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**Erratum: “Boundedness of families of canonically polarized manifolds: a higher dimensional analogue of Shafarevich’s conjecture”.**
*(English)*
Zbl 1234.14029

This article was first printed in [Ann. Math. (2) 172, No. 3, 1719–1748 (2010; Zbl 1223.14040)]. The paper under review should be considered the version of record as it corrects a font error that resulted in notational problems throughout the first printing.

Let \(k\) be an algebraically closed field of characteristic zero. Consider a smooth projective curve \(B\) of genus \(g\) over \(k\) and a finite subset \(\Delta\) of \(B\). Then a family over \(B\) is a flat projective morphism \(f: X\to B\) with connected fibers, where \(X\) is a smooth projective variety over \(k\). Such a family is called isotrivial if any two fibers of \(f\) are isomorphic for general base points \(a\) and \(b\) in \(B\), while \(f: X\to B\) is admissible (with respect to the pair \((B,\Delta)\)) if it is not isotrivial and \(\Delta\) contains the discriminant locus of \(f\), i.e., the map \(f: X\setminus f^{-1}(\Delta)\to B\setminus\Delta\) is smooth.

At the 1962 ICM in Stockholm, I. R. Shafarevich formulated the following conjecture:

Let \((B,\Delta)\) be given as above and \(q\geq 2\) an integer. Then

(1) there exist only finitely many isomorphism classes of admissible families of curves of genus \(q\) over \(B\);

(2) there exist no such families if the inequality \(2g-2+ \#\Delta\leq 0\) holds.

Shafarevich proved (2) in the special case \(B=\mathbb{P}^1_k\), whereas (1) was confirmed by A. N. Parshin for \(\Delta=\phi\) [Izv. Akad. Nauk SSSR, Ser. Mat. 32, 1191–1219 (1968; Zbl 0181.23902)] and by S. J. Arakelov [Izv. Akad. Nauk SSSR, Ser. Mat. 35, 1269–1293 (1971; Zbl 0238.14012)] in general. Actually, Parshin and Arakelov used a reformulation of Shafarevich’s conjecture in terms of deformation types of families. More precisely, they proved the following version: Let \((B,\Delta)\) be fixed as above and \(q\geq 2\) an integer. Then the following statements hold.

(B) (Boundedness): There exist only finitely many deformation types of admissible families of curves of genus \(q\) over \(B\) with respect to \(B\setminus\Delta\);

(R) (Rigidity): There exist no non-trivial deformations of admissible families of the above type;

(H) (Hyperbolicity): if \(2g- 2+\#\Delta\leq 0\), then no admissible families of the above type exist.

As for the study of this version of Shafarevich’s conjecture for families of higher-dimensional varieties, the past decade has seen an avalanche of results concerning both (B) and (H), whereas (R) definitely fails in higher dimensions, due to E. Viehweg [Positivity of direct image sheaves and applications to families of higher-dimensional manifolds. Trieste: The Abdus Salam International Centre for Theoretical Physics. ICTP Lect. Notes 6, 249–284 (2001; Zbl 1092.14044)] and S. Kovacs [in: Higher dimensional varieties and rational points. Lectures of the summer school and conference, Budapest, Hungary, September 3–21, 2001. Berlin: Springer; Budapest: János Bolyai Mathematical Society. Bolyai Soc. Math. Stud. 12, 133–167 (2003; Zbl 1058.14057)].

In the present paper, the authors investigate the validity of the boundedness property (B) for families of higher-dimensional canonically polarized projective varieties from a stack-theoretic point of view. Their approach is based on the following philosophy: If there existed an algebraic stack \({\mathcal D}\) parametrizing families of canonically polarized varieties over the base \(B\setminus\Delta\), and if furthermore \({\mathcal D}\) could be shown to be of finite type, then the boundedness condition (B) would follow for families of canonically polarized varieties.

However, as E. Bedulev and E. Viehweg proved several years ago in their seminal paper [Invent. Math. 139, No. 3, 603–615 (2000; Zbl 1057.14044)], such stack \({\mathcal D}\) almost never exists. In view of this fact, the bulk of work done in the current paper is devoted to pointing out that a suitable proxy for \({\mathcal D}\) can be constructed by standard stack-theoretic methods. Their main result (Theorem 1.7.) concerns so-called weakly bounded compactifiable Deligne-Mumford stacks over a quasi-compact and quasi-separated \(\mathbb{Q}\)-scheme \(T\). At the end of the paper, this general framework is applied to the Deligne-Mumford stack of canonically polarized varieties with given Hilbert polynomial \(h\), thereby showing that the number of deformation types of canonically polarized varieties over an arbitrary variety with proper singular locus is finite, and that this number is uniformly bounded in any finite type family of base varieties. Also, the authors obtain a direct generalization of Shafarevich’s original conjecture to certain families of canonically polarized varieties, namely to the newly introduced, so-called infinitesimally rigid families of this type.

As for the precise contents of this paper, the first four sections provide the general stack-theoretic framework, together with the main theorems, while the last section discusses the above-mentioned applications to canonically polarized varieties and to the higher-dimensional analogue of Shafarevich’s conjecture.

Let \(k\) be an algebraically closed field of characteristic zero. Consider a smooth projective curve \(B\) of genus \(g\) over \(k\) and a finite subset \(\Delta\) of \(B\). Then a family over \(B\) is a flat projective morphism \(f: X\to B\) with connected fibers, where \(X\) is a smooth projective variety over \(k\). Such a family is called isotrivial if any two fibers of \(f\) are isomorphic for general base points \(a\) and \(b\) in \(B\), while \(f: X\to B\) is admissible (with respect to the pair \((B,\Delta)\)) if it is not isotrivial and \(\Delta\) contains the discriminant locus of \(f\), i.e., the map \(f: X\setminus f^{-1}(\Delta)\to B\setminus\Delta\) is smooth.

At the 1962 ICM in Stockholm, I. R. Shafarevich formulated the following conjecture:

Let \((B,\Delta)\) be given as above and \(q\geq 2\) an integer. Then

(1) there exist only finitely many isomorphism classes of admissible families of curves of genus \(q\) over \(B\);

(2) there exist no such families if the inequality \(2g-2+ \#\Delta\leq 0\) holds.

Shafarevich proved (2) in the special case \(B=\mathbb{P}^1_k\), whereas (1) was confirmed by A. N. Parshin for \(\Delta=\phi\) [Izv. Akad. Nauk SSSR, Ser. Mat. 32, 1191–1219 (1968; Zbl 0181.23902)] and by S. J. Arakelov [Izv. Akad. Nauk SSSR, Ser. Mat. 35, 1269–1293 (1971; Zbl 0238.14012)] in general. Actually, Parshin and Arakelov used a reformulation of Shafarevich’s conjecture in terms of deformation types of families. More precisely, they proved the following version: Let \((B,\Delta)\) be fixed as above and \(q\geq 2\) an integer. Then the following statements hold.

(B) (Boundedness): There exist only finitely many deformation types of admissible families of curves of genus \(q\) over \(B\) with respect to \(B\setminus\Delta\);

(R) (Rigidity): There exist no non-trivial deformations of admissible families of the above type;

(H) (Hyperbolicity): if \(2g- 2+\#\Delta\leq 0\), then no admissible families of the above type exist.

As for the study of this version of Shafarevich’s conjecture for families of higher-dimensional varieties, the past decade has seen an avalanche of results concerning both (B) and (H), whereas (R) definitely fails in higher dimensions, due to E. Viehweg [Positivity of direct image sheaves and applications to families of higher-dimensional manifolds. Trieste: The Abdus Salam International Centre for Theoretical Physics. ICTP Lect. Notes 6, 249–284 (2001; Zbl 1092.14044)] and S. Kovacs [in: Higher dimensional varieties and rational points. Lectures of the summer school and conference, Budapest, Hungary, September 3–21, 2001. Berlin: Springer; Budapest: János Bolyai Mathematical Society. Bolyai Soc. Math. Stud. 12, 133–167 (2003; Zbl 1058.14057)].

In the present paper, the authors investigate the validity of the boundedness property (B) for families of higher-dimensional canonically polarized projective varieties from a stack-theoretic point of view. Their approach is based on the following philosophy: If there existed an algebraic stack \({\mathcal D}\) parametrizing families of canonically polarized varieties over the base \(B\setminus\Delta\), and if furthermore \({\mathcal D}\) could be shown to be of finite type, then the boundedness condition (B) would follow for families of canonically polarized varieties.

However, as E. Bedulev and E. Viehweg proved several years ago in their seminal paper [Invent. Math. 139, No. 3, 603–615 (2000; Zbl 1057.14044)], such stack \({\mathcal D}\) almost never exists. In view of this fact, the bulk of work done in the current paper is devoted to pointing out that a suitable proxy for \({\mathcal D}\) can be constructed by standard stack-theoretic methods. Their main result (Theorem 1.7.) concerns so-called weakly bounded compactifiable Deligne-Mumford stacks over a quasi-compact and quasi-separated \(\mathbb{Q}\)-scheme \(T\). At the end of the paper, this general framework is applied to the Deligne-Mumford stack of canonically polarized varieties with given Hilbert polynomial \(h\), thereby showing that the number of deformation types of canonically polarized varieties over an arbitrary variety with proper singular locus is finite, and that this number is uniformly bounded in any finite type family of base varieties. Also, the authors obtain a direct generalization of Shafarevich’s original conjecture to certain families of canonically polarized varieties, namely to the newly introduced, so-called infinitesimally rigid families of this type.

As for the precise contents of this paper, the first four sections provide the general stack-theoretic framework, together with the main theorems, while the last section discusses the above-mentioned applications to canonically polarized varieties and to the higher-dimensional analogue of Shafarevich’s conjecture.

Reviewer: Werner Kleinert (Berlin)

### MSC:

14J10 | Families, moduli, classification: algebraic theory |

14D15 | Formal methods and deformations in algebraic geometry |

14C20 | Divisors, linear systems, invertible sheaves |

14D05 | Structure of families (Picard-Lefschetz, monodromy, etc.) |

14H10 | Families, moduli of curves (algebraic) |

14A20 | Generalizations (algebraic spaces, stacks) |

14D23 | Stacks and moduli problems |

14D06 | Fibrations, degenerations in algebraic geometry |

### Keywords:

families of curves; families of varieties; Shafarevich’s conjecture; stacks; deformations; principally polarized varieties### Citations:

Zbl 1223.14040; Zbl 0238.14012; Zbl 1092.14044; Zbl 1057.14044; Zbl 0181.23902; Zbl 1058.14057
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\textit{S. J. Kovács} and \textit{M. Lieblich}, Ann. Math. (2) 173, No. 1, 585--617 (2011; Zbl 1234.14029)

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### References:

[1] | D. Abramovich, M. Olsson, and A. Vistoli, ”Tame stacks in positive characteristic,” Ann. Inst. Fourier \((\)Grenoble\()\), vol. 58, iss. 4, pp. 1057-1091, 2008. · Zbl 1222.14004 |

[2] | D. Abramovich and A. Vistoli, ”Compactifying the space of stable maps,” J. Amer. Math. Soc., vol. 15, iss. 1, pp. 27-75, 2002. · Zbl 0991.14007 |

[3] | Aesop, Fables, 600 B.C. |

[4] | M. A. Van Opstall, Open problems. |

[5] | J. S. Arakelov, ”Families of algebraic curves with fixed degeneracies,” Izv. Akad. Nauk SSSR Ser. Mat., vol. 35, pp. 1269-1293, 1971. · Zbl 0248.14004 |

[6] | E. Bedulev and E. Viehweg, ”On the Shafarevich conjecture for surfaces of general type over function fields,” Invent. Math., vol. 139, iss. 3, pp. 603-615, 2000. · Zbl 1057.14044 |

[7] | C. Birker, P. Cascini, C. D. Hacon, and J. McKernan, ”Existence of minimal models for varieties of log general type,” J. Amer. Math. Soc., vol. 23, pp. 405-468, 2010. · Zbl 1210.14019 |

[8] | C. Cadman, ”Using stacks to impose tangency conditions on curves,” Amer. J. Math., vol. 129, iss. 2, pp. 405-427, 2007. · Zbl 1127.14002 |

[9] | L. Caporaso, ”On certain uniformity properties of curves over function fields,” Compositio Math., vol. 130, iss. 1, pp. 1-19, 2002. · Zbl 1067.14022 |

[10] | Arithmetic Geometry, Cornell, G. and Silverman, J. H., Eds., New York: Springer-Verlag, 1986. · Zbl 0596.00007 |

[11] | A. J. de Jong, ”Smoothness, semi-stability and alterations,” Inst. Hautes Études Sci. Publ. Math., iss. 83, pp. 51-93, 1996. · Zbl 0916.14005 |

[12] | G. Faltings, ”Endlichkeitssätze für abelsche Varietäten über Zahlkörpern,” Invent. Math., vol. 73, iss. 3, pp. 349-366, 1983. · Zbl 0588.14026 |

[13] | G. Faltings, ”Erratum: “Finiteness theorems for abelian varieties over number fields”,” Invent. Math., vol. 75, iss. 2, p. 381, 1984. · Zbl 0588.14026 |

[14] | A. Grothendieck, ”Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV,” Inst. Hautes Études Sci. Publ. Math., vol. 32, p. 361, 1967. · Zbl 0153.22301 |

[15] | G. Heier, ”Uniformly effective Shafarevich conjecture on families of hyperbolic curves over a curve with prescribed degeneracy locus,” J. Math. Pures Appl., vol. 83, iss. 7, pp. 845-867, 2004. · Zbl 1066.14028 |

[16] | L. Illusie, Complexe Cotangent et Déformations. I, New York: Springer-Verlag, 1971, vol. 239. · Zbl 0224.13014 |

[17] | G. Kempf, F. F. Knudsen, D. Mumford, and B. Saint-Donat, Toroidal Embeddings. I, New York: Springer-Verlag, 1973, vol. 339. · Zbl 0271.14017 |

[18] | J. Kollár, ”Projectivity of complete moduli,” J. Differential Geom., vol. 32, iss. 1, pp. 235-268, 1990. · Zbl 0684.14002 |

[19] | J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge: Cambridge Univ. Press, 1998, vol. 134. · Zbl 0926.14003 |

[20] | S. J. Kovács, ”Families of varieties of general type: the Shafarevich conjecture and related problems,” in Higher Dimensional Varieties and Rational Points, New York: Springer-Verlag, 2003, vol. 12, pp. 133-167. · Zbl 1058.14057 |

[21] | S. J. Kovács, ”Subvarieties of moduli stacks of canonically polarized varieties: generalizations of Shafarevich’s conjecture,” in Algebraic Geometry-Seattle 2005. Part 2, Providence, RI: Amer. Math. Soc., 2009, vol. 80, pp. 685-709. · Zbl 1183.14002 |

[22] | S. J. Kovács, ”Young person’s guide to moduli of higher dimensional varieties,” in Algebraic Geometry-Seattle 2005. Part 2, Providence, RI: Amer. Math. Soc., 2009, vol. 80, pp. 711-743. · Zbl 1182.14034 |

[23] | A. Kresch, ”On the geometry of Deligne-Mumford stacks,” in Algebraic Geometry-Seattle 2005. Part 1, Providence, RI: Amer. Math. Soc., 2009, vol. 80, pp. 259-271. · Zbl 1169.14001 |

[24] | S. Lang, Number Theory. III. Diophantine Geometry, New York: Springer-Verlag, 1991, vol. 60. · Zbl 0744.14012 |

[25] | G. Laumon and L. Moret-Bailly, Champs Algébriques, New York: Springer-Verlag, 2000, vol. 39. · Zbl 0945.14005 |

[26] | M. Lieblich, ”Twisted sheaves and the period-index problem,” Compos. Math., vol. 144, iss. 1, pp. 1-31, 2008. · Zbl 1133.14018 |

[27] | K. Matsuki and M. Olsson, ”Kawamata-Viehweg vanishing as Kodaira vanishing for stacks,” Math. Res. Lett., vol. 12, iss. 2-3, pp. 207-217, 2005. · Zbl 1080.14023 |

[28] | H. Matsumura, Commutative Ring Theory, Second ed., Cambridge: Cambridge Univ. Press, 1989, vol. 8. · Zbl 0666.13002 |

[29] | T. Matsusaka, ”On canonically polarized varieties. II,” Amer. J. Math., vol. 92, pp. 283-292, 1970. · Zbl 0195.22802 |

[30] | T. Matsusaka, ”Polarized varieties with a given Hilbert polynomial,” Amer. J. Math., vol. 94, pp. 1027-1077, 1972. · Zbl 0256.14004 |

[31] | M. Möller, E. Viehweg, and K. Zuo, ”Special families of curves, of abelian varieties, and of certain minimal manifolds over curves,” in Global Aspects of Complex Geometry, New York: Springer-Verlag, 2006, pp. 417-450. · Zbl 1112.14027 |

[32] | D. Mumford, Abelian Varieties, Published for the Tata Institute of Fundamental Research, Bombay, 1970, vol. 5. · Zbl 0223.14022 |

[33] | C. Okonek, M. Schneider, and H. Spindler, Vector Bundles on Complex Projective Spaces, Mass.: Birkhäuser, 1980, vol. 3. · Zbl 0438.32016 |

[34] | M. Olsson, ”A stacky semistable reduction theorem,” Int. Math. Res. Not., vol. 2004, iss. 29, pp. 1497-1509, 2004. · Zbl 1093.14002 |

[35] | A. N. Parvsin, ”Algebraic curves over function fields. I,” Izv. Akad. Nauk SSSR Ser. Mat., vol. 32, pp. 1191-1219, 1968. · Zbl 0181.23902 |

[36] | J. Serre, Local Fields, New York: Springer-Verlag, 1979, vol. 67. · Zbl 0423.12016 |

[37] | Schémas en Groupes. I: Propriétés Générales des Schémas en Groupes, New York: Springer-Verlag, 1970, vol. 151. · Zbl 0207.51401 |

[38] | Théorie des Intersections et Théorème de Riemann-Roch, New York: Springer-Verlag, 1971, vol. 225. · Zbl 0218.14001 |

[39] | M. Spivakovsky, ”A new proof of D. Popescu’s theorem on smoothing of ring homomorphisms,” J. Amer. Math. Soc., vol. 12, iss. 2, pp. 381-444, 1999. · Zbl 0919.13009 |

[40] | R. W. Thomason and T. Trobaugh, ”Higher algebraic \(K\)-theory of schemes and of derived categories,” in The Grothendieck Festschrift, Vol. III, Boston, MA: Birkhäuser, 1990, vol. 88, pp. 247-435. · Zbl 0731.14001 |

[41] | E. Viehweg, ”Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces,” in Algebraic Varieties and Analytic Varieties, Amsterdam: North-Holland, 1983, vol. 1, pp. 329-353. · Zbl 0513.14019 |

[42] | E. Viehweg, Quasi-Projective Moduli for Polarized Manifolds, New York: Springer-Verlag, 1995, vol. 30. · Zbl 0844.14004 |

[43] | 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, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001, vol. 6, pp. 249-284. · Zbl 1092.14044 |

[44] | E. Viehweg, ”Compactifications of smooth families and of moduli spaces of polarized manifolds,” Ann. of Math., vol. 172, pp. 809-910, 2010. · Zbl 1238.14009 |

[45] | E. Viehweg and K. Zuo, ”On the isotriviality of families of projective manifolds over curves,” J. Algebraic Geom., vol. 10, iss. 4, pp. 781-799, 2001. · Zbl 1079.14503 |

[46] | E. Viehweg and K. Zuo, ”Base spaces of non-isotrivial families of smooth minimal models,” in Complex Geometry, New York: Springer-Verlag, 2002, pp. 279-328. · Zbl 1006.14004 |

[47] | E. Viehweg and K. Zuo, ”Discreteness of minimal models of Kodaira dimension zero and subvarieties of moduli stacks,” in Surveys in Differential Geometry, Vol. VIII, Internat. Press, Somerville, MA, 2003, vol. VIII, pp. 337-356. · Zbl 1085.14016 |

[48] | A. Vistoli, ”Grothendieck topologies, fibered categories and descent theory,” in Fundamental Algebraic Geometry, Providence, RI: Amer. Math. Soc., 2005, vol. 123, pp. 1-104. · Zbl 1085.14001 |

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