Abstract

In this talk, we present a Tian-type approximation theorem for Grassmannian embeddings associated with  $H^0(X,L^p\otimes E)$, where L is a positive line bundle and E is a holomorphic vector bundle over a compact complex manifold X. In particular, we establish a full asymptotic expansion of the induced Grassmannian Chern forms and compute the first coefficients explicitly. When X is Kähler, we combine the first-order asymptotics with Dinh–Sibony’s theory of meromorphic transforms to prove that normalized currents of integration over the loci where several random sections become linearly dependent converge almost surely to the corresponding powers of the first Chern form of  L.