Bouche, Thierry Convergence de la mêtrique de Fubini-Study d’un fibré linéaire positif. (Convergence of the Fubini-Study metric of a positive line bundle.). (French) Zbl 0685.32015 Ann. Inst. Fourier 40, No. 1, 117-130 (1990). Let E be a positive line bundle over a compact complex manifold. We show that the distortion function defined by the quotient of the initial metric by the Fubini-Study metric of \(E^{\otimes k}\) admits an equivalent when k tends to infinity. This improves the previous inequalities given by Kempf and Ji over Abelian varieties, and extends them to any projective manifold. The proof rests upon the computation of an equivalent for the heat kernel, with control of the convergence with respect to the time parameter. Reviewer: Th.Bouche Cited in 3 ReviewsCited in 43 Documents MSC: 32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results 35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs Keywords:positive holomorphic line bundle; Fubini-Study metric; kernel × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML References: [1] [1] , Demailly’s asymptotic inequalities : a heat equation proof, J. Funct. Anal., 72 (1987), 263-278. · Zbl 0649.58030 [2] [2] , Inégalités de Morse holomorphes pour un fibré linéaire à courbure dégénérée, à paraître, Prépublication de l’Institut Fourier, n°142, Grenoble (1990). [3] [3] , Champs magnétiques et inégalités de Morse pour la \(d``\)-cohomologie, Ann. Inst. Fourier, Grenoble, 35-4 (1985), 189-229. · Zbl 0565.58017 [4] [4] , Holomorphic Morse Inequalities, Lectures given at the AMS Summer Institute, July 1989, to appear. [5] [5] , , , Kato’s inequality and the spectral distribution of Laplacians on compact Riemannian manifolds, J. Differential Geom., 15 (1980), 7-38. · Zbl 0442.58032 [6] [6] , Inequality for distortion function of invertible sheaves on Abelian varieties, Duke Math. J., 58-3 (1989), 657-667. · Zbl 0711.14024 [7] [7] , Metric on invertible sheaves on Abelian varieties, preprint 1988. [8] [8] ., , Curvature and the eigenvalues of the laplacian, J. Differential Geom., 1 (1967), 43-69. · Zbl 0198.44301 [9] [9] , , Some properties of the eigenfunctions of the Laplace Operator on Riemannian manifolds, Can. J. Math., 1 (1949), 242-256. · Zbl 0041.42701 [10] [10] , Kato’s inequality and the comparison of semi-groups, J. Funct. Anal., 32 (1979), 97-101. · Zbl 0413.47037 [11] [11] , Kähler metrics on algebraic manifolds, Ph. D. Thesis, Harvard University, 57 p., 1988. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.