Kähler manifold
with "f" a C∞ function and the "z"s and "w"s holomorphic functions. On a Kähler manifold, the components of a harmonic form are again harmonic. Therefore, for any compact Kähler manifold "X", the Hodge theorem gives a decomposition of the cohomology of "X" with complex coefficients as a direct sum of complex vector spaces: