A Characterization of Varieties whose Universal Cover is the Polydisk or a Tube Domain

TL;DR

This paper characterizes when a complex algebraic variety's universal cover is either a polydisk or a tube domain, using special tensor sections and properties of the canonical bundle.

Contribution

It provides new criteria linking the existence of semispecial and slope zero tensors to the universal cover being a polydisk or a tube domain, extending previous results.

Findings

Universal cover is a polydisk iff certain semispecial tensor conditions hold.

A variety admits a slope zero tensor iff its universal cover is a tube type bounded symmetric domain.

Results extend to Galois conjugates of the variety, preserving the universal cover type.

Abstract

Catanese and Franciosi defined a semispecial tensor as a (non zero) section of the n-th symmetric power of the cotangent bundle twisted by the anticanonical divisor and by a 2-torsion line bundle. A slope zero tensor is instead a section of the nm-th symmetric power of the cotangent bundle twisted by m times the anticanonical divisor. With these definitions we have: Theorem 1. The universal cover of X is the polydisk iff 1) holds. 1) X has ample canonical bundle and admits a semispecial tensor such that at some point p the corresponding hypersurface in the projectivized tangent space is reduced. Theorem 2. If X has ample canonical bundle it admits a slope zero tensor if and only if the universal cover of X is a bounded symmetric domain D of tube type. The domain D is determined by the multiplicities of the irreducible components of the corresponding tangential hypersurface. We have then a corollary which extends previous results by Kazhdan. Corollary. Assume that the universal covering of X is a bounded symmetric domain D of tube type. Let X^s be a Galois conjugate of X : then also the universal cover of X^s is biholomorphic to D.