# Dolbeault cohomology of complex manifolds with torus action

## TL;DR

This paper analyzes the Dolbeault cohomology algebra of complex manifolds with torus symmetry, including moment-angle and LVM-manifolds, providing models and Hodge decomposition results.

## Contribution

It introduces a dga model for basic Dolbeault cohomology and proves Hodge decomposition for manifolds with maximal holomorphic torus action.

## Key findings

- Provides a description of the Dolbeault cohomology algebra for the class of manifolds considered.
- Establishes a dga model for the ordinary Dolbeault cohomology algebra.
- Proves Hodge decomposition for the basic Dolbeault cohomology.

## Abstract

We describe the basic Dolbealut cohomology algebra of the canonical foliation on a class of complex manifolds with a torus symmetry group. This class includes complex moment-angle manifolds, LVM- and LVMB-manifolds and, in most generality, complex manifolds with a maximal holomorphic torus action. We also provide a dga model for the ordinary Dolbeault cohomology algebra. The Hodge decomposition for the basic Dolbeault cohomology is proved by reducing to the transversely Kaehler (equivalently, polytopal) case using a foliated analogue of toric blow-up.