  We develop a geometric approach toward an interplay between a pair of quantum
Schur algebras of arbitrary finite type. Then by Beilinson-Lusztig-MacPherson's
stabilization procedure in the setting of partial flag varieties of type A
(resp. type B/C), the Howe duality between a pair of quantum general linear
groups (resp. a pair of $\imath$quantum groups of type AIII/IV) is established.
The Howe duality for quantum general linear groups has been provided via
quantum coordinate algebras in [Z02]. We also generalize this algebraic
approach to $\imath$quantum groups of type AIII/IV, and prove that the quantum
Howe duality derived from partial flag varieties coincides with the one
constructed by quantum coordinate (co)algebras. Moreover, the explicit
multiplicity-free decompositions for these Howe dualities are obtained.
