Pluripotential theory and special holonomy, I: Hodge decompositions and $\partial\bar\partial$ lemmata
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916818898124800 |
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| author | Pacini, Tommaso Raffero, Alberto |
| author_facet | Pacini, Tommaso Raffero, Alberto |
| contents | Let $M$ be a compact torsion-free $G_2$ 7-manifold or Calabi-Yau 6-manifold. We prove Hodge decomposition theorems for the $dd^ϕ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_03778 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pluripotential theory and special holonomy, I: Hodge decompositions and $\partial\bar\partial$ lemmata Pacini, Tommaso Raffero, Alberto Differential Geometry Let $M$ be a compact torsion-free $G_2$ 7-manifold or Calabi-Yau 6-manifold. We prove Hodge decomposition theorems for the $dd^ϕ$ 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. |
| title | Pluripotential theory and special holonomy, I: Hodge decompositions and $\partial\bar\partial$ lemmata |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2501.03778 |