Canonical heights, invariant currents, and dynamical eigensystems of morphisms for line bundles

Abstract

We construct canonical heights of subvarieties for dynamical systems of several morphisms associated with line bundles defined over a number field, and study some of their properties. We also construct invariant currents for such systems over C.1991 Mathematics Subject Classification. 11G50, 14G40, 58F23.

Canonical height functions for affine plane automorphisms

“…On certain K3 surfaces with two involutions, Silverman [14] developed the theory of canonical height functions that behave well relative to the two involutions. For the theory of canonical height functions on some other projective varieties, see for example [1], [16], [7]. In this paper, we show the existence of canonical height functions on the affine plane associated with polynomial automorphisms of dynamical degree ≥ 2.…”

On variation of dynamical canonical heights for semigroups of morphisms

“…We can now give a similar estimate for the difference between the canonical local heightλ Vt,Et,(Q)t defined by Kawaguchi in theorem 4.2.1 of [16] and a given Weil local height λ V,E , generalizing Lang's result [20] for abelian varieties. Here we make extensive use of the notation in theorem 7.3 and corollary 7.4 of [27], and obtain a local version of theorem 2.1.…”

“…. From this and from theorem 1.2.1 of [16], we have that for each t ∈ T 0 there is a canonical heightĥ Vt,ηt,(Q)t : V t (K) → R. We now fix a Weil height h V,η : V(K) → R associated to η. It follows from the properties of the height functions of Kawaguchi [16], and functoriality of the Weil height function, that…”

“…It was an idea of Kawaguchi [16] to consider polarized dynamical systems of several maps, namely, given X/K a projective variety, f 1 , ...f k : X → X morphisms on defined over K, L an invertible sheaf on X and a real number d > k so that f * 1 L ⊗ ... ⊗ f * k L ∼ = L ⊗d , he constructed a canonical height function associated to the polarized dynamical system (X, f 1 , ..., f k , L) that generalizes the earlier constructions mentioned above. In the Wheler's K3 surfaces' case above, for example, the canonical height defined by Silverman arises from the system formed by (σ 1 , σ 2 ) by Kawaguchi's method.…”

Equidistribution of Periodic Points of Some Automorphisms on K3 Surfaces

“…He [8] suggested the polarizable dynamical system of several morphisms: Definition 1.1. Let W be a projective variety, let L be an ample line bundle on W and let M = {ϕ 1 , .…”