An Aubin-Yau theorem for transversally K\"ahler foliations

TL;DR

This paper extends the Aubin-Yau theorem to transversally K"ahler foliations, providing a new proof under homological orientability, and simplifies the proof of the Vaisman Aubin-Yau theorem.

Contribution

It adapts classical Aubin-Yau methods to the transversally K"ahler foliation setting under homological orientability, offering a new proof and simplifying existing results.

Findings

Established an Aubin-Yau theorem for transversally K"ahler foliations

Provided a self-contained proof adapting classical methods

Simplified the proof of the Vaisman Aubin-Yau theorem

Abstract

Transversally K\"ahler foliations are a generalisation of K\"ahler manifolds, appearing naturally in the complex non-K\"ahler setting. We give a self-contained proof of how the classical methods used in the proof of the Aubin-Yau theorem adapt to the transversally K\"ahler case under the homological orientability condition. We apply this result to obtain a new, simpler proof of the already known Vaisman Aubin-Yau theorem.