Riemann-Roch Theorem and Index Theorem in Non-commutative Geometry

高维复非交换环面上的黎曼-罗赫定理

The Riemann-Roch Theorem on higher dimensional complex noncommutative tori

本文证明了配备复结构的任意维非交换环面上的黎曼-罗赫定理与霍奇定理的类似物，并探讨了区分非交换阿贝尔簇与非代数非交换复环面的理论意义。

A Riemann-Roch theorem for the noncommutative two torus

We prove the analogue of the Riemann-Roch formula for the noncommutative two torus $ A_θ = C(\mathbb{T}_θ^2)$ equipped with an arbitrary translation invariant complex structure and a Weyl factor represented by a positive element $k\in C^{\infty}(\mathbb{T}_θ^2)$. We consider a topologically trivial line bundle equipped with a general holomorphic structure and the corresponding twisted Dolbeault Laplacians. We define an spectral triple ($A_θ, \mathcal{H}, D)$ that encodes the twisted Dolbeault complex of $ A_θ$ and whose index gives the left hand side of the Riemann-Roch formula. Using Connes' …

An Analytic Grothendieck Riemann Roch Theorem

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 $\m…