Expected Euler characteristic of excursion sets of random holomorphic sections on complex manifolds

Abstract

We prove a formula for the expected euler characteristic of excursion sets of random sections of powers of an ample bundle (L, h), where h is a Hermitian metric, over a Kahler manifold (M, ω). We then prove that the critical radius of the Kodaira embedding ΦN : M → CPn given by an orthonormal basis of H0( M, LN) is bounded below when N → ∞. This result also gives conditions about when the preceding formula is valid.