An analytic Grothendieck Riemann Roch theorem

Abstract

Abstract. We extend the Boutet de Monvel Toeplitz index theorem to complex manifold with isolated singularities following the relative K-homology theory of Baum, Douglas, and Taylor for manifold 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 S 2m−1 transversely, the representations of C[z 1 , · · · , z m ] on the closure of I in L 2 a (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 Z I ∩ S 2m−1 . In the appendix, we prove with Kai Wang that if f ∈ L 2 a (B m ) vanishes on Z I ∩ B m , then f is contained inside the closure of the ideal I in L 2 a (B m ).

Geometric Arveson-Douglas conjecture and holomorphic extension

“…The methods of this paper differ from those in [13] and [14]. We obtain essential normality by proving that the projection operator onto the submodule is in the Toeplitz algebra.…”

“…This approach is new and allows us to analyse the conjecture using tools from complex harmonic analysis. Part of our ideas come from Suárez's paper [27] and the first author, Tang and Yu's paper [13].…”

“…Shalit and Kennedy [20] proved the case when M can be decomposed into varieties having "good" properties; Engliš and Eschmeier [14] proved the case when M is a homogeneous subvariety that is smooth away from the origin and the case when M is a (not necessarily homogeneous) smooth submanifold that intersects ∂B n transversally; the first author, Tang and Yu [13] proved the weaker form under the assumption that M is a (not necessarily homogenous) complete intersection space that intersects ∂B n transversally and has no singular point on ∂B n .…”

“…[13]), especially when I is radical, one can prove that [I], the closure of the ideal I, consists of all the holomorphic functions in H that vanish on V (I). Here V (I) = {z ∈ B n : p(z) = 0, ∀p ∈ I}.…”

“…[6] and section 4.2 in [13]) implies the essential normality of the quotient module. (See the remark in the end of section 3).…”