Isoparametric hypersurfaces and complex structures

TL;DR

This paper constructs almost complex and complex structures on specific isoparametric hypersurfaces in spheres, leading to new complex structures on certain product manifolds with vanishing first Chern class.

Contribution

It introduces methods to build complex structures on isoparametric hypersurfaces and specific product manifolds in spheres, expanding the understanding of complex geometry in these contexts.

Findings

Complex structures on $S^1\times S^7\times S^6$ are constructed.

Complex structures on $S^1\times S^3\times S^2$ with vanishing first Chern class are established.

New examples of complex structures on product manifolds in spheres are provided.

Abstract

The main purpose of this note is to construct almost complex or complex structures on certain isoparametric hypersurfaces in unit spheres. As a consequence, complex structures on $S^1\times S^7\times S^6$, and on $S^1\times S^3\times S^2$ with vanishing first Chern class, are built.