On the existence of logarithmic and orbifold jet differentials
Annales Henri Lebesgue, Volume 7 (2024), pp. 1-67.


Keywords Projective variety, directed variety, orbifold, ramification divisor, entire curve, jet differential, Green–Griffiths conjecture, algebraic differential operator, holomorphic Morse inequalities, Chern curvature, Chern form


We introduce the concept of directed orbifold, namely triples (X,V,D) formed by a directed algebraic or analytic variety (X,V), and a ramification divisor D, where V is a coherent subsheaf of the tangent bundle T X . In this context, we introduce an algebra of orbifold jet differentials and their sections. These jet sections can be seen as algebraic differential operators acting on germs of curves, with meromorphic coefficients, whose poles are supported by D and multiplicities are bounded by the ramification indices of the components of D. We estimate precisely the curvature tensor of the corresponding directed structure VD in the general orbifold case – with a special attention to the compact case D=0 and to the logarithmic situation where the ramification indices are infinite. Using holomorphic Morse inequalities on the tautological line bundle of the projectivized orbifold Green–Griffiths bundle, we finally obtain effective sufficient conditions for the existence of global orbifold jet differentials.


[BD19] Brotbek, Damian; Deng, Ya Kobayashi hyperbolicity of the complements of general hypersurfaces of high degree, Geom. Funct. Anal., Volume 29 (2019) no. 3, pp. 690-750 | DOI | MR | Zbl

[Bon93] Bonavero, Laurent Singular holomorphic Morse inequalities, C. R. Math., Volume 317 (1993) no. 12, pp. 1163-1166 | MR | Zbl

[BT76] Bedford, Eric; Taylor, Bert A. The Dirichlet problem for a complex Monge–Ampère equation, Invent. Math., Volume 37 (1976), pp. 1-44 | DOI | Zbl

[Cad19] Cadorel, Benoit Generalized algebraic Morse inequalities and jet differentials (2019) | arXiv

[Cam04] Campana, Frédéric Orbifolds, special varieties and classification theory., Ann. Inst. Fourier, Volume 54 (2004) no. 3, pp. 499-630 | DOI | Numdam | MR | Zbl

[CDR20] Campana, Frédéric; Darondeau, Lionel; Rousseau, Erwan Orbifold hyperbolicity, Compos. Math., Volume 156 (2020) no. 8, pp. 1664-1698 | DOI | MR | Zbl

[Dar16] Darondeau, Lionel On the logarithmic Green–Griffiths conjecture, Int. Math. Res. Not., Volume 2016 (2016) no. 6, pp. 1871-1923 | DOI | MR | Zbl

[Dem82] Demailly, Jean-Pierre Relations entre les différentes notions de fibrés et de courants positifs, Semin. P. Lelong – H. Skoda, Analyse, Annees 1980/81, et: Les fonctions plurisousharmoniques en dimension finie ou infinie, Colloq. Wimereux 1981 (Lecture Notes in Mathematics), Volume 919, Springer (1982), pp. 56-76 | Zbl

[Dem85] Demailly, Jean-Pierre Champs magnétiques et inégalités de Morse pour la d -cohomologie. (Magnetic fields and Morse inequalities for d -cohomology), C. R. Math., Volume 301 (1985), pp. 119-122 | MR | Zbl

[Dem96] Demailly, Jean-Pierre L 2 vanishing theorems for positive line bundles and adjunction theory, Transcendental methods in algebraic geometry. Lectures given at the 3rd session of the Centro Internazionale Matematico Estivo (CIME), Cetraro, Italy, July 4–12, 1994 (Lecture Notes in Mathematics), Volume 1646, Cetraro: Springer, 1996, pp. 1-97 | MR | Zbl

[Dem97] Demailly, Jean-Pierre Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, Algebraic geometry. Proceedings of the Summer Research Institute, Santa Cruz, CA, USA, July 9–29, 1995, American Mathematical Society, 1997, pp. 285-360 | DOI | Zbl

[Dem11] Demailly, Jean-Pierre Holomorphic Morse inequalities and the Green–Griffiths–Lang conjecture, Pure Appl. Math. Q., Volume 7 (2011) no. 4, pp. 1165-1207 | DOI | MR | Zbl

[Dem12] Demailly, Jean-Pierre Hyperbolic algebraic varieties and holomorphic differential equations, Acta Math. Vietnam., Volume 37 (2012) no. 4, pp. 441-512 | MR | Zbl

[Dem15] Demailly, Jean-Pierre Towards the Green–Griffiths–Lang conjecture, Analysis and geometry. MIMS-GGTM, Tunis, Tunisia, March 24–27, 2014. Proceedings of the international conference. In honour of Mohammed Salah Baouendi (Springer Proceedings in Mathematics & Statistics), Volume 127, Springer, 2015, pp. 141-159 | DOI | MR | Zbl

[Dem20] Demailly, Jean-Pierre Recent results on the Kobayashi and Green–Griffiths–Lang conjectures, Jpn. J. Math. (3), Volume 15 (2020) no. 1, pp. 1-120 | DOI | MR | Zbl

[DR23] Darondeau, Lionel; Rousseau, Erwan Quasi-positive orbifold cotangent bundles. Pushing further an example by Junjiro Noguchi, Epijournal Geom. Algebr., Volume 8 (2023), 3

[GG80] Green, Mark; Griffiths, Phillip Two applications of algebraic geometry to entire holomorphic mappings, Differential geometry, Proc. int. Chern Symp., Berkeley 1979 (1980), pp. 41-74 | DOI | Zbl

[Lan05] Landau, E. Sur quelques théorèmes de M. Pétrovitch relatifs aux zéros des fonctions analytiques., Bull. Soc. Math. Fr., Volume 33 (1905), pp. 251-261 | DOI | Numdam | MR | Zbl

[MT22] Merker, Joël; Ta, The-Anh Degrees d(nlogn) n and d(nlogn) n in the conjectures of Green-Griffiths and of Kobayashi, Acta Math. Vietnam., Volume 47 (2022) no. 1, pp. 305-358 | DOI | Zbl

[Sem54] Semple, J. G. Some investigations in the geometry of curve and surface elements, Proc. Lond. Math. Soc., Volume 4 (1954), pp. 24-49 | DOI | MR | Zbl

[Tra95] Trapani, Stefano Numerical criteria for the positivity of the difference of ample divisors, Math. Z., Volume 219 (1995), pp. 387-401 | DOI | MR | Zbl