Geometry of complex bounded domains with finite-volume quotients (1801.00459v2)

Abstract: We first show that for a bounded pseudoconvex domain with a manifold quotient of finite-volume in the sense of Kahler-Einstein measure, the identity component of the automorphism group of this domain is semi-simple without compact factors. This partially answers an open question in [Fra95]. Then we apply this result in different settings to solve several open problems, for examples, (1). We prove that the automorphism group of the Griffiths domain [Gri71] in $\mathbb{C}^2$ is discrete. This gives a complete answer to an open question raised four decades ago. (2). We show that for a contractible HHR/USq complex manifold $D$ with a finite-volume manifold quotient $M$, if $D$ contains a one-parameter group of holomorphic automorphisms and the fundamental group of $M$ is irreducible, then $D$ is biholomorphic to a bounded symmetric domain. This theorem can be viewed as a finite-volume version of Nadel-Frankel's solution for the Kahzdan conjecture, which has been open for years. (3). We show that for a bounded convex domain $D\subset \mathbb{C}^n$ of $C^2$-smooth boundary, if $D$ has a finite-volume manifold quotient with an irreducible fundamental group, then $D$ is biholomorphic to the unit ball in $\mathbb{C}^n$, which partially solves an old conjecture of Yau. For (2) and (3) above, if the complex dimension is equal to $2$, more refined results will be provided.