Constraints on the cohomological correspondence associated to a self map

Shuddhodan, K. V.

doi:10.1112/s0010437x19007188

Compositio Math.

2019

DOI: 10.1112/s0010437x19007188

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Constraints on the cohomological correspondence associated to a self map

K. V. Shuddhodan¹

Abstract: In this article we establish some constraints on the eigenvalues for the action of a self map of a proper variety on its ℓ-adic cohomology. The essential ingredients are a trace formula due to Fujiwara, and the theory of weights.

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Cited by 2 publications

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Eigenvalues and dynamical degrees of self-maps on abelian varieties

2023

J. Algebraic Geom.

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Let X X be a smooth projective variety over an algebraically closed field, and f : X → X f\colon X\to X a surjective self-morphism of X X . The i i -th cohomological dynamical degree χ i ( f ) \chi _i(f) is defined as the spectral radius of the pullback f ∗ f^{*} on the étale cohomology group H e ´ t i ( X , Q ℓ ) H^i_{\acute {\mathrm {e}}\mathrm {t}}(X, \mathbf {Q}_\ell ) and the k k -th numerical dynamical degree λ k ( f ) \lambda _k(f) as the spectral radius of the pullback f ∗ f^{*} on the vector space N k ( X ) R \mathsf {N}^k(X)_{\mathbf {R}} of real algebraic cycles of codimension k k on X X modulo numerical equivalence. Truong conjectured that χ 2 k ( f ) = λ k ( f ) \chi _{2k}(f) = \lambda _k(f) for all 0 ≤ k ≤ dim ⁡ X 0 \le k \le \dim X as a generalization of Weil’s Riemann hypothesis. We prove this conjecture in the case of abelian varieties. In the course of the proof we also obtain a new parity result on the eigenvalues of self-maps of abelian varieties in prime characteristic, which is of independent interest.

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Eigenvalues and dynamical degrees of self-maps on abelian varieties

2023

J. Algebraic Geom.

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Cohomological and numerical dynamical degrees on abelian varieties

2019

Alg. Number Th.

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Constraints on the cohomological correspondence associated to a self map

Abstract: In this article we establish some constraints on the eigenvalues for the action of a self map of a proper variety on its ℓ-adic cohomology. The essential ingredients are a trace formula due to Fujiwara, and the theory of weights.

Cited by 2 publications

References 9 publications

Eigenvalues and dynamical degrees of self-maps on abelian varieties

Eigenvalues and dynamical degrees of self-maps on abelian varieties

Cohomological and numerical dynamical degrees on abelian varieties

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