Varieties with $\mathbb P$-units

TL;DR

This paper investigates special classes of compact Kähler manifolds with trivial canonical bundle, focusing on those whose cohomology is generated by a single element, revealing connections to Calabi–Yau, hyperkähler, and Enriques varieties.

Contribution

It classifies and provides structure results for Kähler manifolds with trivial canonical bundle and cohomology generated by one element, especially with higher nilpotency index.

Findings

Identification of strict Calabi–Yau and hyperkähler cases based on generator properties.

Examples and structure theorems for higher nilpotency index cases.

Connection between these varieties and higher-dimensional Enriques varieties.

Abstract

We study the class of compact K\"ahler manifolds with trivial canonical bundle and the property that the cohomology of the trivial line bundle is generated by one element. If the square of the generator is zero, we get the class of strict Calabi--Yau manifolds. If the generator is of degree 2, we get the class of compact hyperk\"ahler manifolds. We provide some examples and structure results for the cases where the generator is of higher nilpotency index and degree. In particular, we show that varieties of this type are closely related to higher-dimensional Enriques varieties.