Siu's curvature positivity and $L^2$ extension theorems for $(p,q)$-forms

In this paper, we introduce Siu’s curvature operator $A_{p,q}^E$ for vector-bundle-valued differential forms on K"ahler manifolds. When $p=n$, this operator reduces to the classical Akizuki–Nakano curvature operator. We first characterize the semipositivity of $A_{p,q}^E$ in terms of an optimal $L^2$-estimate condition for the $\bar{\mathrm{\partial }}$-operator, and then prove an Ohsawa–Takegoshi-type extension theorem for $E$-valued $(p,q)$-forms under the curvature condition $A_{p,q+1}^E\ge 0$, using a new twisted basic estimate adapted to this setting. As an application, we prove the local freeness of the higher direct image sheaf $R^q{s}_{\ast }({\mathrm{\Omega }}_{X\mathrm{/}{B}_m}^p\otimes E)$ under the curvature conditions $A_{p,q+1}^E\ge 0$ and $A_{p,q}^E\ge 0$, where $s:X\to B_m:=\{t\in {\mathbb{C}}^m:\mathrm{\mid }t\mathrm{\mid }<1\}$ is a proper holomorphic submersion from a K"ahler manifold $X$, and $E$ is a Hermitian holomorphic vector bundle.