THE LOCAL INDEX FORMULA FOR A HERMITIAN MANIFOLD

Let M be a compact complex manifold of real dimension m = 2 (m) over b ar with a Hermitian metric. Let a(n)(x,Delta(p,q)) be the heat equatio n asymptotics of the complex Laplacian Delta(p,q). Then Tr(L)2(fe-(t D elta p,q)) similar to Sigma(n=0)(infinity)t((n-m)/2) integral(M) fa(n) (x, Delta(p,q)) for any f is an element of C-infinity(M); the a, vanis h for n odd. Let ag(M) be the arithmetic genus and let a(n)(x, partial derivative) := Sigma(q)(-1)(q)a(n)(x, Delta(0,q)) be the supertrace o f the heat equation asymptotics. Then integral(M)a(n)(x, partial deriv ative)dx = 0 if n not equal m while integral(M)a(m)(x, partial derivat ive)dx = ag(M). The Todd polynomial Td((m) over bar) is the integrand of the Riemann Roch Hirzebruch formula. If the metric on M is Kaehler, then the local index theorem holds: (1) a(n)(x,partial derivative) = 0 for n < m, and a(m)(x, partial derivative) = Td((m) over bar) (x). I n this note, we show Equation (1) fails if the metric on M is not Kaeh ler.