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 $\C P^n$ is larger than $d^{-\frac{1}{2}(3n+2)}$. 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 $d^{\frac{3}{2}(n+2)}$, and that its spectral gap is bounded below by $\exp (-d^{\frac{1}{4}(3n+15)})$. 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 in~\cite{feng1999diameter} for planar curves.

References in corpus (1)

• Friedman-Ramanujan functions in random hyperbolic geometry and application to spectral gaps I