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.
Intersection of (1,1)-currents and the domain of definition of the Monge-Ampere operator
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abstract
We study the Monge-Amp\` ere operator within the framework of Dinh-Sibony's intersection theory defined via density currents. We show that if $u$ is a plurisubharmonic function belonging to the Blocki-Cegrell class, then the Dinh-Sibony $n$-fold self-product of $\text{dd}^c u$ exists and coincides with the classically defined Monge-Amp\`ere measure $(\text{dd}^c u)^n$.
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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.