2016 Elsevier Inc. 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 B m be the unit ball in C m , and I an ideal in the polynomial algebra C[z 1 ,...,z m ]. We prove that when the zero variety Z I is a complete intersection space with only isolated singularities and intersects with the unit sphere S2m-1 transversely, the representations of C[z 1 ,...,z m ] on the closure of I in L a2 (B m ) and also the corresponding quotient space Q I are essentially normal. Furthermore, we prove an index theorem for Toeplitz operators on Q I by showing that the representation of C[z 1 ,...,z m ] on the quotient space Q I gives the fundamental class of the boundary ZIS 2m-1 . In the appendix, we prove with Kai Wang that if fLa2(B m ) vanishes on ZIB m , then f is contained inside the closure of the ideal I in La2(B m ).
