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

In this paper, we introduce Siu's curvature operator $A^E_{p,q}$ 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 $A^E_{p,q}$ in terms of an optimal $L^2$-estimate condition for the $\bar\partial$-operator, and then prove an Ohsawa--Takegoshi-type extension theorem for $E$-valued $(p,q)$-forms under the curvature condition $A^E_{p,q+1}\geq0$, using a new twisted basic estimate adapted to this setting. As an application, we prove the local freeness of the higher direct image sheaf $R^q s_*(\Omega^p_{X/ B_m}\otimes E)$ under the curvature conditions $A^E_{p,q+1}\geq0$ and $A^E_{p,q}\geq0$, where $s: X \to B_m:=\{t\in\mathbb C^m:\ |t|<1\}$ is a proper holomorphic submersion from a K\"ahler manifold $X$, and $E$ is a Hermitian holomorphic vector bundle.