On degree growth and stabilization of three-dimensional monomial maps

Abstract

Given a rational self-map f : X X on an n-dimensional Kahler manifold X, one can define a pullback map f ∗ : H(X) → H(X) for 0 ≤ p ≤ n. In general, the pullback does not commute with iteration; that is, (f ∗)k = (f k)∗. Following Sibony ([Si]; see also [FoSi]), we call the map f (algebraically) stable if the action on the cohomology of X is compatible with iterations. More precisely, f is called p-stable if the pullback on H(X) satisfies (f ∗)k = (f k)∗ for all k ∈N. If f is not p-stable on X, one might try to find a birational change of coordinate h : X ′ X such that f = h−1 f h is p-stable. This is not always possible even for p = 1, as shown by Favre [Fa]. However, for p = 1 and n = 2, one can find such a stable model (with at worst quotient singularities) for quite a few classes of surface maps [DFa; FaJ]. Also for p = 1, such a model can be obtained for certain monomial maps [Fa; JW; L]. For an n × n integer matrix A = (ai,j ), the associated monomial map fA : (C∗)n → (C∗)n is defined by fA(x1, . . . , xn) = (∏