[Paper Review] An Analytic Grothendieck Riemann Roch Theorem

Ronald G. Douglas, Xiang Tang+1 more|arXiv (Cornell University)|Apr 16, 2014

Holomorphic and Operator Theory24 references3 citations

TL;DR

This paper extends the Grothendieck-Riemann-Roch theorem to complex manifolds with isolated singularities and boundaries by developing an analytic index theory for Toeplitz operators on quotient spaces of Bergman spaces. It proves that when the zero variety of a radical ideal in the polynomial ring is a complete intersection with isolated singularities intersecting the unit sphere transversely, the corresponding representations on the closure of the ideal and its quotient space are essentially normal, and the quotient representation realizes the fundamental class of the boundary via the Connes-Chern character, generalizing Boutet de Monvel's index theorem.

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 $\ball^m$ 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 $Z_I\cap \mathbb{S}^{2m-1}$. In the appendix, we prove with Kai Wang that if $f\in L^2_a(\ball^m)$ vanishes on $Z_I\cap \ball ^m$, then $f$ is contained inside the closure of the ideal $I$ in $L^2_a(\ball^m)$.

Motivation & Objective

• To generalize the Grothendieck-Riemann-Roch theorem to complex manifolds with isolated singularities and boundaries.
• To extend Boutet de Monvel's Toeplitz index theorem to singular complex manifolds with boundary via relative K-homology.
• To establish the Arveson-Douglas conjecture for ideals whose zero varieties are complete intersections with isolated singularities intersecting the unit sphere transversely.
• To prove that the closure of an ideal in the Bergman space equals the kernel of the restriction map to the zero variety, under specific geometric conditions.
• To show that the quotient representation on the Bergman space quotient realizes the fundamental class of the boundary via the Connes-Chern character.

Proposed method

• Uses relative K-homology theory of Baum, Douglas, and Taylor for manifolds with boundary to define a K-homology class from the ∂̄-operator with Neumann boundary conditions.
• Applies the Connes-Chern character to relate K-homology classes to periodic cyclic cohomology, enabling a noncommutative generalization of Grothendieck-Riemann-Roch.
• Constructs extension and restriction operators between Bergman spaces on scaled balls and their intersections with the zero variety, ensuring uniform operator norms.
• Employs a limiting argument using holomorphic functions on shrinking and expanding balls to approximate functions in the kernel of the restriction map.
• Uses the Nevanlinna-type approximation via dilated holomorphic functions to show that functions vanishing on the zero variety are in the closure of the ideal.
• Establishes that the boundary map in relative K-homology sends the ∂̄-Neumann class to the fundamental class of the CR boundary, generalizing Boutet de Monvel's index theorem.

Experimental results

Research questions

• RQ1 Does the representation of the polynomial algebra on the closure of an ideal in the Bergman space remain essentially normal when the zero variety has isolated singularities and intersects the unit sphere transversely?
• RQ2 Can the analytic Grothendieck-Riemann-Roch theorem be extended to complex manifolds with singularities and boundaries using relative K-homology?
• RQ3 Is the quotient representation on the Bergman space quotient of the ideal unitarily equivalent to the fundamental class of the boundary CR manifold?
• RQ4 Does the kernel of the restriction map from the Bergman space to the zero variety coincide with the closure of the ideal in the Bergman norm?
• RQ5 What is the index-theoretic meaning of the quotient representation in terms of the Connes-Chern character and the boundary fundamental class?

Key findings

• The representation of the polynomial algebra on the closure of the ideal I in L²_a(𝐵^m) is essentially normal when Z_I is a complete intersection with isolated singularities and intersects ∂𝐵^m transversely.
• The quotient representation on Q_I = L²_a(𝐵^m)/cl(I) realizes the fundamental class of the boundary Z_I ∩ ∂𝐵^m as a K-homology class via the Connes-Chern character.
• The kernel of the restriction map R: L²_a(𝐵^m) → L²_a,M(Ω_I) equals the closure of the ideal I in L²_a(𝐵^m), under the given geometric assumptions.
• The index of the Toeplitz operator on Q_I is computed via the boundary fundamental class, generalizing Boutet de Monvel's index theorem to singular settings.
• The extension and restriction operators between scaled Bergman spaces are uniformly bounded, enabling approximation arguments via holomorphic dilation.
• The closure of the ideal I in L²_a(𝐵^m) is precisely the space of holomorphic functions vanishing on Z_I ∩ 𝐵^m, confirming a key case of the Arveson-Douglas conjecture.