Slope zero tensors, uniformizing variations of Hodge structure and quotients of tube domains

Abstract

We prove an equivalence between two approaches to characterizing complex-projective varieties $X$ with klt singularities and ample canonical divisor that are uniformized by bounded symmetric domains. In order to do so, we show how to construct a uniformizing variation of Hodge structure from a slope zero tensor and vice versa. As a consequence, we generalize various uniformization results of Catanese and Di Scala to the singular setting. For example, we prove that $X$ is a quotient of a bounded symmetric domain of tube type by a group acting properly discontinuously and freely in codimension one if and only if $X$ admits a slope zero tensor. As a key step in the proof, we establish the compactness of the holonomy group of the singular K\"{a}hler--Einstein metric on $X_{\mathrm{reg}}$.