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Siu's curvature positivity and L2 extension theorems for (p,q)-forms

  • Gang Huang
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Abstract

In this paper, we introduce Siu's curvature operator AEp,q for vector-bundle-valued differential forms on Kähler manifolds. When p=n, this operator reduces to the classical Akizuki--Nakano curvature operator. We first characterize the semipositivity of AEp,q in terms of an optimal L2-estimate condition for the ∂̅-operator, and then prove an Ohsawa--Takegoshi-type extension theorem for E-valued (p,q)-forms under the curvature condition AEp,q+1≥0, using a new twisted basic estimate adapted to this setting. As an application, we prove the local freeness of the higher direct image sheaf Rq s*(ΩpX/ Bm⊗ E) under the curvature conditions AEp,q+1≥0 and AEp,q≥0, where s: X → Bm:={t∈ℂm: |t|<1} is a proper holomorphic submersion from a Kähler manifold X, and E is a Hermitian holomorphic vector bundle.

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Cite

BibTeXRIS
BibTeX
@misc{huang2026sius,
  title = {{Siu's curvature positivity and $L^2$ extension theorems for $(p,q)$-forms}},
  author = {Gang Huang},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2607.23094},
  eprint = {2607.23094},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.23094},
}
RIS
TY  - UNPB
TI  - Siu's curvature positivity and L^2 extension theorems for (p,q)-forms
AU  - Gang Huang
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2607.23094
UR  - https://arxiv.org/abs/2607.23094
AB  - In this paper, we introduce Siu's curvature operator A^E_p,q for vector-bundle-valued differential forms on Kähler manifolds. When p=n, this operator reduces to the classical Akizuki--Nakano curvature operator. We first characterize the semipositivity of A^E_p,q in terms of an optimal L^2-estimate condition for the ∂̅-operator, and then prove an Ohsawa--Takegoshi-type extension theorem for E-valued (p,q)-forms under the curvature condition A^E_p,q+1≥0, using a new twisted basic estimate adapted to this setting. As an application, we prove the local freeness of the higher direct image sheaf R^q s_*(Ω^p_X/ B_m⊗ E) under the curvature conditions A^E_p,q+1≥0 and A^E_p,q≥0, where s: X → B_m:={t∈ℂ^m: |t|<1} is a proper holomorphic submersion from a Kähler manifold X, and E is a Hermitian holomorphic vector bundle.
ER  -