We construct a Brody hyperbolic Horikawa surface that is a double cover of branched along a smooth curve of degree 10. We also construct Brody hyperbolic double covers of Hirzebruch surfaces with branch loci of the lowest possible bidegree.
Nous construisons une surface de Horikawa Brody-hyperbolique qui est un revêtement double de ramifié le long d’une courbe lisse de degré 10. Nous construisons également des surfaces Brody-hyperboliques qui sont des revêtements doubles de surfaces de Hirzebruch, dont l’ensemble de ramification est de bidegré minimal.
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Accepted:
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Keywords: Brody hyperbolicity, Horikawa surfaces, cyclic covers
Mot clés : Hyperbolicité au sens de Brody, surface de Horikawa, revêtements cycliques
Liu, Yuchen 1
@article{AIF_2018__68_2_541_0, author = {Liu, Yuchen}, title = {Construction of hyperbolic {Horikawa} surfaces}, journal = {Annales de l'Institut Fourier}, pages = {541--561}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {68}, number = {2}, year = {2018}, doi = {10.5802/aif.3169}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3169/} }
TY - JOUR AU - Liu, Yuchen TI - Construction of hyperbolic Horikawa surfaces JO - Annales de l'Institut Fourier PY - 2018 SP - 541 EP - 561 VL - 68 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3169/ DO - 10.5802/aif.3169 LA - en ID - AIF_2018__68_2_541_0 ER -
%0 Journal Article %A Liu, Yuchen %T Construction of hyperbolic Horikawa surfaces %J Annales de l'Institut Fourier %D 2018 %P 541-561 %V 68 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3169/ %R 10.5802/aif.3169 %G en %F AIF_2018__68_2_541_0
Liu, Yuchen. Construction of hyperbolic Horikawa surfaces. Annales de l'Institut Fourier, Volume 68 (2018) no. 2, pp. 541-561. doi : 10.5802/aif.3169. https://aif.centre-mersenne.org/articles/10.5802/aif.3169/
[1] Families of curves on a surface of general type, Dokl. Akad. Nauk SSSR, Volume 236 (1977) no. 5, pp. 1041-1044 | MR | Zbl
[2] Compact manifolds and hyperbolicity, Trans. Am. Math. Soc., Volume 235 (1978), pp. 213-219 | DOI | MR | Zbl
[3] Hyperbolicity of generic surfaces of high degree in projective 3-space, Am. J. Math., Volume 122 (2000) no. 3, pp. 515-546 http://muse.jhu.edu/journals/american_journal_of_mathematics/v122/122.3demailly.pdf | DOI | MR | Zbl
[4] Théorie nouvelle des familles complexes normales. Applications a l’étude des fonctions algébroïdes, Ann. Sci. Éc. Norm. Supér., Volume 61 (1944), pp. 1-44 | DOI | MR | Zbl
[5] Une sextique hyperbolique dans , Math. Ann., Volume 330 (2004) no. 3, pp. 473-476 | DOI | MR | Zbl
[6] Around Brody lemma (2017) (http://arxiv.org/abs/1703.01850)
[7] Two applications of algebraic geometry to entire holomorphic mappings, The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979), Springer, 1980, pp. 41-74 | MR | Zbl
[8] Algebraic surfaces of general type with small I, Ann. Math., Volume 104 (1976) no. 2, pp. 357-387 | DOI | MR | Zbl
[9] Examples of hyperbolic hypersurfaces of low degree in projective spaces, Int. Math. Res. Not. IMRN (2016) no. 18, pp. 5518-5558 | DOI | MR
[10] Curves in quadric and cubic surfaces whose complements are Kobayashi hyperbolically imbedded, Ann. Inst. Fourier (Grenoble), Volume 65 (2015) no. 5, pp. 2057-2068 http://aif.cedram.org/item?id=AIF_2015__65_5_2057_0 | DOI | MR | Zbl
[11] Hyperbolic manifolds and holomorphic mappings, Pure and Applied Mathematics, 2, Marcel Dekker, Inc., 1970, ix+148 pages | MR | Zbl
[12] Hyperbolic complex spaces, Grundlehren der Mathematischen Wissenschaften, 318, Springer, 1998, xiv+471 pages | DOI | MR | Zbl
[13] Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 32, Springer, 1996, viii+320 pages | DOI | MR | Zbl
[14] Hyperbolic and Diophantine analysis, Bull. Am. Math. Soc., Volume 14 (1986) no. 2, pp. 159-205 | DOI | MR | Zbl
[15] Hyperbolicity of cyclic covers and complements (2016) (to appear in Trans. Am. Math. Soc., available at http://arxiv.org/abs/1601.07514)
[16] Holomorphic curves in surfaces of general type, Proc. Nat. Acad. Sci. U.S.A., Volume 87 (1990) no. 1, pp. 80-82 | DOI | MR | Zbl
[17] Diophantine approximations and foliations, Publ. Math., Inst. Hautes Étud. Sci. (1998) no. 87, pp. 121-174 | DOI | MR | Zbl
[18] On the hyperbolicity of surfaces of general type with small , J. Lond. Math. Soc., Volume 87 (2013) no. 2, pp. 453-477 | DOI | MR | Zbl
[19] Constructing low degree hyperbolic surfaces in , Houston J. Math., Volume 28 (2002) no. 2, pp. 377-388 | MR | Zbl
[20] Subvarieties of general hypersurfaces in projective space, J. Differ. Geom., Volume 39 (1994) no. 1, pp. 139-172 http://projecteuclid.org/euclid.jdg/1214454680 | DOI | MR | Zbl
[21] Stability of hyperbolic embeddedness and the construction of examples, Mat. Sb. (N.S.), Volume 135(177) (1988) no. 3, p. 361-372, 415 | MR | Zbl
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