On the hyperbolicity of base spaces for maximally variational families of smooth projective varieties (with an appendix by Dan Abramovich). (English) Zbl 1487.32146

Deng, Ya

J. Eur. Math. Soc. (JEMS) 24, No. 7, 2315-2359 (2022).

Summary: For maximal variational smooth families of projective manifolds whose general fibers have semi-ample canonical bundle, the Viehweg hyperbolicity conjecture states that the base spaces of such families are of log-general type. This deep conjecture was recently proved by Campana-Păun and was later generalized by Popa-Schnell. In this paper we prove that those base spaces are pseudo Kobayashi hyperbolic, as predicted by the Lang conjecture: any complex quasi-projective manifold is pseudo Kobayashi hyperbolic if it is of log-general type. As a consequence, we prove the Brody hyperbolicity of moduli spaces of polarized manifolds with semi-ample canonical bundle. This proves a 2003 conjecture by E. Viehweg and K. Zuo [Duke Math. J. 118, No. 1, 103–150 (2003; Zbl 1042.14010)]. We also prove the Kobayashi hyperbolicity of base spaces for effectively parametrized families of minimal projective manifolds of general type. This generalizes previous work by To-Yeung, who further assumed that these families are canonically polarized.

MSC:

32Q45	Hyperbolic and Kobayashi hyperbolic manifolds
14D07	Variation of Hodge structures (algebro-geometric aspects)
14E99	Birational geometry

Keywords:

pseudo Kobayashi hyperbolicity; Brody hyperbolicity; moduli spaces

Citations:

Zbl 1042.14010

References:

[1]	Abramovich, D., Hassett, B.: Stable varieties with a twist. In: Classification of Algebraic Vari-eties, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 1-38 (2011) Zbl 1223.14039 MR 2779465 · Zbl 1223.14039
[2]	Abramovich, D., Karu, K.: Weak semistable reduction in characteristic 0. Invent. Math. 139, 241-273 (2000) Zbl 0958.14006 MR 1738451 · Zbl 0958.14006
[3]	Abramovich, D., Vistoli, A.: Compactifying the space of stable maps. J. Amer. Math. Soc. 15, 27-75 (2002) Zbl 0991.14007 MR 1862797 · Zbl 0991.14007
[4]	Arapura, D.: Frobenius amplitude and strong vanishing theorems for vector bundles. Duke Math. J. 121, 231-267 (2004) Zbl 1067.14018 MR 2034642 · Zbl 1067.14018
[5]	Berndtsson, B., Pȃun, M.: Bergman kernels and the pseudoeffectivity of relative canonical bundles. Duke Math. J. 145, 341-378 (2008) Zbl 1181.32025 MR 2449950 · Zbl 1181.32025
[6]	Berndtsson, B., Pȃun, M.: Bergman kernels and subadjunction. arXiv:1002.4145 (2010)
[7]	Berndtsson, B., Pȃun, M., Wang, X.: Algebraic fiber spaces and curvature of higher direct images. arXiv:1704.02279 (2017)
[8]	Bhatt, B., Ho, W., Patakfalvi, Z., Schnell, C.: Moduli of products of stable varieties. Compos. Math. 149, 2036-2070 (2013) Zbl 1300.14012 MR 3143705 · Zbl 1300.14012
[9]	Boucksom, S., Diverio, S.: A note on Lang’s conjecture for quotients of bounded domains. Épijournal Géom. Algébrique 5, art. 5, 10 pp. (2021) Zbl 07341100 MR 4275254 · Zbl 1464.32036
[10]	Brotbek, D.: On the hyperbolicity of general hypersurfaces. Publ. Math. Inst. Hautes Études Sci. 126, 1-34 (2017) Zbl 1458.32022 MR 3735863 · Zbl 1458.32022
[11]	Brotbek, D., Deng, Y.: Kobayashi hyperbolicity of the complements of general hypersurfaces of high degree. Geom. Funct. Anal. 29, 690-750 (2019) Zbl 1452.32034 MR 3962877 · Zbl 1452.32034
[12]	Cadorel, B., Rousseau, E., Taji, B.: Hyperbolicity of singular spaces. J. École Polytech. Math. 6, 1-18 (2019) Zbl 1427.32017 MR 3882579 · Zbl 1427.32017
[13]	Cadorel, B., Diverio, S., Guenancia, H.: On subvarieties of singular quotients of bounded domains. arXiv:1905.04212 (2019)
[14]	Campana, F., Pȃun, M.: Orbifold generic semi-positivity: an application to families of canon-ically polarized manifolds. Ann. Inst. Fourier (Grenoble) 65, 835-861 (2015) Zbl 1338.14012 MR 3449168 · Zbl 1338.14012
[15]	Campana, F., Pȃun, M.: Positivity properties of the bundle of logarithmic tensors on compact Kähler manifolds. Compos. Math. 152, 2350-2370 (2016) Zbl 1388.32018 MR 3577897 · Zbl 1388.32018
[16]	Campana, F., Pȃun, M.: Foliations with positive slopes and birational stability of orbifold cotangent bundles. Publ. Math. Inst. Hautes Études Sci. 129, 1-49 (2019) Zbl 1423.14109 MR 3949026 · Zbl 1423.14109
[17]	Cao, J.: Albanese maps of projective manifolds with nef anticanonical bundles. Ann. Sci. École Norm. Sup. (4) 52, 1137-1154 (2019) Zbl 1442.32026 MR 4057779 · Zbl 1442.32026
[18]	Cao, J., Pȃun, M.: Kodaira dimension of algebraic fiber spaces over abelian varieties. Invent. Math. 207, 345-387 (2017) Zbl 1360.14050 MR 3592759 · Zbl 1360.14050
[19]	Catanese, F., Kawamata, Y.: Fujita decomposition over higher dimensional base. Eur. J. Math. 5, 720-728 (2019) Zbl 1423.14071 MR 3993260 · Zbl 1423.14071
[20]	Cattani, E., Kaplan, A., Schmid, W.: Degeneration of Hodge structures. Ann. of Math. (2) 123, 457-535 (1986) Zbl 0617.14005 MR 840721 · Zbl 0617.14005
[21]	Demailly, J.-P.: Algebraic criteria for Kobayashi hyperbolic projective varieties and jet dif-ferentials. In: Algebraic Geometry-Santa Cruz 1995, Proc. Sympos. Pure Math. 62, Amer. Math. Soc., Providence, RI, 285-360 (1997) Zbl 0919.32014 MR 1492539 · Zbl 0919.32014
[22]	Demailly, J.-P.: Recent results on the Kobayashi and Green-Griffiths-Lang conjectures. Japan J. Math. 15, 1-120 (2020) Zbl 1436.32086 MR 4068832 · Zbl 1436.32086
[23]	Deng, Y.: Applications of the Ohsawa-Takegoshi extension theorem to direct image problems. arXiv:1703.07279 (2017)
[24]	Deng, Y.: Kobayashi hyperbolicity of moduli spaces of minimal projective manifolds of gen-eral type (with an appendix by Dan Abramovich). arXiv:1806.01666 (2018)
[25]	Deng, Y.: Pseudo Kobayashi hyperbolicity of base spaces of families of minimal projective manifolds with maximal variation. arXiv:1809.05891 (2018)
[26]	Deng, Y.: Hyperbolicity of bases of log Calabi-Yau families. arXiv:1901.04423 (2019)
[27]	Deng, Y.: Big Picard theorem and algebraic hyperbolicity for varieties admitting a variation of Hodge structures. arXiv:2001.04426 (2020)
[28]	Deng, Y., Lu, S., Sun, R., Zuo, K.:, Base spaces of non-isotrivial families of smooth minimal models. In: Complex Geometry (Göttingen, 2000), Springer, Berlin, 279-328 (2002) MR 1922109 · Zbl 1006.14004
[29]	Deng, Y., Lu, S.: Sun, R., Zuo, K.: Picard theorems or moduli spaces of polarized varieties. arXiv:1911.02973 (2019)
[30]	Dervan, R., Ross, J.: Stable maps in higher dimensions. Math. Ann. 374, 1033-1073 (2019) Zbl 07087149 MR 3985107 · Zbl 1505.14025
[31]	Griffiths, P., Tu, L.: Curvature properties of the Hodge bundles. In: Topics in Transcendental Algebraic Geometry (Princeton, NJ, 1981/1982), Ann. of Math. Stud. 106, Princeton Univ. Press, Princeton, NJ, 29-49 (1984) Zbl 0579.32045 MR 756844 · Zbl 0579.32045
[32]	Karu, K.: Minimal models and boundedness of stable varieties. J. Algebraic Geom. 9, 93-109 (2000) Zbl 0980.14008 MR 1713521 · Zbl 0980.14008
[33]	Kashiwara, M.: The asymptotic behavior of a variation of polarized Hodge structure. Publ. RIMS Kyoto Univ. 21, 853-875 (1985) Zbl 0594.14012 MR 817170 · Zbl 0594.14012
[34]	Kawamata, Y.: Minimal models and the Kodaira dimension of algebraic fiber spaces. J. Reine Angew. Math. 363, 1-46 (1985) Zbl 0589.14014 MR 814013 · Zbl 0589.14014
[35]	Kebekus, S., Kovács, S. J.: Families of canonically polarized varieties over surfaces. Invent. Math. 172, 657-682 (2008) Zbl 1140.14031 MR 2393082 · Zbl 1140.14031
[36]	Kebekus, S., Kovács, S. J.: Families of varieties of general type over compact bases. Adv. Math. 218, 649-652 (2008) Zbl 1137.14027 MR 2414316 · Zbl 1137.14027
[37]	Kebekus, S., Kovács, S. J.: The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties. Duke Math. J. 155, 1-33 (2010) Zbl 1208.14027 MR 2730371 · Zbl 1208.14027
[38]	Kobayashi, S.: Hyperbolic Complex Spaces. Grundlehren Math. Wiss. 318, Springer, Berlin (1998) Zbl 0917.32019 MR 1635983 · Zbl 0917.32019
[39]	Kollár, J., Shepherd-Barron, N. I.: Threefolds and deformations of surface singularities. Invent. Math. 91, 299-338 (1988) Zbl 0642.14008 MR 922803 · Zbl 0642.14008
[40]	Kovács, S. J., Patakfalvi, Z.: Projectivity of the moduli space of stable log-varieties and sub-additivity of log-Kodaira dimension. J. Amer. Math. Soc. 30, 959-1021 (2017) Zbl 1393.14034 MR 3671934 · Zbl 1393.14034
[41]	Kresch, A., Vistoli, A.: On coverings of Deligne-Mumford stacks and surjectivity of the Brauer map. Bull. London Math. Soc. 36, 188-192 (2004) Zbl 1062.14004 MR 2026412 · Zbl 1062.14004
[42]	Lang, S.: Number Theory. III. Encyclopaedia Math. Sci. 60, Springer, Berlin (1991) Zbl 0744.14012 MR 1112552 · Zbl 0744.14012
[43]	Mori, S.: Classification of higher-dimensional varieties. In: Algebraic Geometry, Bowdoin, 1985 (Brunswick, ME, 1985), Proc. Sympos. Pure Math. 46, Amer. Math. Soc., Providence, RI, 269-331 (1987) Zbl 0656.14022 MR 927961 · Zbl 0656.14022
[44]	Patakfalvi, Z.: Viehweg’s hyperbolicity conjecture is true over compact bases. Adv. Math. 229, 1640-1642 (2012) Zbl 1235.14031 MR 2871152 · Zbl 1235.14031
[45]	Pȃun, M., Takayama, S.: Positivity of twisted relative pluricanonical bundles and their direct images. J. Algebraic Geom. 27, 211-272 (2018) Zbl 1430.14017 MR 3764276 · Zbl 1430.14017
[46]	Peters, C. A. M.: A criterion for flatness of Hodge bundles over curves and geometric applic-ations. Math. Ann. 268, 1-19 (1984) Zbl 0548.14004 MR 744325 · Zbl 0548.14004
[47]	Popa, M., Schnell, C.: Viehweg’s hyperbolicity conjecture for families with maximal vari-ation. Invent. Math. 208, 677-713 (2017) Zbl 1375.14043 MR 3648973 · Zbl 1375.14043
[48]	Popa, M., Taji, B., Wu, L.: Brody hyperbolicity of base spaces of certain families of varieties. Algebra Number Theory 13, 2205-2242 (2019) Zbl 1444.14035 MR 4039502 · Zbl 1444.14035
[49]	Rousseau, E.: Hyperbolicity, automorphic forms and Siegel modular varieties. Ann. Sci. École Norm. Sup. (4) 49, 249-255 (2016) Zbl 1361.32032 MR 3465981 · Zbl 1361.32032
[50]	Royden, H. L.: Intrinsic metrics on Teichmüller space. In: Proc. Int. Congress of Mathem-aticians (Vancouver, BC, 1974), Vol. 2, 217-221 (1975) Zbl 0344.32005 MR 0447636 · Zbl 0344.32005
[51]	Schmid, W.: Variation of Hodge structure: the singularities of the period mapping. Invent. Math. 22, 211-319 (1973) Zbl 0278.14003 MR 382272 · Zbl 0278.14003
[52]	Schnell, C.: On a theorem of Campana and Pȃun. Épijournal Géom. Algébrique 1, art. 8, 9 pp. (2017) Zbl 1409.14032 MR 3743111 · Zbl 1409.14032
[53]	Schumacher, G.: Positivity of relative canonical bundles and applications. Invent. Math. 190, 1-56 (2012) Zbl 1258.32005 MR 2969273 · Zbl 1258.32005
[54]	Schumacher, G.: Moduli of canonically polarized manifolds, higher order Kodaira-Spencer maps, and an analogy to Calabi-Yau manifolds. In: Uniformization, Riemann-Hilbert Corres-pondence, Calabi-Yau Manifolds & Picard-Fuchs Equations, Adv. Lect. Math. 42, Int. Press, Somerville, MA, 369-399 (2018) Zbl 1411.32018 MR 3822912 · Zbl 1411.32018
[55]	Siu, Y. T.: Curvature of the Weil-Petersson metric in the moduli space of compact Kähler-Einstein manifolds of negative first Chern class. In: Contributions to Several Complex Variables, Aspects Math. E9, Vieweg, Braunschweig, 261-298 (1986) Zbl 0598.32020 MR 859202 · Zbl 0598.32020
[56]	Siu, Y.-T.: Hyperbolicity of generic high-degree hypersurfaces in complex projective space. Invent. Math. 202, 1069-1166 (2015) Zbl 1333.32020 MR 3425387 · Zbl 1333.32020
[57]	Steenbrink, J. H. M.: Mixed Hodge structure on the vanishing cohomology. In: Real and Com-plex Singularities (Oslo, 1976), 525-563 (1977) Zbl 0373.14007 MR 0485870 · Zbl 0373.14007
[58]	To, W.-K., Yeung, S.-K.: Finsler metrics and Kobayashi hyperbolicity of the moduli spaces of canonically polarized manifolds. Ann. of Math. (2) 181, 547-586 (2015) Zbl 1319.32024 MR 3275846 · Zbl 1319.32024
[59]	To, W.-K., Yeung, S.-K.: Augmented Weil-Petersson metrics on moduli spaces of polarized Ricci-flat Kähler manifolds and orbifolds. Asian J. Math. 22, 705-727 (2018) Zbl 1411.32014 MR 3862058 · Zbl 1411.32014
[60]	Viehweg, E.: Weak positivity and the stability of certain Hilbert points. II. Invent. Math. 101, 191-223 (1990) Zbl 0721.14007 MR 1055715 · Zbl 0721.14007
[61]	Viehweg, E.: Quasi-Projective Moduli for Polarized Manifolds. Ergeb. Math. Grenzgeb. 30, Springer, Berlin (1995) Zbl 0844.14004 MR 1368632 · Zbl 0844.14004
[62]	Viehweg, E., Zuo, K.: On the isotriviality of families of projective manifolds over curves. J. Algebraic Geom. 10, 781-799 (2001) Zbl 1079.14503 MR 1838979 · Zbl 1079.14503
[63]	Viehweg, E., Zuo, K.: On the Brody hyperbolicity of moduli spaces for canonically polarized manifolds. Duke Math. J. 118, 103-150 (2003) Zbl 042.14010 MR 1978884 · Zbl 1042.14010
[64]	Wolpert, S. A.: Chern forms and the Riemann tensor for the moduli space of curves. Invent. Math. 85, 119-145 (1986) Zbl 0595.32031 MR 842050 · Zbl 0595.32031
[65]	Yamanoi, K.: Pseudo Kobayashi hyperbolicity of subvarieties of general type on abelian vari-eties. J. Math. Soc. Japan 71, 259-298 (2019) Zbl 1417.32029 MR 3909921 · Zbl 1417.32029
[66]	Zucker, S.: Degeneration of Hodge bundles (after Steenbrink). In: Topics in Transcendental Algebraic Geometry (Princeton, NJ, 1981/1982), Ann. of Math. Stud. 106, Princeton Univ. Press, Princeton, NJ, 121-141 (1984) Zbl 0574.14008 MR 756849 · Zbl 0574.14008
[67]	Zuo, K.: On the negativity of kernels of Kodaira-Spencer maps on Hodge bundles and applic-ations. Asian J. Math. 4, 279-301 (2000) Zbl 0983.32020 MR 1803724 · Zbl 0983.32020