This paper is devoted to the study of the asymptotics of Monge-Amp{\`e}re volumes of direct images associated with high tensor powers of an ample line bundle. We study the leading term of this asymptotics and provide a classification of bundles saturating the topological bound of Demailly. In the special case of high symmetric powers of ample vector bundles, this provides a characterization of vector bundles admitting projectively flat Hermitian structures.
Infinite-dimensional flats in the space of positive metrics on an ample line bundle
Semiclassical spectral analysis of Toeplitz operators on symplectic manifolds: The case of discrete wells
Characterization on projective submanifolds of codimensions 2 and 3
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