An analytic Grothendieck Riemann Roch theorem

Abstract

We extend the Boutet de Monvel Toeplitz index theorem to complex manifolds with isolated singularities following the relative K-homology theory of Baum, Douglas, and Taylor for manifolds with boundary. We apply this index theorem to study the Arveson–Douglas conjecture. Let Bm be the unit ball in Cm, and I an ideal in the polynomial algebra C[z1,⋯,zm]. We prove that when the zero variety ZI is a complete intersection space with only isolated singularities and intersects with the unit sphere S2m−1 transversely, the representations of C[z1,⋯,zm] on the closure of I in La2(Bm) and also the corresponding quotient space QI are essentially normal. Furthermore, we prove an index theorem for Toeplitz operators on QI by showing that the representation of C[z1,⋯,zm] on the quotient space QI gives the fundamental class of the boundary ZI∩S2m−1. In the appendix, we prove with Kai Wang that if f∈La2(Bm) vanishes on ZI∩Bm, then f is contained inside the closure of the ideal I in La2(Bm).