Arakelov inequalities in higher dimensions

References

[1] D. Abramovich and K. Karu, Weak semistable reduction in characteristic 0, Invent. Math. 139 (2000), no. 2, 241–273. 10.1007/s002229900024Search in Google Scholar

[2] K. Adiprasito, G. Liu and M. Temkin, Semistable reduction in characteristic 0, Sém. Lothar. Combin. 82B (2020), Article ID 25. Search in Google Scholar

[3] S. J. Arakelov, Families of algebraic curves with fixed degeneracies, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 1269–1293. Search in Google Scholar

[4] E. Bedulev and E. Viehweg, On the Shafarevich conjecture for surfaces of general type over function fields, Invent. Math. 139 (2000), no. 3, 603–615. 10.1007/s002229900046Search in Google Scholar

[5] B. Bhatt, W. Ho, Z. Patakfalvi and C. Schnell, Moduli of products of stable varieties, Compos. Math. 149 (2013), no. 12, 2036–2070. 10.1112/S0010437X13007288Search in Google Scholar

[6] S. Bloch and D. Gieseker, The positivity of the Chern classes of an ample vector bundle, Invent. Math. 12 (1971), 112–117. 10.1007/BF01404655Search in Google Scholar

[7] S. B. Bradlow, O. García-Prada and P. B. Gothen, Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces, Geom. Dedicata 122 (2006), 185–213. 10.1007/s10711-007-9127-ySearch in Google Scholar

[8] D. Brotbek and Y. Brunebarbe, Arakelov–Nevanlinna inequalities for variations of Hodge structures and applications, preprint (2020), https://arxiv.org/abs/2007.12957. Search in Google Scholar

[9] F. Campana and M. Păun, Foliations with positive slopes and birational stability of orbifold cotangent bundles, Publ. Math. Inst. Hautes Études Sci. 129 (2019), 1–49. 10.1007/s10240-019-00105-wSearch in Google Scholar

[10] B. Conrad, Grothendieck duality and base change, Lecture Notes in Math. 1750, Springer, Berlin 2000. 10.1007/b75857Search in Google Scholar

[11] P. Deligne, Un théorème de finitude pour la monodromie, Discrete groups in geometry and analysis (New Haven 1984), Progr. Math. 67, Birkhäuser, Boston (1987), 1–19. 10.1007/978-1-4899-6664-3_1Search in Google Scholar

[12] H. Esnault and E. Viehweg, Lectures on vanishing theorems, DMV Seminar 20, Birkhäuser, Basel 1992. 10.1007/978-3-0348-8600-0Search in Google Scholar

[13] O. Fujino, Semipositivity theorems for moduli problems, Ann. of Math. (2) 187 (2018), no. 3, 639–665. 10.4007/annals.2018.187.3.1Search in Google Scholar

[14] T. Fujita, On Kähler fiber spaces over curves, J. Math. Soc. Japan 30 (1978), no. 4, 779–794. 10.2969/jmsj/03040779Search in Google Scholar

[15] C. D. Hacon and C. Xu, Existence of log canonical closures, Invent. Math. 192 (2013), no. 1, 161–195. 10.1007/s00222-012-0409-0Search in Google Scholar

[16] R. Hartshorne, Algebraic geometry, Grad. Texts in Math. 52, Springer, New York 1977. 10.1007/978-1-4757-3849-0Search in Google Scholar

[17] D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves, 2nd ed., Cambridge Math. Libr., Cambridge University, Cambridge 2010. 10.1017/CBO9780511711985Search in Google Scholar

[18] J. Jost and K. Zuo, Arakelov type inequalities for Hodge bundles over algebraic varieties. I. Hodge bundles over algebraic curves, J. Algebraic Geom. 11 (2002), no. 3, 535–546. 10.1090/S1056-3911-02-00299-0Search in Google Scholar

[19] K. Karu, Minimal models and boundedness of stable varieties, J. Algebraic Geom. 9 (2000), no. 1, 93–109. Search in Google Scholar

[20] Y. Kawamata, Characterization of abelian varieties, Compos. Math. 43 (1981), no. 2, 253–276. Search in Google Scholar

[21] Y. Kawamata, K. Matsuda and K. Matsuki, Introduction to the minimal model problem, Algebraic geometry (Sendai 1985), Adv. Stud. Pure Math. 10, North-Holland, Amsterdam (1987), 283–360. 10.2969/aspm/01010283Search in Google Scholar

[22] S. Kebekus and S. J. Kovács, Families of canonically polarized varieties over surfaces, Invent. Math. 172 (2008), no. 3, 657–682. 10.1007/s00222-008-0128-8Search in Google Scholar

[23] S. Kebekus and S. J. Kovács, The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties, Duke Math. J. 155 (2010), no. 1, 1–33. 10.1215/00127094-2010-049Search in Google Scholar

[24] G. Kempf, F. F. Knudsen, D. Mumford and B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Math. 339, Springer, Berlin 1973. 10.1007/BFb0070318Search in Google Scholar

[25] J. Kollár, Higher direct images of dualizing sheaves. I, Ann. of Math. (2) 123 (1986), no. 1, 11–42. 10.2307/1971351Search in Google Scholar

[26] J. Kollár, Projectivity of complete moduli, J. Differential Geom. 32 (1990), no. 1, 235–268. 10.4310/jdg/1214445046Search in Google Scholar

[27] J. Kollár, Moduli of varieties of general type, Handbook of moduli. Vol. II, Adv. Lect. Math. (ALM) 25, International Press, Somerville (2013), 131–157. Search in Google Scholar

[28] J. Kollár, Log-plurigenera in stable families, Peking Math. J. 1 (2018), no. 1, 81–107. 10.1007/s42543-018-0002-6Search in Google Scholar

[29] J. Kollár, Families of varieties of general type, Cambridge Tracts in Math. 231, Cambridge University, Cambridge 2023. 10.1017/9781009346115Search in Google Scholar

[30] J. Kollár and S. J. Kovács, Log canonical singularities are Du Bois, J. Amer. Math. Soc. 23 (2010), no. 3, 791–813. 10.1090/S0894-0347-10-00663-6Search in Google Scholar

[31] J. Kollár and S. J. Kovács, Deformations of log canonical and 𝐹-pure singularities, Algebr. Geom. 7 (2020), no. 6, 758–780. 10.14231/AG-2020-027Search in Google Scholar

[32] J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Math. 134, Cambridge University, Cambridge 1998. 10.1017/CBO9780511662560Search in Google Scholar

[33] J. Kollár and N. I. Shepherd-Barron, Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), no. 2, 299–338. 10.1007/BF01389370Search in Google Scholar

[34] S. J. Kovács, Smooth families over rational and elliptic curves, J. Algebraic Geom. 5 (1996), no. 2, 369–385; Erratum: J. Algebraic Geom. 6 (1997), no. 2, 391. Search in Google Scholar

[35] 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. 10.1515/crll.1997.487.171Search in Google Scholar

[36] S. J. Kovács, Algebraic hyperbolicity of fine moduli spaces, J. Algebraic Geom. 9 (2000), no. 1, 165–174. Search in Google Scholar

[37] S. J. Kovács, Logarithmic vanishing theorems and Arakelov–Parshin boundedness for singular varieties, Compos. Math. 131 (2002), no. 3, 291–317. Search in Google Scholar

[38] S. J. Kovács, Singularities of stable varieties, Handbook of moduli. Vol. II, Adv. Lect. Math. (ALM) 25, International Press, Somerville (2013), 159–203. Search in Google Scholar

[39] S. J. Kovács and M. Lieblich, Boundedness of families of canonically polarized manifolds: A higher dimensional analogue of Shafarevich’s conjecture, Ann. of Math. (2) 172 (2010), no. 3, 1719–1748. Search in Google Scholar

[40] S. J. Kovács and Z. Patakfalvi, Projectivity of the moduli space of stable log-varieties and subadditivity of log-Kodaira dimension, J. Amer. Math. Soc. 30 (2017), no. 4, 959–1021. 10.1090/jams/871Search in Google Scholar

[41] S. J. Kovács and K. Schwede, Inversion of adjunction for rational and Du Bois pairs, Algebra Number Theory 10 (2016), no. 5, 969–1000. 10.2140/ant.2016.10.969Search in Google Scholar

[42] S. J. Kovács and B. Taji, Hodge sheaves underlying flat projective families, Math. Z. 303 (2023), no. 3, Paper No. 75. 10.1007/s00209-023-03219-4Search in Google Scholar

[43] V. Koziarz and J. Maubon, Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type, Geom. Dedicata 137 (2008), 85–111. 10.1007/s10711-008-9288-3Search in Google Scholar

[44] V. Koziarz and J. Maubon, The Toledo invariant on smooth varieties of general type, J. reine angew. Math. 649 (2010), 207–230. 10.1515/crelle.2010.093Search in Google Scholar

[45] R. Lazarsfeld, Positivity in algebraic geometry. I. Classical setting: Line bundles and linear series, Ergeb. Math. Grenzgeb. (3) 48, Springer, Berlin 2004. 10.1007/978-3-642-18808-4_4Search in Google Scholar

[46] T. Matsusaka, Polarized varieties with a given Hilbert polynomial, Amer. J. Math. 94 (1972), 1027–1077. 10.2307/2373563Search in Google Scholar

[47] A. N. Paršin, Algebraic curves over function fields. I, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 1191–1219. Search in Google Scholar

[48] Z. Patakfalvi, Viehweg’s hyperbolicity conjecture is true over compact bases, Adv. Math. 229 (2012), no. 3, 1640–1642. 10.1016/j.aim.2011.12.013Search in Google Scholar

[49] C. A. M. Peters, Arakelov-type inequalities for hodge bundles, preprint (2000), https://arxiv.org/pdf/math/0007102.pdf. Search in Google Scholar

[50] I. R. Šafarevič, Algebraic number fields, Proceedings of the International Congress of Mathematicians (Stockholm 1962), Institut Mittag-Leffler, Djursholm (1963), 163–176. Search in Google Scholar

[51] Y.-T. Siu, Invariance of plurigenera, Invent. Math. 134 (1998), no. 3, 661–673. 10.1007/s002220050276Search in Google Scholar

[52] B. Taji, On the Kodaira dimension of base spaces of families of manifolds, J. Pure Appl. Algebra 225 (2021), no. 11, Paper No. 106729. 10.1016/j.jpaa.2021.106729Search in Google Scholar

[53] B. Taji, Birational geometry of smooth families of varieties admitting good minimal models, Eur. J. Math. 9 (2023), no. 4, Article ID 88. 10.1007/s40879-023-00681-6Search in Google Scholar

[54] E. Viehweg, Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces, Algebraic varieties and analytic varieties (Tokyo 1981), Adv. Stud. Pure Math. 1, North-Holland, Amsterdam (1983), 329–353. 10.2969/aspm/00110329Search in Google Scholar

[55] E. Viehweg, Quasi-projective moduli for polarized manifolds, Ergeb. Math. Grenzgeb. (3) 30, Springer, Berlin 1995. 10.1007/978-3-642-79745-3Search in Google Scholar

[56] E. Viehweg, Arakelov inequalities, Surveys in differential geometry. Vol. XIII, Surv. Differ. Geom. 13, International Press, Somerville (2009), 245–275. 10.4310/SDG.2008.v13.n1.a8Search in Google Scholar

[57] E. Viehweg, Compactifications of smooth families and of moduli spaces of polarized manifolds, Ann. of Math. (2) 172 (2010), no. 2, 809–910. 10.4007/annals.2010.172.809Search in Google Scholar

[58] E. Viehweg and K. Zuo, Base spaces of non-isotrivial families of smooth minimal models, Complex geometry (Göttingen 2000), Springer, Berlin (2002), 279–328. 10.1007/978-3-642-56202-0_16Search in Google Scholar

[59] E. Viehweg and K. Zuo, On the Brody hyperbolicity of moduli spaces for canonically polarized manifolds, Duke Math. J. 118 (2003), no. 1, 103–150. 10.1215/S0012-7094-03-11815-3Search in Google Scholar

[60] E. Viehweg and K. Zuo, Arakelov inequalities and the uniformization of certain rigid Shimura varieties, J. Differential Geom. 77 (2007), no. 2, 291–352. 10.4310/jdg/1191860396Search in Google Scholar

[61] S. Zucker, Remarks on a theorem of Fujita, J. Math. Soc. Japan 34 (1982), no. 1, 47–54. 10.2969/jmsj/03410047Search in Google Scholar