It is well known that the kernel of the heat equation gives a formula for the index of an elliptic operator and also for the Lefschetz fixed point formulas. For the de Rham, signature, and SPIN-c complexes this formula agrees with the classical formula of the Atiyah-Singer index theorem in terms of characteristic classes. For the Dolbeault complex, however, this formula does not agree in general with the formula of the index theorem. Toledo and Tong have given a formula for the Lefschetz number in the holomorphic case and it is of interest to what extent the heat equation formula agrees with this formula in the non-Kähler case (the two agree for Kähler manifolds and isometries). The author assumes that the fixed point set is Kähler and that the group action in question is semi- simple. He then constructs a metric in the normal direction so that the formulas of the heat equation agree with those of Toledo and Tong.