Curvature, macroscopic dimensions, and symmetric products of surfaces

Abstract

We present a detailed study of the curvature and symplectic asphericity properties of symmetric products of surfaces. We show that these spaces can be used to answer nuanced questions arising in the study of closed Riemannian manifolds with positive scalar curvature. For example, we prove that symmetric products of surfaces sharply distinguish between two distinct notions of macroscopic dimension introduced by Gromov and the second-named author. As a natural generalization of this circle of ideas, we address the Gromov--Lawson and Gromov conjectures in the Kaehler projective setting and draw new connections between the theories of the minimal model, positivity in algebraic geometry, and macroscopic dimensions.