  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 $\mathbb{C} \mathit{P}^n$ is larger than $d^{-\frac{3}2(n+1)}$.
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 $d^{3(n+1)}$,
and that its spectral gap is bounded below by $\exp(-d^{\frac{3}2(n+3)})$.
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.
