The Local Index Formula for a Hermitian Manifold

Let M be a compact complex manifold of real dimension m = 2m̄ with a Hermitian metric. Let an(x,∆) be the heat equation asymptotics of the complex Laplacian ∆. Then TrL2(fe−t∆ p,q ) ∼ Σn=0t ∫ M fan(x,∆) for any f ∈ C∞(M); the an vanish for n odd. Let ag(M) be the arithmetic genus and let an(x, ∂̄) := Σq(−1)an(x,∆) be the supertrace of the heat equation asymptotics. Then ∫ M an(x, ∂̄)dx = 0 if n 6= m while ∫ M am(x, ∂̄)dx = ag(M). The Todd polynomial Tdm̄ is the integrand of the Riemann Roch Hirzebruch formula. If the metric on M is Kaehler, then the local index theorem holds: