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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 Kähler manifold $(M,ω)$. We then prove that the critical radius of the Kodaira embedding $Φ_N:M\rightarrow \CP^n$ given by an orthonormal basis of $H^0(M,L^N)$ is bounded below when $N\rightarrow \infty$. This result also gives conditions about when the preceding formula is valid.