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

TL;DR

This paper derives a formula for the expected Euler characteristic of excursion sets of random holomorphic sections on complex manifolds and investigates the asymptotic behavior of the Kodaira embedding's critical radius.

Contribution

It introduces a new formula for the expected Euler characteristic of excursion sets of random sections and analyzes the asymptotic lower bound of the Kodaira embedding's critical radius.

Findings

Derived a formula for the expected Euler characteristic of excursion sets.

Proved the critical radius of the Kodaira embedding is bounded below as N approaches infinity.

Provided conditions under which the formula for the expected Euler characteristic holds.

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 K\"{a}hler manifold $(M,\omega)$. We then prove that the critical radius of the Kodaira embedding $\Phi_N:M\rightarrow \CP^n$ 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.