Tian’s theorem for Grassmannian embeddings and degeneracy sets of random sections

Abstract Let ( X , ω ) (X,\omega) be a compact Kähler manifold, ( L , h L ) (L,h^{L}) a positive line bundle, and ( E , h E ) (E,h^{E}) a Hermitian holomorphic vector bundle of rank 𝑟 on 𝑋. We prove that the pullback by the Kodaira embedding associated to L p ⊗ E L^{p}\otimes E of the 𝑘-th Chern form of the dual of the universal bundle over the Grassmannian converges as p → ∞ p\to\infty to the 𝑘-th power of the Chern form c 1 ⁢ ( L , h L ) c_{1}(L,h^{L}) for 0 ≤ k ≤ r 0\leq k\leq r . If c 1 ⁢ ( L , h L ) = ω c_{1}(L,h^{L})=\omega , we also determine the second term in the semi-classical expansion, which involves c 1 ⁢ ( E , h E ) c_{1}(E,h^{E}) . As a consequence, we show that the limit distribution of zeros of random sequences of holomorphic sections of high powers