zbMATH — the first resource for mathematics

Canonical metrics of commuting maps. (English) Zbl 1173.14306
Summary: Let $$\varphi : X \rightarrow X$$ be a map on an projective variety. It is known that whenever the map $$\varphi ^* : \text{Pic}(X) \rightarrow \text{Pic}(X)$$ has an eigenvalue $$\alpha > 1$$, we can build a canonical measure, a canonical height and a canonical metric associated to $$\varphi$$. In the present work, we establish the following fact: if two commuting maps $$\varphi , \psi : X \rightarrow X$$ satisfy these conditions, for eigenvalues $$\alpha$$ and $$\beta$$ and the same eigenvector $$\mathcal L$$, then the canonical metric, the canonical measure, and the canonical height associated to both maps, are identical.
MSC:
 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 14K22 Complex multiplication and abelian varieties 14H52 Elliptic curves 16W22 Actions of groups and semigroups; invariant theory (associative rings and algebras) 28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures
Full Text: