Riemannian structures and point-counting

Abstract

Suppose $(X_n)$ is a sequence of positive-dimensional smooth projective complete intersections over $\mathbb{F}_q$ with dimensions bounded from above and with characteristic zero lifts $(\tilde{X}_n)$ to smooth projective geometrically connected varieties. Suppose each complex variety $\tilde{X}^{an}_n$ has (underlying real manifold equipped with) a Riemannian metric $g_n$ of sectional curvature at least $-\kappa_n^2$, $\kappa_n\geq 0$, and diameter at most $D_n$. In this note, we show that if \[\lim_{n\rightarrow\infty}\min\{d:X_n(\mathbb{F}_{q^d})\neq\emptyset\}=+\infty,\] then $\kappa_nD_n\rightarrow+\infty$. We deduce this theorem by proving a more general theorem estimating the number of points over finite fields of the above varieties in terms of the sectional curvature and diameter of Riemannian structures on the analytification of characteristic zero lifts. We also prove a version of this estimate for non-projective varieties equipped with complete metrics of non-negative sectional curvature. Finally, we prove a characteristic zero analogue of the above estimate by relating fixed-points of algebraic endomorphisms of smooth projective complex complete intersections to Riemannian structures on them.