Metric and spectral aspects of random complex divisors

Abstract

For any integer $n\ge 2$, 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 $C{d}^3$, which generalizes and amends the bound given by Feng–Schumacher [Compositio Math. 119 (1999), 331–334] for planar curves.