Abstract
For a given complex projective variety, the existence of entire curves is strongly constrained by the positivity properties of the cotangent bundle. The Green–Griffiths–Lang conjecture stipulates that entire curves drawn on a variety of general type should all be contained in a proper algebraic subvariety. We present here new results on the existence of differential equations that strongly restrain the locus of entire curves in the general context of foliated or directed varieties, under appropriate positivity conditions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bonavero L, Inégalités de Morse holomorphes singulières, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993) 1163–1166
Demailly J-P, Champs magnétiques et inégalités de Morse pour la \(d^{\prime \prime }\)-cohomologie, Ann. Inst. Fourier (Grenoble) 35 (1985) 189–229
Demailly J-P, Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, AMS Summer School on Algebraic Geometry, Santa Cruz 1995, Proceeding Symposia in Pure Math., vol. 062.2 (eds) J Kollár and R Lazarsfeld (1997) (Providence, RI: Amer. Math. Soc.) pp. 285–360
Demailly J-P, Holomorphic Morse inequalities and the Green–Griffiths–Lang conjecture, Pure Appl. Math. Quart. 7 (2011) 1165–1208
Demailly J-P, Towards the Green–Griffiths–Lang conjecture, Conference “Analysis and Geometry”, Tunis, March 2014, in honor of Mohammed Salah Baouendi, Springer Proc. Math. Stat., vol. 127 (eds) A Baklouti, A El Kacimi, S Kallel and N Mir (2015) (Springer-Verlag) pp. 141–159
Demailly J-P, Recent results on the Kobayashi and Green-Griffiths-Lang conjectures, Jpn. J. Math. 15 (2020) 1–120
Etesse A, Ampleness of Schur powers of cotangent bundles and \(k\)-hyperbolicity, Res. Math. Sci. 8(2021) article # 7
Green M and Griffiths P A, Two applications of algebraic geometry to entire holomorphic map**s, The Chern Symposium 1979, Proc. International Symposiyum Berkeley, CA, 1979 (1980) (New York: Springer-Verlag) pp. 41–74
Lang S, Hyperbolic and Diophantine analysis, Bull. Amer. Math. Soc. 14 (1986) 159–205
Lang S, Introduction to complex hyperbolic spaces (1987) (New York: Springer-Verlag)
Laytimi F and Nahm W, Vanishing theorems for products of exterior and symmetric powers, ar**v:math/9809064v2 [math.AG], 24 Feb. 1999
McQuillan M, Diophantine approximations and foliations, Publ. Math. I.H.É.S. 87 (1998) 121–174
Semple J G, Some investigations in the geometry of curves and surface elements, Proc. London Math. Soc. (3) 4 (1954) 24–49
Siu Y T and Yeung S K, Defects for ample divisors of Abelian varieties, Schwarz lemma and hyperbolic hypersurfaces of low degree, Am. J. Math. 119 (1997) 1139–1172
Acknowledgements
This work is supported by the Advanced ERC Grant ALKAGE number 670846, from September 2015, attributed by the European Research Council.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicating Editor: D S Nagaraj
This article is part of the “Special Issue in Memory of Professor C S Seshadri”.
Rights and permissions
About this article
Cite this article
Demailly, JP. On the locus of higher order jets of entire curves in complex projective varieties. Proc Math Sci 132, 56 (2022). https://doi.org/10.1007/s12044-022-00681-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12044-022-00681-8
Keywords
- Projective variety
- directed variety
- entire curve
- jet differential
- Green–Griffiths bundle
- simple bundle
- exceptional locus
- algebraic differential operator
- holomorphic Morse inequalities