Metric and spectral aspects of random complex divisors

For any integer n≥ 2 , we prove that for any large enough integer d , with large probability the injectivity radius of a random degree d complex hypersurface in CPⁿ is larger than {d-(n+2)/2} . Here the hypersurface is endowed with the restriction of the ambient Fubini–Study metric, and the probability measure is induced by the Fubini–Study L² -Hermitian product on the space of homogeneous complex polynomials of degree d in (n+1) -variables. We also prove that with high probability, the sectional curvatures of the random hypersurface are bounded by {dⁿ⁺²} , and that its spectral gap is bounded below by exp(-d(n+8)/2) . These results extend to random submanifolds of higher codimension in any complex projective manifold. Independently, we prove that the diameter of a degree d divisor is bounded by Cd³ , which generalizes and amends the bound given by Feng–Schumacher [Compositio Math. 119 (1999), 331–334] for planar curves.