The Kollar-Pardon conjecture on projective varieties with semialgebraic universal cover

Theorem 1.1: for a connected normal projective complex variety $X$, $\widetilde X$ is biholomorphic to a semialgebraic open subset of a projective variety if and only if $\widetilde X\cong D\times\mathbb C^m\times F$ with $D$ bounded symmetric and $F$ simply connected normal projective (either may be a point); no algebraicity of deck transformations is assumed and the compact factor may be singular. The cover is quasi-projective exactly when $D$ is a point; a smooth projective variety covered by $\mathbb C^n$ is finitely etale covered by an abelian variety. The companion proves that a connected bounded semialgebraic open subset of an affine variety with a proper cocompact holomorphic group action is smooth and a bounded symmetric domain. Not covered: non-normal or non-projective (e.g. compact Kahler) $X$.

Theorem 1.1: for a connected normal projective complex variety $X$, $\widetilde X$ is biholomorphic to a semialgebraic open subset of a projective variety if and only if $\widetilde X\cong D\times\mathbb C^m\times F$ with $D$ bounded symmetric and $F$ simply connected normal projective (either may be a point); no algebraicity of deck transformations is assumed and the compact factor may be singular. The cover is quasi-projective exactly when $D$ is a point; a smooth projective variety covered by $\mathbb C^n$ is finitely etale covered by an abelian variety. The companion proves that a connected bounded semialgebraic open subset of an affine variety with a proper cocompact holomorphic group action is smooth and a bounded symmetric domain. Not covered: non-normal or non-projective (e.g. compact Kahler) $X$.