The m-genus of a compact complex manifold is the complex dimen-sion of the vector space of all holomorphic sections of the m-th tensor power of its canonical line bundle. The deformational invariance of the m-genus for any positive m is known to hold for compact complex algebraic mani-folds. When m is 1, such a deformational invariance for all compact Kahler manifolds is a just direct consequence of the Hodge decomposition. The question naturally arises whether the deformational invariance of m-genus for m greater than 1 can also be understood in the context of some form of Hodge decomposition with the vector space of all holomorphic m-canonical sections as a summand. We discuss the results and the developments in the study of this problem by starting with the simplest case of compact Riemann
surfaces.