We prove an index theorem for the quotient module of a monomial ideal. We obtain this result by resolving the monomial ideal by a sequence of Bergman space like essentially normal Hilbert modules.1 Arveson [2, Corollary 2.2] proved the similar result on the Drury-Arveson space.

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In this paper we prove a K-homology index theorem for the Toeplitz operators obtained from the multishifts of the Bergman space on several classes of egg-like domains. This generalizes our theorem with Douglas and Yu on the unit ball [21].

This paper studies the essential normality of Bergman modules over the intersection of complex ellipsoids, as well as their quotients by monomial ideals.

We study the geometry in the perturbations of principal submodules in the Drury-Arveson space. We show that the perturbations give rise to smooth vector bundles of Hilbert spaces which are equipped with natural Hermitian connections. We compute the associated parallel transport operators and explore properties of the monodromy.1 The smoothness of the vector bundle will be established in Theorem 2.3. 2 Namely, we are momentarily putting aside continuity or smoothness considerations.C such that limn→∞ an/bn = C.