Canonical Heights, Invariant Currents, and Dynamical Systems of Morphisms Associated with Line Bundles

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. Introduction Let X be a projective variety over a field K and fi : X → X (i = 1, · · · , k) morphisms over K. Let L be a line bundle on X, and d > k a real number. We say that a pair (X; f1, · · · , fk) is a dynamical system of k morphisms over K associated with L of degree d if ⊗k i=1 f ∗ i (L) ≃ L⊗d holds. The purpose of this paper to construct canonical heights of subvarieties for such systems when K is a number field and study some of their properties, and construct invariant currents for such systems when K is C. We also remark on distribution of points of small heights for Lattès examples, which are certain endomorphisms of P . Firstly, we explain canonical heights. Weil heights play one of the key roles in Diophantine geometry, and particular Weil heights that enjoy nice properties (called “canonical” heights) are sometimes of great use. Over abelian varieties A defined over a number field K, Néron and Tate constructed height functions (called canonical height functions or Néron–Tate height functions) ĥL : A(K) → R with respect to symmetric ample line bundles L which enjoys nice properties. More generally, in [7] Call and Silverman constructed canonical height functions on projective varieties X defined over a number field which admit a morphism f : X → X with f ∗(L) ≃ L⊗d for some line bundle L and some d > 1. In another direction, Silverman [20] constructed canonical height functions on certain K3 surfaces S with two involutions σ1, σ2 (called Wheler’s K3 surfaces, cf. [13]) and developed an arithmetic theory analogous to the arithmetic theory on abelian varieties. Regarding canonical heights of subvarieties of projective varieties, Philippon [17], Gubler [8] and Kramer [8] constructed canonical heights of subvarieties of abelian varieties. In [26], Zhang constructed canonical heights of subvarieties of projective varieties X which admit a morphism f : X → X with f ∗(L) ≃ L⊗d for some line bundle L and some d > 1. For Wheler’s K3 surfaces, however, canonical heights of subvarieties seem not to have been constructed. Our first results are construction of canonical heights of subvarieties for dynamical system of morphisms associated with line bundles. 1991 Mathematics Subject Classification. 11G50, 14G40, 58F23.