2026/07/25 by Gang Huang
#math.CV #math.AG #math.DG
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∈\mathbb Cm: |t|<1\ is a proper holomorphic submersion from a Kähler manifold X, and E is a Hermitian holomorphic vector bundle.