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Asymptotics of eigensections on toric varieties. (Asymptotes de sections propres sur des variétés toriques.) (English. French summary) Zbl 1281.32017

Authors’ abstract: Using exhaustion properties of invariant plurisubharmonic functions along with basic combinatorial information on toric varieties, we prove convergence results for sequences of densities \(|\varphi _{ n }|^2 = |s_{ N }|^2/\parallel s_{ N }\parallel_{L^2}^2\) for eigensections \(s_{ N }\in \Gamma (X,L^{ N })\) approaching a semiclassical ray. Here \(X\) is a normal compact toric variety and \(L\) is an ample line bundle equipped with an arbitrary positive bundle metric which is invariant with respect to the compact form of the torus. Our work was motivated by and extends that of Shiffman, Tate and Zelditch.

MSC:

32M12 Almost homogeneous manifolds and spaces
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
22E70 Applications of Lie groups to the sciences; explicit representations
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