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Expected Euler characteristic of excursion sets of random holomorphic sections on complex manifolds. (English) Zbl 1271.32031

Summary: 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 \((M,\omega)\). 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.

MSC:

32Q99 Complex manifolds
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