The local index formula for a Hermitian manifold

Peter B. Gilkey, S. Nikčević and J. Pohjanpelto

Vol. 180 (1997), No. 1, 51–56

DOI: 10.2140/pjm.1997.180.51

Abstract

Let M be a compact complex manifold of real dimension m = 2m with a Hermitian metric. Let a_n(x,Δ^p,q) be the heat equation asymptotics of the complex Laplacian Δ^p,q. Then Tr_L²(fe^−tΔp,q ) ∼ Σ_n=0^∞t^(n−m)∕2 ∫ _Mfa_n(x,Δ^p,q) for any f ∈ C^∞(M); the a_n vanish for n odd. Let ag(M) be the arithmetic genus and let a_n(x,∂) := Σ_q(−1)^qa_n(x,Δ^0,q) be the supertrace of the heat equation asymptotics. Then ∫ _Ma_n(x,∂)dx = 0 if n≠m while ∫ _Ma_m(x,∂)dx = ag(M). The Todd polynomial Td_m is the integrand of the Riemann Roch Hirzebruch formula. If the metric on M is Kaehler, then the local index theorem holds:

In this note, we show Equation (1) fails if the metric on M is not Kaehler.

Milestones

Received: 17 May 1996

Published: 1 September 1997