Pluripotential theory and special holonomy, I: Hodge decompositions and $\partial\bar\partial$ lemmata

TL;DR

This paper extends classical pluripotential theory to special holonomy manifolds, proving Hodge decompositions and $ ext{dd}^ ext{phi}$ lemmas, and relating cohomology to geometric structures like coassociative submanifolds.

Contribution

It introduces Hodge decomposition theorems for $dd^ ext{phi}$ operators on $G_2$ and Calabi-Yau manifolds, generalizing classical $ ext{d} ext{d}^c$ results and relating cohomology to geometric entities.

Findings

Established Hodge decompositions for $dd^ ext{phi}$ operators.

Derived analogues of the $ ext{d} ext{d}^c$ lemma in special holonomy contexts.

Connected cohomology classes to geometric structures like coassociative submanifolds.

Abstract

Let $M$ be a compact torsion-free $G_2$ 7-manifold or Calabi-Yau 6-manifold. We prove Hodge decomposition theorems for the $dd^\phi$ operators, introduced by Harvey and Lawson, which generalize the $i\partial\bar\partial$ operator used in classical pluripotential theory. We then obtain analogues of the $\partial\bar\partial$ lemma in this context. We formalize this by defining cohomology spaces analogous to Bott-Chern cohomology and we relate them to harmonic forms on $M$. In the $G_2$ case we provide a geometric interpretation of the corresponding cohomology classes in terms of coassociative submanifolds and gerbes: this is analogous to the classical interpretation of Bott-Chern cohomology classes in terms of divisors and holomorphic line bundles.