New numerical invariants called hyperbolic indices, defined via liftable directed currents on Demailly-Semple towers, give a sufficient condition for Kobayashi hyperbolicity and grow at least linearly in degree for general hypersurfaces.
Generalized Okounkov Bodies, Hyperbolicity Related and Direct Image Problems
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abstract
The aim of this work is to deal with effective questions related to the Kobayashi and Debarre conjectures, and based on the work of Damian Brotbek and Lionel Darondeau. We first show that if a line bundle $L$ generates $k$-jets, the $k$-th Wronskian ideal sheaf associated with global sections of $L$ reaches its largest possible value. Then we obtain an effective estimate for a weaker version of a theorem of Nakamaye on certain universal Grassmannians. We finally derive from this explicit effective bounds for the Kobayashi and Debarre conjectures.
years
2026 2representative citing papers
Introduces generalized algebraic Morse inequalities to give a fully algebraic proof of Demailly's theorem on Green-Griffiths jet differentials for manifolds of general type, with an extension to Hasse-Schmidt jet differentials over arbitrary algebraically closed fields.
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Quantitative hyperbolicity for complex manifolds via numerical invariants
New numerical invariants called hyperbolic indices, defined via liftable directed currents on Demailly-Semple towers, give a sufficient condition for Kobayashi hyperbolicity and grow at least linearly in degree for general hypersurfaces.
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Generalized algebraic Morse inequalities and Hasse-Schmidt jet differentials
Introduces generalized algebraic Morse inequalities to give a fully algebraic proof of Demailly's theorem on Green-Griffiths jet differentials for manifolds of general type, with an extension to Hasse-Schmidt jet differentials over arbitrary algebraically closed fields.