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.