On Rigid Manifolds of Kodaira Dimension 1

TL;DR

This paper investigates rigid compact complex manifolds with Kodaira dimension 1, focusing on product-quotient varieties, group actions, and classifications of minimal group orders, providing new insights into their structure and singularity resolutions.

Contribution

It offers a classification of rigid manifolds of Kodaira dimension 1 arising from product-quotient constructions, including group descriptions and singularity resolutions.

Findings

No free rigid action on product of (n-1) elliptic curves and a genus ≥ 2 curve.

Describes groups and quotients involved in these manifolds.

Provides a complete classification for minimal group orders.

Abstract

We discuss rigid compact complex manifolds of Kodaira dimension 1, arising as product-quotient varieties. First, we show that there is no free rigid action on the product of $(n-1)$ elliptic curves and a curve of genus at least two. Then, we describe the occurring groups, study the quotients and prove that there is always a suitable resolution of singularities preserving rigidity. Finally, we give a complete classification of the cases arising for minimal group orders.