A rigid Calabi--Yau 3-fold

TL;DR

This paper investigates the geometric and cohomological properties of a specific rigid Calabi--Yau threefold derived from a quotient of an elliptic curve product, providing new models and insights into its mirror symmetry.

Contribution

It analyzes the cohomology, describes projective models, and introduces a new smoothing of a singular model of the rigid Calabi--Yau threefold.

Findings

Cohomology of the threefold is explicitly described.

A simple formula for the trilinear form on Picard group is provided.

A new smoothing of a singular model with small Hodge numbers is constructed.

Abstract

The aim of this paper is to analyze some geometric properties of the rigid Calabi--Yau threefold $\mathcal{Z}$ obtained by a quotient of $E^3$, where $E$ is a specific elliptic curve. We describe the cohomology of $\mathcal{Z}$ and give a simple formula for the trilinear form on $Pic(\mathcal{Z})$. We describe some projective models of $\mathcal{Z}$ and relate these to its generalized mirror. A smoothing of a singular model is a Calabi--Yau threefold with small Hodge numbers which was not known before.