Metric and spectral aspects of random complex divisors

TL;DR

The paper develops a probabilistic metric-geometry framework for random complex divisors in projective manifolds, focusing on the induced Riemannian geometry of their zero loci. It combines Bergman-kernel asymptotics, discriminant-transversality, and local graph representations to obtain high-probability bounds on injectivity radius, sectional curvature, spectral gap, and diameter, with precise degree-dependent scalings in terms of $d$ and the ambient dimension $n$. For random complex hypersurfaces in $\mathbb{C}P^n$, the authors prove injectivity radius $\gtrsim d^{-\frac{1}{2}(3n+2)}$, curvature $\lesssim d^{\frac{3}{2}(n+2)}$, and spectral gap $\gtrsim \exp(-d^{\frac{1}{4}(3n+15)})$, and establish a diameter bound $\mathrm{diam}\le C d^3$ (extending to $d^{3r}$ in codimension $r$). In the planar case ($n=2$) they also obtain probabilistic systolic and curvature/eigenvalue results, illustrating the connection to related random-surface models and deterministic bounds. Overall, the work provides a rigorous probabilistic counterpart to deterministic geometric bounds for complex algebraic submanifolds and links Bergman kernel asymptotics to global metric properties.

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 $\C P^n$ 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 $Cd^3$, which generalizes and amends the bound given in~\cite{feng1999diameter} for planar curves.

Figures (2)

• Figure 1: Deterministic parameters of the two different models of real surfaces, the Weil-Petersson one with hyperbolic surfaces, and the Fubini-Study model with complex algebraic curves equipped with the induced rescaled induced metric $\sqrt {2\pi d} g_{FS}$ on $\mathbb{C} P^2$.
• Figure 2: Statistics of some metric parameters. The complex algebraic curves are equipped with the induced rescaled induced metric $\sqrt {2\pi d} g_{FS}$ on $\mathbb{C} P^2$.

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Example 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Theorem 2.1