For any smooth map \(u_0\) from \(\mathbb{S}^1 \times \mathbb{R}\) into a K”ahler manifold \(N\) with complex structure \(J\) and metric \(h\), the authors define a KdV geometric flow as the Cauchy problem: \[ \frac{\partial u}{\partial t} = \nabla^2_x u_x + \frac{1}{2} R(u_x, J_uu_x)J_uu_x,\quad u(x,0)=u_0(x). \] Here \(R\) is the Riemannian curvature tensor on \(N\) and \(J_u=J(u)\). It is shown that the above Cauchy problem has short time existence of solutions in suitable Sobolev spaces. Moreover, long time existence of solutions is shown when \((N, J, h)\) is a certain type of locally Hermitian space due to conservation laws additional to the ones which hold on Kähler manifolds.