (1999; Zbl 0955.14031)], the methods of this paper are more conceptual and might generalise to higher dimension.

Reviewer: Andreas Höring (Freiburg)

MSC:

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

Keywords:

variation; moduli stack, canonically polarized manifold; Shafarevich’s conjecture; Viehweg’s conjecture

Citations:

Zbl 0248.14004; Zbl 1079.14503; Zbl 1140.14031; Zbl 0955.14031

References:

[1]	M. C. Beltrametti and A. J. Sommese, The Adjunction Theory of Complex Projective Varieties , de Gruyter Exp. Math. 16 , Walter de Gruyter, Berlin, 1995. · Zbl 0845.14003
[2]	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. · Zbl 1210.14019
[3]	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 , · Zbl 1267.32017
[4]	L. Caporaso, J. Harris, and B. Mazur, Uniformity of rational points , J. Amer. Math. Soc. 10 (1997), 1–35. JSTOR: · Zbl 0872.14017
[5]	A. Corti, ed., Flips for 3-Folds and 4-Folds , Oxford Lecture Ser. Math. Appl. 35 , Oxford Univ. Press, Oxford, 2007. · Zbl 05175029
[6]	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.
[7]	H. Esnault and E. Viehweg, Lectures on Vanishing Theorems , DMV Seminar 20 , Birkhäuser, Basel, 1992. · Zbl 0779.14003
[8]	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. · Zbl 1194.14056
[9]	D. Greb, S. Kebekus, S. J. Kovács, and T. Peternell, Differential forms on log canonical spaces , · Zbl 1258.14021
[10]	R. Hartshorne, Algebraic Geometry , Grad. Texts in Math. 52 , Springer, New York, 1977. · Zbl 0367.14001
[11]	D. Huybrechts and M. Lehn, The Geometry of Moduli Spaces of Sheaves , Aspects Math. E31 , Friedr. Vieweg, Braunschweig, 1997. · Zbl 0872.14002
[12]	S. Iitaka, Algebraic Geometry , Grad. Texts in Math. 76 , Springer, New York, 1982 · Zbl 0491.14006
[13]	S. Kebekus and S. J. Kovács, Families of canonically polarized varieties over surfaces , Invent. Math. 172 (2008), 657–682. · Zbl 1140.14031
[14]	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. · Zbl 1121.14012
[15]	S. Kebekus, L. Solá Conde, and M. Toma, Rationally connected foliations after Bogomolov and McQuillan, J. Algebraic Geom. 16 (2007), 65–81. · Zbl 1120.14011
[16]	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. \({\!}\); Mathematical Reviews (MathSciNet): · Zbl 0818.14007
[17]	S. Keel and J. Mckernan, Rational curves on quasi-projective surfaces , Mem. Amer. Math. Soc. 140 (1999), no. 669. · Zbl 0955.14031
[18]	J. Kollár, ed., Flips and abundance for algebraic threefolds , Astérisque 211 , Soc. Math. France, Montrouge, 1992.
[19]	-, Lectures on resolution of singularities , Ann. of Math. Stud. 166 , Princeton Univ. Press, Princeton, 2007. · Zbl 1113.14013
[20]	J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties , Cambridge Tracts in Math. 134 , Cambridge Univ. Press, Cambridge, 1998.
[21]	S. J. Kovács, Algebraic hyperbolicity of fi ne moduli spaces, J. Algebraic Geom. 9 (2000), 165–174. · Zbl 0970.14008
[22]	R. Lazarsfeld, Positivity in Algebraic Geometry, II , Ergeb. Math. Grenzgeb. 3, Folge, A Series of Modern Surveys in Mathematics 49 , Springer, Berlin, 2004. · Zbl 1093.14500
[23]	Y. Miyaoka, Deformations of a morphism along a foliation and applications , Proc. Sympos. Pure Math. 46 , Amer. Math. Soc., Providence, 1987, 245–268. · Zbl 0659.14008
[24]	D. Mumford, “Picard groups of moduli problems” in Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963) , Harper & Row, New York, 1965, 33–81. · Zbl 0187.42801
[25]	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. · Zbl 0634.14003
[26]	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.
[27]	-, Basic Algebraic Geometry, 1 , 2nd ed., Springer, Berlin, 1994
[28]	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. · Zbl 1092.14044
[29]	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. · Zbl 1006.14004
[30]	-, On the isotriviality of families of projective manifolds over curves , J. Algebraic Geom. 10 (2001), 781–799. · Zbl 1079.14503