Volumes of complex analytic subvarieties of Hermitian symmetric spaces

Abstract

We give lower bounds of volumes of k-dimensional complex analytic subvarieties of certain naturally defined domains in n-dimensional complex space forms of constant (positive, zero, or negative) holomorphic sectional curvature. For each 1 ≤ k ≤ n, the lower bounds are sharp in the sense that these bounds are attained by k-dimensional complete totally geodesic complex submanifolds. Such lower bounds are obtained by constructing singular potential functions corresponding to blow-ups of the Kähler metrics involved. Similar lower bounds are also obtained in the case of Hermitian symmetric spaces of noncompact type. In this case, the lower bounds are sharp for those values of k at which the Hermitian symmetric space contains k-dimensional complete totally geodesic complex submanifolds which are complex hyperbolic spaces of minimum holomorphic sectional curvature.