Hyperbolicity of cyclic covers and complements
HTML articles powered by AMS MathViewer
- by Yuchen Liu PDF
- Trans. Amer. Math. Soc. 370 (2018), 5341-5357 Request permission
Abstract:
We prove that a cyclic cover of a smooth complex projective variety is Brody hyperbolic if its branch divisor is a generic small deformation of a large enough multiple of a Brody hyperbolic base-point-free ample divisor. We also show the hyperbolicity of complements of those branch divisors. As an application, we find new examples of Brody hyperbolic hypersurfaces in $\mathbb {P}^{n+1}$ that are cyclic covers of $\mathbb {P}^n$.References
- Damian Brotbek, On the hyperbolicity of general hypersurfaces, preprint available at arXiv:1604.00311.
- Herbert Clemens, Curves on generic hypersurfaces, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 4, 629–636. MR 875091, DOI 10.24033/asens.1521
- Ciro Ciliberto and Mikhail Zaidenberg, 3-fold symmetric products of curves as hyperbolic hypersurfaces in ${\Bbb P}^4$, Internat. J. Math. 14 (2003), no. 4, 413–436. MR 1984661, DOI 10.1142/S0129167X0300182X
- C. Ciliberto and M. Zaidenberg, Scrolls and hyperbolicity, Internat. J. Math. 24 (2013), no. 4, 1350026, 25. MR 3062966, DOI 10.1142/S0129167X13500262
- Jean-Pierre Demailly and Jawher El Goul, Hyperbolicity of generic surfaces of high degree in projective 3-space, Amer. J. Math. 122 (2000), no. 3, 515–546. MR 1759887, DOI 10.1353/ajm.2000.0019
- Jean-Pierre Demailly, Proof of the Kobayashi conjecture on the hyperbolicity of very general hypersurfaces, preprint available at arXiv:1501.07625.
- Ya Deng, Effectivity in the hyperbolicity-related problems, preprint available at arXiv:1606.03831.
- Simone Diverio, Joël Merker, and Erwan Rousseau, Effective algebraic degeneracy, Invent. Math. 180 (2010), no. 1, 161–223. MR 2593279, DOI 10.1007/s00222-010-0232-4
- Julien Duval, Une sextique hyperbolique dans $\textrm {P}^3(\mathbf C)$, Math. Ann. 330 (2004), no. 3, 473–476 (French, with English and French summaries). MR 2099189, DOI 10.1007/s00208-004-0551-0
- Jawher El Goul, Logarithmic jets and hyperbolicity, Osaka J. Math. 40 (2003), no. 2, 469–491. MR 1988702
- Lawrence Ein, Subvarieties of generic complete intersections, Invent. Math. 94 (1988), no. 1, 163–169. MR 958594, DOI 10.1007/BF01394349
- Lawrence Ein, Subvarieties of generic complete intersections. II, Math. Ann. 289 (1991), no. 3, 465–471. MR 1096182, DOI 10.1007/BF01446583
- Hirotaka Fujimoto, A family of hyperbolic hypersurfaces in the complex projective space, Complex Variables Theory Appl. 43 (2001), no. 3-4, 273–283. The Chuang special issue. MR 1820928, DOI 10.1080/17476930108815318
- Mark Green and Phillip Griffiths, Two applications of algebraic geometry to entire holomorphic mappings, The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979), Springer, New York-Berlin, 1980, pp. 41–74. MR 609557
- Mark L. Green, The hyperbolicity of the complement of $2n+1$ hyperplanes in general position in $P_{n}$ and related results, Proc. Amer. Math. Soc. 66 (1977), no. 1, 109–113. MR 457790, DOI 10.1090/S0002-9939-1977-0457790-9
- Dinh Tuan Huynh, Examples of hyperbolic hypersurfaces of low degree in projective spaces, Int. Math. Res. Not. IMRN 18 (2016), 5518–5558. MR 3567250, DOI 10.1093/imrn/rnv306
- Dinh Tuan Huynh, Construction of hyperbolic hypersurfaces of low degree in $\Bbb {P}^n(\Bbb {C})$, Internat. J. Math. 27 (2016), no. 8, 1650059, 9. MR 3530280, DOI 10.1142/S0129167X16500592
- Atsushi Ito and Yusaku Tiba, Curves in quadric and cubic surfaces whose complements are Kobayashi hyperbolically imbedded, Ann. Inst. Fourier (Grenoble) 65 (2015), no. 5, 2057–2068 (English, with English and French summaries). MR 3449206, DOI 10.5802/aif.2982
- Shoshichi Kobayashi, Hyperbolic manifolds and holomorphic mappings, Pure and Applied Mathematics, vol. 2, Marcel Dekker, Inc., New York, 1970. MR 0277770
- Shoshichi Kobayashi, Hyperbolic complex spaces, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 318, Springer-Verlag, Berlin, 1998. MR 1635983, DOI 10.1007/978-3-662-03582-5
- János Kollár, Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol. 200, Cambridge University Press, Cambridge, 2013. With a collaboration of Sándor Kovács. MR 3057950, DOI 10.1017/CBO9781139547895
- Serge Lang, Hyperbolic and Diophantine analysis, Bull. Amer. Math. Soc. (N.S.) 14 (1986), no. 2, 159–205. MR 828820, DOI 10.1090/S0273-0979-1986-15426-1
- Robert Lazarsfeld, Positivity in algebraic geometry. I, 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], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. MR 2095471, DOI 10.1007/978-3-642-18808-4
- M. McQuillan, Holomorphic curves on hyperplane sections of $3$-folds, Geom. Funct. Anal. 9 (1999), no. 2, 370–392. MR 1692470, DOI 10.1007/s000390050091
- Kazuo Masuda and Junjiro Noguchi, A construction of hyperbolic hypersurface of $\textbf {P}^n(\textbf {C})$, Math. Ann. 304 (1996), no. 2, 339–362. MR 1371771, DOI 10.1007/BF01446298
- Shigefumi Mori, On a generalization of complete intersections, J. Math. Kyoto Univ. 15 (1975), no. 3, 619–646. MR 393054, DOI 10.1215/kjm/1250523007
- Mihai Păun, Vector fields on the total space of hypersurfaces in the projective space and hyperbolicity, Math. Ann. 340 (2008), no. 4, 875–892. MR 2372741, DOI 10.1007/s00208-007-0172-5
- Erwan Rousseau, Weak analytic hyperbolicity of complements of generic surfaces of high degree in projective 3-space, Osaka J. Math. 44 (2007), no. 4, 955–971. MR 2383820
- Erwan Rousseau, Weak analytic hyperbolicity of generic hypersurfaces of high degree in $\Bbb P^4$, Ann. Fac. Sci. Toulouse Math. (6) 16 (2007), no. 2, 369–383 (English, with English and French summaries). MR 2331545, DOI 10.5802/afst.1152
- Erwan Rousseau, Logarithmic vector fields and hyperbolicity, Nagoya Math. J. 195 (2009), 21–40. MR 2552951, DOI 10.1017/S0027763000009685
- Xavier Roulleau and Erwan Rousseau, On the hyperbolicity of surfaces of general type with small $c^2_1$, J. Lond. Math. Soc. (2) 87 (2013), no. 2, 453–477. MR 3046280, DOI 10.1112/jlms/jds053
- Yum-Tong Siu, Hyperbolicity of generic high-degree hypersurfaces in complex projective space, Invent. Math. 202 (2015), no. 3, 1069–1166. MR 3425387, DOI 10.1007/s00222-015-0584-x
- Yum-Tong Siu and Sai-kee Yeung, Hyperbolicity of the complement of a generic smooth curve of high degree in the complex projective plane, Invent. Math. 124 (1996), no. 1-3, 573–618. MR 1369429, DOI 10.1007/s002220050064
- Yum-Tong Siu and Sai-Kee Yeung, Defects for ample divisors of abelian varieties, Schwarz lemma, and hyperbolic hypersurfaces of low degrees, Amer. J. Math. 119 (1997), no. 5, 1139–1172. MR 1473072, DOI 10.1353/ajm.1997.0033
- Bernard Shiffman and Mikhail Zaidenberg, Two classes of hyperbolic surfaces in $\textbf {P}^3$, Internat. J. Math. 11 (2000), no. 1, 65–101. MR 1757892, DOI 10.1142/S0129167X00000064
- Bernard Shiffman and Mikhail Zaidenberg, Hyperbolic hypersurfaces in $\Bbb P^n$ of Fermat-Waring type, Proc. Amer. Math. Soc. 130 (2002), no. 7, 2031–2035. MR 1896038, DOI 10.1090/S0002-9939-01-06417-6
- Claire Voisin, On a conjecture of Clemens on rational curves on hypersurfaces, J. Differential Geom. 44 (1996), no. 1, 200–213. MR 1420353
- Geng Xu, Subvarieties of general hypersurfaces in projective space, J. Differential Geom. 39 (1994), no. 1, 139–172. MR 1258918
- M. G. Zaĭdenberg, The complement to a general hypersurface of degree $2n$ in $\textbf {CP}^n$ is not hyperbolic, Sibirsk. Mat. Zh. 28 (1987), no. 3, 91–100, 222 (Russian). MR 904640
- Mikhail Zaidenberg, Hyperbolicity in projective spaces, Sūrikaisekikenkyūsho K\B{o}kyūroku 819 (1993), 136–156. International Symposium “Holomorphic Mappings, Diophantine Geometry and Related Topics” (Kyoto, 1992). MR 1247074
- M. G. Zaĭdenberg, Hyperbolicity of generic deformations, Funktsional. Anal. i Prilozhen. 43 (2009), no. 2, 39–46 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 43 (2009), no. 2, 113–118. MR 2542273, DOI 10.1007/s10688-009-0015-0
- M. G. Zaĭdenberg, Stability of hyperbolic embeddedness and the construction of examples, Mat. Sb. (N.S.) 135(177) (1988), no. 3, 361–372, 415 (Russian); English transl., Math. USSR-Sb. 63 (1989), no. 2, 351–361. MR 937646, DOI 10.1070/SM1989v063n02ABEH003278
- M. Zaĭdenberg and B. Shiffman, New examples of Kobayashi hyperbolic surfaces in $\Bbb C\rm P^3$, Funktsional. Anal. i Prilozhen. 39 (2005), no. 1, 90–94 (Russian); English transl., Funct. Anal. Appl. 39 (2005), no. 1, 76–79. MR 2132443, DOI 10.1007/s10688-005-0020-x
Additional Information
- Yuchen Liu
- Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544-1000
- Email: yuchenl@math.princeton.edu
- Received by editor(s): May 26, 2016
- Received by editor(s) in revised form: October 13, 2016
- Published electronically: December 29, 2017
- Additional Notes: The author was partially supported by NSF grant DMS-0968337.
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 5341-5357
- MSC (2010): Primary 32Q45; Secondary 14J70, 14J29
- DOI: https://doi.org/10.1090/tran/7097
- MathSciNet review: 3812107