  A V-cycle multigrid method for the Hellan-Herrmann-Johnson (HHJ)
discretization of the Kirchhoff plate bending problems is developed in this
paper. It is shown that the contraction number of the V-cycle multigrid HHJ
mixed method is bounded away from one uniformly with respect to the mesh size.
The uniform convergence is achieved for the V-cycle multigrid method with only
one smoothing step and without full elliptic regularity. The key is a stable
decomposition of the kernel space which is derived from an exact sequence of
the HHJ mixed method, and the strengthened Cauchy Schwarz inequality. Some
numerical experiments are provided to confirm the proposed V-cycle multigrid
method. The exact sequences of the HHJ mixed method and the corresponding
commutative diagram is of some interest independent of the current context.
