  The $N=2$ topological Yang-Mills and holomorphic Yang-Mills theories on
simply connected compact K\"{a}hler surfaces with $p_g\geq 1$ are reexamined.
The $N=2$ symmetry is clarified in terms of a Dolbeault model of the
equivariant cohomology. We realize the non-algebraic part of Donaldson's
polynomial invariants as well as the algebraic part. We calculate Donaldson's
polynomials on $H^{2,0}(S,\BZ)\oplus H^{0,2}(S,\BZ)$.
