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

Official URL: https://dx.doi.org/10.1515/crelle-2026-0047

Let (X; ω) be a compact Kahler manifold, (L; hL) a positive line bundle, and (E; hE) a Hermitian holomorphic vector bundle of rank r on X. We prove that the pullback by the Kodaira embedding associated to Lp ⊗E of the k-th Chern form of the dual of the universal bundle over the Grassmannian converges as p ω 1 to the k-th power of the Chern form c1(L; hL) for 0 ≤ k ≤ r. If c1(L; hL) D ω, we also determine the second term in the semi-classical expansion, which involves c1(E; hE). As a consequence, we show that the limit distribution of zeros of random sequences of holomorphic sections of high powers Lp ⊗E is c1(L; hL)r . Furthermore, we compute the expectation of the currents of integration along degeneracy sets of random holomorphic sections.