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This paper investigates the fundamental groups of normal projective Calabi-Yau varieties with klt singularities and their regular loci, proving both are almost abelian. The ranks of these groups are shown to be dictated by the irregularities of specific finite etale and quasi-etale covers. The methodology leverages differential geometry and recent breakthroughs in the geometric theory of complex Monge-Ampere equations.
Calabi-Yau fundamental groups are proven to be almost abelian, with ranks directly linked to cover irregularities.
We study the fundamental groups of a normal projective Calabi-Yau variety $X$ with klt singularities and of its regular locus $X_{reg}$. We prove that both groups are almost abelian, and that their ranks are determined by the irregularities of finite etale and quasi-etale covers, respectively. Our approach is differential geometric, and relies on recent advances in the geometric theory of complex Monge-Ampere equations.