Complex structures on the product of two Sasakian manifolds

TL;DR

This paper constructs a family of complex structures on the product of two compact Sasakian manifolds, analyzes their properties, and computes their Dolbeault cohomology groups, revealing new insights into their geometric structure.

Contribution

It introduces a novel family of complex structures on Sasakian manifold products and compares them with existing structures, expanding understanding of their geometric and cohomological properties.

Findings

None of the new complex structures admit compatible locally conformally K"ahler metrics for dimensions greater than 1

Computed Dolbeault cohomology groups of the product manifolds

Compared new complex structures with previously studied families

Abstract

A Sasakian manifold is a Riemannian manifold whose metric cone admits a certain K\"ahler structure which behaves well under homotheties. We show that the product of two compact Sasakian manifolds admits a family of complex structures indexed by a complex nonreal parameter, none of whose members admits any compatible locally conformally K\"ahler metrics if both Sasakian manifolds are of dimension greater than $1$. We compare this family with another family of complex structures which has been studied in the literature. We compute the Dolbeault cohomology groups of these products of compact Sasakian manifolds.