On the equidistribution of totally geodesic submanifolds in compact locally symmetric spaces and application to boundedness results for negative curves and exceptional divisors

TL;DR

This paper establishes an equidistribution theorem for totally geodesic submanifolds in compact locally symmetric spaces, with applications to finiteness results for negative curves and exceptional divisors in complex surfaces.

Contribution

It proves a new equidistribution result for totally geodesic submanifolds and derives finiteness theorems for negative curves and exceptional divisors in certain Hermitian locally symmetric spaces.

Findings

Finiteness of totally geodesic negative curves on certain complex surfaces

Finiteness of exceptional totally geodesic divisors in higher-dimensional spaces

Convergence of currents of integration along totally geodesic subvarieties

Abstract

We prove an equidistribution result for totally geodesic submanifolds in a compact locally symmetric space. In the case of Hermitian locally symmetric spaces, this gives a convergence theorem for currents of integration along totally geodesic subvarieties. As a corollary, we obtain that on a complex surface which is a compact quotient of the bidisc or of the 2-ball, there is at most a finite number of totally geodesic curves with negative self intersection. More generally, we prove that there are only finitely many exceptional totally geodesic divisors in a compact Hermitian locally symmetric space of the noncompact type of dimension at least 2.