  This paper is concerned with a fourth order nonlinear dispersive partial
differential equation for closed curve flow on a K\"ahler manifold. The main
results is that the initial value problem has a solution locally in time if the
K\"ahler manifold is a compact locally hermitian symmetric space. The proof is
based on the geometric energy method combined with a nice gauge transformation
to eliminate the loss of derivatives. Interestingly, the results can be applied
to construct a generalized bi-Schr\"odinger flow proposed by Ding and Wang. The
assumption on the manifold plays a crucial role both to enjoy a good solvable
structure of the problem and to reduce the generalized bi-Schr\"odinger flow
equation to the one considered in the present paper.
