Expected Euler characteristic of excursion sets of random holomorphic sections on complex manifolds Jing Sun 32Q99excursion setcritical radius 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 K\"ahler manifold $(M,\omega)$. We then prove that the critical radius of the Kodaira embedding $\Phi_N:M\to\mathbb{C}\mathbb{P}^n$ given by an orthonormal basis of $H^0(M,L^N)$ is bounded below when $N\to\infty$. This result also gives conditions about when the preceding formula is valid. Indiana University Mathematics Journal 2012 text pdf 10.1512/iumj.2012.61.4672 10.1512/iumj.2012.61.4672 en Indiana Univ. Math. J. 61 (2012) 1157 - 1174 state-of-the-art mathematics http://iumj.org/access/