Degree growth of meromorphic surface maps

By S. Boucksom, C. Favre, Mattias Jonsson. 2006

71 citations recorded by the source corpus.

We study the degree growth of iterates of meromorphic self-maps of compact Kahler surfaces. Using cohomology classes on the Riemann-Zariski space, we show that the degrees grow similarly to those of mappings that are algebraically stable on some bimeromorphic model. 0. Introduction Let X be a compact Kahler surface, and let F : X X be a dominant meromorphic mapping. Fix a Kahler class ω on X, normalized by (ω)X = 1, and define the degree of F with respect to ω to be the positive real number degω(F ) := (F ∗ω · ω)X = (ω · F∗ω)X, where (·)X denotes the intersection form on H 1,1 R (X). When X = P2 and ω is the class of a line, this coincides with the usual algebraic degree of F . One can show that degω(F n+m) ≤ 2 degω(F ) degω(F ) for all m, n. Hence the limit λ1 := lim n→∞ degω(F ) exists. We refer to it as the asymptotic degree of F . It follows from standard arguments (see Proposition 3.1) that λ1 does not depend on the choice of ω, that λ1 is invariant under bimeromorphic conjugacy, and that λ1 ≥ λ2, where λ2 is the topological degree of F .

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1edc83ee3f49840be8f2d808f9e42a3f1ee0ba68

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Periodic points of birational transformations on projective surfaces (2015, methodology): “. It only depends on the class of E in D, by the projection formula. Let RD denote the space of all real-valued R functions on D. Then we obtain a map W(X) →N1(X) R ×RD, which is a homeomorphism (see [4] a the proof). Then by this homeomorphism, any Weil class α corresponds to a point (ωα,(αE)E∈D) ∈”
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Normal subgroups in the Cremona group (2010, background): “The (completed) Picard-Manin space Z̄(X) of X is the L2-completion of Z(X) (see [4, 6] for details); in other words”