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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
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