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 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