Numerical action for endomorphisms

TL;DR

The paper develops a cone-based spectral framework for surjective endomorphisms $f$ of projective varieties, linking dynamical growth in cohomology to the geometry of positivity cones via ${\rm Sp}(f^*,{\rm Big}(X))$ and ${\rm Sp}(f^*,{\rm Amp}(X))$. It introduces cohomological Lyapunov exponents $\mu_i(f)$ and proves that big-cone dynamics are governed by these exponents, with quasi-amplified equating to cohomological hyperbolicity. A key contribution is the algebraic analogue of currents: generated cycles ${\mathcal{G}}_i(X)$ and their positive version ${\mathcal{G}}^+_i(X)$, together with a measure-theoretic framework on the constructible topology, enabling decomposition, restriction, and intersection theories. The work also establishes a dynamical application to growth and amplification across subvarieties, culminating in a transfer principle: the spectrum on the ample cone is determined by spectra on periodic subvarieties, mirroring the big-cone results and informing conjectures in algebraic dynamics.

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.

Theorems & Definitions (46)

• Example 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6: c.f. Corollary \ref{['corfactoramplified']} and \ref{['corfactoramplifiedquasi']}
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Definition 2.5