Expected Euler Characteristic of Excursion Sets of Random Holomorphic Sections on Complex Manifolds

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, omega). We then prove that the critical radius of the Kodaira embedding Phi(N) : M -> CPn given by an orthonormal basis of H-0 (M, L-N) is bounded below when N -> infinity. This result also gives conditions about when the preceding formula is valid.