Abstract. We classify bimeromorphic self-maps f : X of compact kähler surfaces X and classify them in terms of their actions f * : H 1,1 (X) on cohomology. We observe that the growth rate of ||f n * || is invariant under bimeromorphic conjugacy, and that by conjugating one can always arrange that f n * = f * n . We show that the sequence ||f n * || can be bounded, grow linearly, grow quadratically, or grow exponentially. In the first three cases, we show that after conjugating, f is an automorphism virtually isotopic to the identity, f preserves a rational fibration, or f preserves an elliptic fibration, respectively. In the last case, we show that there is an unique (up to scaling) expanding eigenvector θ+ for f * , that θ+ is nef, and that f is bimeromorphically conjugate to an automorphism if and only if θ 2 + = 0. We go on in this case to construct a dynamically natural positive current representing θ+, and we study the growth rate of periodic orbits of f . We conclude by illustrating our results with a particular family of examples.
We describe the set V of all R ∪ {∞} valued valuations ν on the ring C [[x, y]] normalized by min{ν(x), ν(y)} = 1. It has a natural structure of an R-tree, induced by the order relation ν 1 ≤ ν 2 iff ν 1 (φ) ≤ ν 2 (φ) for all φ. It can also be metrized, endowing it with a metric tree structure. From the algebraic point of view, these structures are obtained by taking a suitable quotient of the Riemann-Zariski variety of C [[x, y]], in order to force it to be a Hausdorff topological space. The tree structure on V also provides an identification of valuations with balls of irreducible curves in a natural ultrametric. We show that the dual graphs of all sequences of blow-ups patch together, yielding an R-tree naturally isomorphic to V. Altogether, this gives many different approaches to the valuative tree V. We then describe a natural Laplace operator on V. It associates to (special) functions of V a complex Borel measure. Using this operator, we show how measures on the valuative tree can be used to encode naturally both integrally closed ideals in R and cohomology classes of the voute etoilee of (C 2 , 0).
We give an algebraic construction of the positive intersection products of pseudo-effective classes and use them to prove that the volume function on the Néron-Severi space of a projective variety is C 1 -differentiable, expressing its differential as a positive intersection product. We also relate the differential to the restricted volumes. We then apply our differentiability result to prove an algebro-geometric version of the Diskant inequality in convex geometry, allowing us to characterize the equality case of the Khovanskii-Teissier inequalities for nef and big classes.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.
Copyright © 2025 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.