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arXiv · 2502.04779

Numerical action for endomorphisms

Junyi Xie
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Abstract

Let $f: X\to X$ be a surjective endomorphism of a projective variety of dimension $d$. The aim of this paper is to study the action of $f$ on the numerical group of divisors. For exmaple, I proved that $f$ is cohomologically hyperbolic if and only if it is quasi-amplified; and it is amplified if and only if every subsystem of $(X,f)$ is cohomologically hyperbolic. For the proofs, I introduced a notion of spectrum in linear algebra for an open and saliant invariant cone. I also introduce a notion of generated (positive) cycles as an algebraic analogy of (positive) closed current.

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Junyi Xie. 2025-02-07. Numerical action for endomorphisms. https://arxiv.org/abs/2502.04779

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