For a complex manifold Σ with C∗-action, we define the m-th C∗ Fourier-Dolbeault cohomology group and consider the m-index on Σ. By applying the method of transversal heat kernel asymptotics, we obtain a local index formula for the m-index. We can reinterpret Kawasaki’s Hirzebruch-Riemann-Roch formula for a compact complex orbifold with an orbifold holomorphic line bundle by our integral formulas over a (smooth) complex manifold and finitely many complex submanifolds arising from singular strata. This is joint work with Chin-Yu Hsiao and I-Hsun Tsai.