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.


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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[1] Arnold, V.; Varchenko, A.; Goussein-Zadé, S., Singularités des applications différentiables, 2 (monodromie et comportement asymptotique des intégrales), (1986), Edition Mir
[2] Barlet, D., Singularités réelles isolées et développements asymptotiques d’intégrales oscillantes, 25-50, (2004), Seminaires et SMF · Zbl 1080.32028
[3] Burns, D.; Guillemin, V.; Wong, Z., Stability functions, Geom. Funct. Anal., 19, 5, 1258-1295, (2010) · Zbl 1186.53101
[4] Fulton, W., Introduction to Toric Varieties, 131, (1983), Princeton Univ. Press, Princeton · Zbl 0813.14039
[5] Heinzner, P., Geometric invariant theory on Stein spaces, Math. Ann., 289, 631-662, (1991) · Zbl 0728.32010
[6] Heinzner, P.; Huckleberry, A., Manuscripta math., Math. Ann., 83, 19-29, (1994) · Zbl 0842.32010
[7] Heinzner, P.; Huckleberry, A., Several Complex Variables, 37, Analytic Hilbert quotients, 309-349, (1999), Cambridge University Press · Zbl 0959.32013
[8] Hörmander, L., The Analysis of Linear Partial Differential Operators, I, (1990), Springer Verlag, New York · Zbl 0712.35001
[9] Jeanquartier, P., Développement asymptotique de la distribution de Dirac, C.r. Acad. Sci. Paris, 271, 1159-1161, (1970) · Zbl 0201.16502
[10] Ma, X.; Zhang, W., Bergman kernels and Symplectic reduction, 318 · Zbl 1171.32001
[11] Neeman, A., The topology of quotient varieties, Ann. of Math. (2), 122, 3, 419-459, (1985) · Zbl 0692.14032
[12] Sebert, H., Semiclassical limits of Kählerian potentials on toric varieties
[13] Shiffman, B.; Tate, T.; Zelditch, S., Distribution laws for integrable eigenfunctions, Ann. Inst. Fourier, 54, 1497-1546, (2004) · Zbl 1081.35063
[14] Song, J.; Zelditch, S., Bergman metrics and geodesics in the space of Kähler metrics on toric varieties, Analysis & PDE, 3, 3, 295-358, (2010) · Zbl 1282.35428
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