Fermat's Cubic, Klein's Quartic and Rigid Complex Manifolds of Kodaira Dimension One

TL;DR

This paper constructs higher-dimensional rigid complex manifolds with Kodaira dimension one by using quotients of Fermat curves and Klein's quartic, and employs toric geometry for resolutions.

Contribution

It introduces a method to produce rigid complex manifolds of dimension n with Kodaira dimension one using singular quotients and toric resolutions.

Findings

Constructed n-dimensional rigid manifolds for all n ≥ 3.

Established the rigidity of the resolved manifolds.

Compared deformation theories of singular models and resolutions.

Abstract

For each $n \geq 3$ the authors provide an $n$-dimensional rigid compact complex manifold of Kodaira dimension $1$. First they construct a series of singular quotients of products of $(n-1)$ Fermat curves with the Klein quartic, which are rigid. Then using toric geometry a suitable resolution of singularities is constructed and the deformation theories of the singular model and of the resolutions are compared, showing the rigidity of the resolutions.