  In this paper, a description of the set-theoretical defining equations of
symplectic (type C) Grassmannian/flag/Schubert varieties in corresponding (type
A) algebraic varieties is given as linear polynomials in Pl$\ddot{u}$cker
coordinates, and it is proved that such equations generate the defining ideal
of variety of type C in those of type A. As applications of this result, the
number of local equations required to obtain the Schubert variety of type C
from the Schubert variety of type A is computed, and further geometric
properties of the Schubert variety of type C are given in the aspect of
complete intersections. Finally, the smoothness of Schubert variety in the
non-minuscule or cominuscule Grassmannian of type C is discussed, filling gaps
in the study of algebraic varieties of the same type.
