Algebraic intermediate hyperbolicities. (English) Zbl 1511.32021
Summary: We extend Lang’s conjectures to the setting of intermediate hyperbolicity and prove two new results motivated by these conjectures. More precisely, we first extend the notion of algebraic hyperbolicity (originally introduced by Demailly) to the setting of intermediate hyperbolicity and show that this property holds if the appropriate exterior power of the cotangent bundle is ample. Then, we prove that this intermediate algebraic hyperbolicity implies the finiteness of the group of birational automorphisms and of the set of surjective maps from a given projective variety. Our work answers the algebraic analogue of a question of Kobayashi on analytic hyperbolicity.
MSC:
| 32Q45 | Hyperbolic and Kobayashi hyperbolic manifolds |
| 14E07 | Birational automorphisms, Cremona group and generalizations |
| 14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |
| 14G99 | Arithmetic problems in algebraic geometry; Diophantine geometry |
| 11G35 | Varieties over global fields |
| 14G05 | Rational points |
References:
| [1] | R. van Bommel, A. Javanpeykar, and L. Kamenova. Boundedness in families of projective varieties. Preprint, 2019, arxiv 1907.11225 |
| [2] | D. Brotbek. On the hyperbolicity of general hypersurfaces. Publ. Math. Inst. Hautes \'{E}tudes Sci. 126:1-34, 2017, DOI 10.1007/s10240-017-0090-3, zbl 1458.32022, MR3735863, arxiv 1604.00311 · Zbl 1458.32022 · doi:10.1007/s10240-017-0090-3 |
| [3] | J. A. Carlson. Some degeneracy theorems for entire functions with values in an algebraic variety. Trans. Amer. Math. Soc. 168:273-301, 1972, DOI 10.2307/1996176, zbl 0246.32023, MR0296356 · Zbl 0246.32023 · doi:10.2307/1996176 |
| [4] | J.-P. Demailly. Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials. In Algebraic geometry, Santa Cruz, CA, 1995, 285-360. Proc. Sympos. Pure Math., 62. Amer. Math. Soc., Providence, RI, 1997, zbl 0919.32014, MR1492539 · Zbl 0919.32014 |
| [5] | Y. Deng. Big Picard theorem and algebraic hyperbolicity for varieties admitting a variation of Hodge structures. Preprint, 2020, arxiv 2001.04426 |
| [6] | D. A. Eisenman. Intrinsic measures on complex manifolds and holomorphic mappings. Memoirs of the American Mathematical Society, 96. American Mathematical Soc., Providence, RI, 1970, zbl 0197.05901, MR0259165 · Zbl 0197.05901 |
| [7] | A. Etesse. Complex-analytic intermediate hyperbolicity, and finiteness properties. Preprint, 2020. Communications in Contemporary Mathematics, to appear, arxiv 2011.12583 |
| [8] | G. Faltings. The general case of S. Lang’s conjecture. In Barsotti symposium in algebraic geometry (Abano Terme, 1991), 175-182. Perspect. Math., 15. Academic Press, San Diego, CA, 1994, zbl 0823.14009, MR1307396 · Zbl 0823.14009 |
| [9] | M. Green and P. Griffiths. Two applications of algebraic geometry to entire holomorphic mappings. In The Chern symposium 1979 (Proc. Internat. Sympos., Berkeley, CA, 1979), 41-74. Springer, New York-Berlin, 1980, zbl 0508.32010, MR0609557 · Zbl 0508.32010 |
| [10] | M. Hanamura. On the birational automorphism groups of algebraic varieties. Compositio Math. 63(1):123-142, 1987, zbl 0655.14007, MR0906382 · Zbl 0655.14007 |
| [11] | J.-M. Hwang, S. Kebekus, and T. Peternell. Holomorphic maps onto varieties of non-negative Kodaira dimension. J. Algebraic Geom. 15(3):551-561, 2006, DOI 10.1090/S1056-3911-05-00411-X, zbl 1112.14014, MR2219848, arxiv math/0307220 · Zbl 1112.14014 · doi:10.1090/S1056-3911-05-00411-X |
| [12] | A. Javanpeykar. Finiteness of non-constant maps over a number field. Preprint, 2021. Mathematical Research Letters, to appear, arxiv 2112.11408 |
| [13] | A. Javanpeykar. The Lang-Vojta conjectures on projective pseudo-hyperbolic varieties. In Arithmetic geometry of logarithmic pairs and hyperbolicity of moduli spaces - hyperbolicity in Montr\'{e}al. CRM Short Courses, 135-196. Springer, Cham, 2020, zbl 1452.14017, MR4294877 · Zbl 1452.14017 |
| [14] | A. Javanpeykar. Arithmetic hyperbolicity: automorphisms and persistence. Math. Ann. 381(1-2):439-457, 2021, DOI 10.1007/s00208-021-02155-0, zbl 1484.11146, MR4322617, arxiv 1809.06818 · Zbl 1484.11146 · doi:10.1007/s00208-021-02155-0 |
| [15] | A. Javanpeykar and L. Kamenova. Demailly’s notion of algebraic hyperbolicity: geometricity, boundedness, moduli of maps. Math. Z. 296(3-4):1645-1672, 2020, DOI 10.1007/s00209-020-02489-6, zbl 07263171, MR4159843, arxiv 1807.03665 · Zbl 1505.32042 · doi:10.1007/s00209-020-02489-6 |
| [16] | A. Javanpeykar and R. Kucharczyk. Algebraicity of analytic maps to a hyperbolic variety. Math. Nachr. 293(8):1490-1504, 2020, DOI 10.1002/mana.201900098, zbl 1469.32006, MR4135823, arxiv 1806.09338 · Zbl 1469.32006 · doi:10.1002/mana.201900098 |
| [17] | A. Javanpeykar and J. Xie. Finiteness properties of pseudo-hyperbolic varieties. Int. Math. Res. Not. IMRN (3):1601-1643, 2022, DOI 10.1093/imrn/rnaa168, zbl 07471360, MR4373220, arxiv 1909.12187 · Zbl 1504.14045 · doi:10.1093/i |
| [18] | Y. Kawamata. On Bloch’s conjecture. Invent. Math. 57(1):97-100, 1980, DOI 10.1007/BF01389820, zbl 0569.32012, MR0564186 · Zbl 0569.32012 · doi:10.1007/BF01389820 |
| [19] | J. Koll{\'a}r and T. Matsusaka. Riemann-Roch type inequalities. Amer. J. Math. 105(1):229-252, 1983, DOI 10.2307/2374387, zbl 0538.14006, MR0692112 · Zbl 0538.14006 · doi:10.2307/2374387 |
| [20] | S. Kobayashi. Some problems in hyperbolic complex analysis. In Complex geometry (Osaka, 1990), 113-120. Lecture Notes in Pure and Appl. Math., 143. Dekker, New York, 1993, zbl 0795.32010, MR1201605 · Zbl 0795.32010 |
| [21] | S. Kobayashi. Hyperbolic complex spaces. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 318. Springer, Berlin, 1998, DOI 10.1007/978-3-662-03582-5, zbl 0917.32019, MR1635983 · Zbl 0917.32019 · doi:10.1007/978-3-662-03582-5 |
| [22] | Myung H. Kwack. Meromorphic mappings into compact complex manifolds with a Grauert positive bundle of \(q\)-forms. Proc. Amer. Math. Soc. 87(4):699-703, 1983, DOI 10.2307/2043363, zbl 0532.32010, MR0687645 · Zbl 0532.32010 · doi:10.2307/2043363 |
| [23] | S. Lang. Hyperbolic and Diophantine analysis. Bull. Amer. Math. Soc. (N.S.) 14(2):159-205, 1986, DOI 10.1090/S0273-0979-1986-15426-1, zbl 0602.14019, MR0828820 · Zbl 0602.14019 · doi:10.1090/S0273-0979-1986-15426-1 |
| [24] | R. Lazarsfeld. Positivity in algebraic geometry. I. Classical setting: line bundles and linear series. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 48. Springer, Berlin, 2004, DOI 10.1007/978-3-642-18808-4, zbl 1093.14501, MR2095471 · Zbl 1093.14501 · doi:10.1007/978-3-642-18808-4 |
| [25] | R. Lazarsfeld. Positivity in algebraic geometry. II. Positivity for vector bundles, and multiplier ideals. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 49. Springer, Berlin, 2004, DOI 10.1007/978-3-642-18810-7, zbl 1093.14500, MR2095472 · Zbl 1093.14500 · doi:10.1007/978-3-642-18810-7 |
| [26] | N. Nitsure. Construction of Hilbert and Quot schemes. In Fundamental algebraic geometry, 105-137. Math. Surveys Monogr., 123. Amer. Math. Soc., Providence, RI, 2005, MR2223407 |
| [27] | J. Noguchi. Meromorphic mappings into a compact complex space. Hiroshima Math. J. 7(2):411-425, 1977, zbl 0365.32017, MR0457791 · Zbl 0365.32017 |
| [28] | Y. Prokhorov and C. Shramov. Jordan property for groups of birational selfmaps. Compos. Math. 150(12):2054-2072, 2014, DOI 10.1112/S0010437X14007581, zbl 1314.14022, MR3292293, arxiv 1307.1784 · Zbl 1314.14022 · doi:10.1112/S0010437X14007581 |
| [29] | E. Rousseau. Hyperbolicity of geometric orbifolds. Trans. Amer. Math. Soc. 362(7):3799-3826, 2010, DOI 10.1090/S0002-9947-10-05019-1, zbl 1196.32018, MR2601610, arxiv 0809.1356 · Zbl 1196.32018 · doi:10.1090/S0002-9947-10-05019-1 |
| [30] | C. Voisin. On some problems of Kobayashi and Lang; algebraic approaches. In Current developments in mathematics, 2003, 53-125. Int. Press, Somerville, MA, 2003, zbl 1215.32014, MR2132645 · Zbl 1215.32014 |
| [31] | K. Yamanoi. Holomorphic curves in algebraic varieties of maximal Albanese dimension. Internat. J. Math. 26(6):1541006, 45 p., 2015, DOI 10.1142/S0129167X15410062, zbl 1319.32015, MR3356877 · Zbl 1319.32015 · doi:10.1142/S0129167X15410062 |
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