Metric and spectral aspects of random complex divisors

Abstract

For any integer n\geq 2 , we prove that for any large enough integer d , with large probability the injectivity radius of a random degree d complex hypersurface in \mathbb{C}P^{n} 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 L^{2} -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 {d^{n+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 Cd^{3} , which generalizes and amends the bound given by Feng–Schumacher [Compositio Math. 119 (1999), 331–334] for planar curves.