Relative dynamical degrees of correspondences over a field of arbitrary characteristic
By Tuyen T. T. Truong. Journal für die Reine und Angewandte Mathematik, 2016
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Abstract Let 𝕂 {\mathbb{K}} be an algebraically closed field of arbitrary characteristic, X and Y irreducible possibly singular algebraic varieties over 𝕂 {\mathbb{K}} . Let f : X ⊢ X {f:X\vdash X} and g : Y ⊢ Y {g:Y\vdash Y} be dominant correspondences, and π : X ⇢ Y {\pi:X\dashrightarrow Y} a dominant rational map which semi-conjugate f and g, i.e. so that π ∘ f = g ∘ π {\pi\circ f=g\circ\pi} . We define relative dynamical degrees λ p ( f | π ) ≥ 1 {\lambda_{p}(f|\pi)\geq 1} for any p = 0 , … , dim ( X ) - dim ( Y ) {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 ) {(\varphi,\psi)} from π 2 : ( X 2 , f 2 ) → ( Y 2 , g 2 ) {\pi_{2}:(X_{2},f_{2})\rightarrow(Y_{2},g_{2})} to π 1 : ( X 1 , f 1 ) → ( Y 1 , g 1 ) {\pi_{1}:(X_{1},f_{1})\rightarrow(Y_{1},g_{1})} we have λ p ( f 1 | π 1 ) ≥ λ p ( f 2 | π 2 ) {\lambda_{p}(f_{1}|\pi_{1})\geq\lambda_{p}(f_{2}|\pi_{2})} for all p. Many of our results are new even when 𝕂 = ℂ {\mathbb{K}=\mathbb{C}} . Self-correspondences are abundant, even on varieties having not many self rational maps, hence these results can be applied in many situations. In the last section of the paper, we will discuss recent new applications of this to algebraic dynamics, in particular to pullback on l-adic cohomology groups in positive characteristics.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/e06bb6bde596edd4efa864fa5440e571202678a2
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Citations in the corpus
- Degrees of iterates of rational maps on normal projective varieties (2017, methodology): “He went further in [31] and obtained Theorem 1 for any normal variety in all characteristic by applying de Jong’s alteration theorem [6].”