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 $ball^m$ be the unit ball in $mathbb{C}^m$, and $I$ an ideal in the polynomial algebra $mathbb{C}[z_1, cdots, z_m]$. We prove that when the zero variety $Z_I$ is a complete intersection space with only isolated singularities and intersects with the unit sphere $mathbb{S}^{2m-1}$ transversely, the representations of $mathbb{C}[z_1, cdots, z_m]$ on the closure of $I$ in $L^2_a(ball^m)$ 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 $mathbb{C}[z_1, cdots, z_m]$ on the quotient space $Q_I$ gives the fundamental class of the boundary $Z_Icap mathbb{S}^{2m-1}$. In the appendix, we prove with Kai Wang that if $fin L^2_a(ball^m)$ vanishes on $Z_Icap ball ^m$, then $f$ is contained inside the closure of the ideal $I$ in $L^2_a(ball^m)$.
تحميل البحث