Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Applications of a Grassmannian technique to hyperbolicity, Chow equivalency, and Seshadri constants

Authors: Eric Riedl and David Yang
Journal: J. Algebraic Geom. 31 (2022), 1-12
Published electronically: July 19, 2021
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Abstract | References | Additional Information

Abstract: In this paper we further develop a Grassmannian technique used to prove results about very general hypersurfaces and present three applications. First, we provide a short proof of the Kobayashi conjecture given previously established results on the Green–Griffiths–Lang conjecture. Second, we completely resolve a conjecture of Chen, Lewis, and Sheng on the dimension of the space of Chow-equivalent points on a very general hypersurface, proving the remaining cases and providing a short, alternate proof for many of the previously known cases. Finally, we relate Seshadri constants of very general points to Seshadri constants of arbitrary points of very general hypersurfaces.

References [Enhancements On Off] (What's this?)

  • Damian Brotbek, On the hyperbolicity of general hypersurfaces, Publ. Math. Inst. Hautes Études Sci. 126 (2017), 1–34. MR 3735863, DOI 10.1007/s10240-017-0090-3
  • Damian Brotbek and Ya Deng, Kobayashi hyperbolicity of the complements of general hypersurfaces of high degree, Geom. Funct. Anal. 29 (2019), no. 3, 690–750. MR 3962877, DOI 10.1007/s00039-019-00496-2
  • G. Bérczi and F. Kirwan, Non-reductive geometric invariant theory and hyperbolicity, arXiv:1909.11417 (2019).
  • Xi Chen, James D. Lewis, and Mao Sheng, Rationally inequivalent points on hypersurfaces in $\Bbb {P}^n$, Adv. Math. 384 (2021), Paper No. 107735, 64. MR 4246097, DOI 10.1016/j.aim.2021.107735
  • Lionel Darondeau, On the logarithmic Green-Griffiths conjecture, Int. Math. Res. Not. IMRN 6 (2016), 1871–1923. MR 3509943, DOI 10.1093/imrn/rnv078
  • Jean-Pierre Demailly, Hyperbolic algebraic varieties and holomorphic differential equations, Acta Math. Vietnam. 37 (2012), no. 4, 441–512. MR 3058660
  • Jean-Pierre Demailly, Recent results on the Kobayashi and Green-Griffiths-Lang conjectures, Jpn. J. Math. 15 (2020), no. 1, 1–120. MR 4068832, DOI 10.1007/s11537-019-1566-3
  • Y. Deng, “Effectivity in the hyperbolicity related problems”, Chapter 4 of the PhD memoir Generalized Okounkov bodies, hyperbolicity-related and direct image problems arXiv:1606.03831 (2016).
  • Shoshichi Kobayashi, Hyperbolic manifolds and holomorphic mappings, Pure and Applied Mathematics, vol. 2, Marcel Dekker, Inc., New York, 1970. MR 0277770
  • J. Merker, Kobayashi hyperbolicity in degree $> n^{2n}$, arXiv:1807.11309 (2018).
  • J. Merker and T. Ta, Degrees $d \geqslant \big ( \sqrt {n}\, \log \, n\big )^n$ and $d \geqslant \big ( n\, \log \, n\big )^n$ in the Conjectures of Green-Griffiths and of Kobayashi, Acta Math. Vietnam. (2021), DOI: 10.1007/s40306-021-00428-z
  • Eric Riedl and David Yang, Rational curves on general type hypersurfaces, J. Differential Geom. 116 (2020), no. 2, 393–403. MR 4168208, DOI 10.4310/jdg/1603936816
  • A. A. Roĭtman, ${G}$-equivalence of zero-dimensional cycles, Mat. Sb. (N.S.) 86 (128) (1971), 557–570 (Russian). MR 0289512
  • A. A. Roĭtman, Rational equivalence of zero-dimensional cycles, Mat. Sb. (N.S.) 89(131) (1972), 569–585, 671 (Russian). MR 0327767
  • 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
  • Claire Voisin, Variations de structure de Hodge et zéro-cycles sur les surfaces générales, Math. Ann. 299 (1994), no. 1, 77–103 (French). MR 1273077, DOI 10.1007/BF01459773
  • Claire Voisin, On a conjecture of Clemens on rational curves on hypersurfaces, J. Differential Geom. 44 (1996), no. 1, 200–213. MR 1420353
  • Claire Voisin, A correction: “On a conjecture of Clemens on rational curves on hypersurfaces” [J. Differential Geom. 44 (1996), no. 1, 200–213; MR1420353 (97j:14047)], J. Differential Geom. 49 (1998), no. 3, 601–611. MR 1669712

Additional Information

Eric Riedl
Affiliation: Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame, Indiana 46556
MR Author ID: 902032

David Yang
Affiliation: Department of Mathematics, Harvard University, 1 Oxford St, Cambridge, Massachusetts 02138
MR Author ID: 1118824

Received by editor(s): July 19, 2018
Received by editor(s) in revised form: February 16, 2021, February 20, 2021, February 24, 2021, April 30, 2021, May 5, 2021, and May 14, 2021
Published electronically: July 19, 2021
Article copyright: © Copyright 2021 University Press, Inc.