Relative dynamical degrees of correspondences over a field of arbitrary characteristic

Tuyen Trung Truong

doi:10.1515/crelle-2017-0052

paper

Relative dynamical degrees of correspondences over a field of arbitrary characteristic

arXiv:1605.05049 · doi:10.1515/crelle-2017-0052

Abstract

Let $K$ be an algebraically closed field of arbitrary characteristic, $X$ an irreducible variety and $Y$ an irreducible projective variety over $K$ , both are not necessarily smooth. Let $f : X \to X$ and $g : Y \to Y$ be dominant correspondences, and $π : X \to Y$ a dominant rational map such that $π \circ f = g \circ π$ . We define relative dynamical degrees $λ_{p} (f ∣ π)$ ( $p = 0, \dots, dim (X) - dim (Y)$ ). These degrees measure the relative growth of positive algebraic cycles, satisfy a product formula when $Y$ is smooth and $g$ is a multiple of a rational map, and are birational invariants. More generally, a weaker product formula is proven for more general semi-conjugacies, and for any generically finite semi-conjugacy $(φ, ψ)$ from $(X_{2}, f_{2}) \to (Y_{2}, g_{2})$ to $(X_{1}, f_{1}) \to (Y_{1}, g_{1})$ we have $λ_{p} (f_{1} ∣ π_{1}) \geq λ_{p} (f_{2} ∣ π_{2})$ for all $p$ . Many of our results are new even when $K = C$ . We make use of de Jong's alterations and Roberts' version of Chow's moving lemma. In the lack of resolution of singularities, the consideration of correspondences is necessary even when $f, g$ are rational maps. The case $K$ is not algebraically closed further requires working with correspondences over reducible varieties.

41 pages. Expositions rewritten, new examples and details of some proofs added, some typos corrected. arXiv admin note: text overlap with arXiv:1501.01523

Relative dynamical degrees of correspondences over a field of arbitrary characteristic

Abstract

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