An $L^2$-$\partial\overline\partial$-Lemma on a class of complete K\"ahler manifolds

Riccardo Piovani

An $L^2$-$\partial\overline\partial$-Lemma on a class of complete K\"ahler manifolds

Differential Geometry 2026-02-10 v1 Complex Variables

Authors: Riccardo Piovani

Abstract

We prove an $L^2$ - $\partial\overline\partial$ -Lemma involving smooth square integrable forms on complete K\"ahler manifolds, provided that the unique self-adjoint extension of the Hodge Laplacian on the Hilbert space of $L^2$ -forms has a gap in its spectrum near zero. This generalises the classical $\partial\overline\partial$ -Lemma on compact K\"ahler manifolds.

Keywords

L2-∂∂̄-lemma complete Kähler manifolds Hodge Laplacian square integrable forms spectral gap

Cite

@article{arxiv.2602.08831,
  title  = {An $L^2$-$\partial\overline\partial$-Lemma on a class of complete K\"ahler manifolds},
  author = {Riccardo Piovani},
  journal= {arXiv preprint arXiv:2602.08831},
  year   = {2026}
}

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