Riemann-Roch for real varieties

相干层的等变陈特征与Riemann-Roch-Grothendieck定理

Equivariant Chern character for coherent sheaves and Riemann-Roch-Grothendieck

本文在有限群作用的紧致复流形上，为相干层建立等变陈特征理论，并证明相应的Riemann-Roch-Grothendieck定理。

由圆作用的奇异约化得到的黎曼-罗赫公式

A Riemann-Roch formula for singular reductions by circle actions

该研究针对允许矩映射取奇异值的可量子化哈密顿S¹-流形，推导了Guillemin-Sternberg原理下的不变黎曼-罗赫数公式，得到了仅依赖辛数据的奇异辛不变量相关结果。

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' …

Local index theory and the Riemann-Roch-Grothendieck theorem for complex flat vector bundles

The purpose of this paper is to give a proof of the real part of the Riemann-Roch-Grothendieck theorem for complex flat vector bundles at the differential form level in the even dimensional fiber case. The proof is, roughly speaking, an application of the local family index theorem for a perturbed twisted spin Dirac operator, a variational formula of the Bismut-Cheeger eta form without the kernel bundle assumption in the even dimensional fiber case, and some properties of the Cheeger-Chern-Simons class of complex flat vector bundle.

Riemann-Roch-Hirzebruch theorem and Topological Quantum Mechanics

In the present paper we discuss an independent on the Grothendieck-Sato isomorphism approach to the Riemann-Roch-Hirzebruch formula for an arbitrary differential operator. Instead of the Grothendieck-Sato isomorphism, we use the Topological Quantum Mechanics (more or less equivalent to the well-known constructions with the Massey operations from [KS], [P], [Me]). The statement that the Massey operations can "produce" the integral in some set-up, has an independent from the RRH theorem interest. We finish the paper by some open questions arising when the main construction is applied to the cycl…

Localization of the Riemann-Roch character

We present a K-theoritic approach to the Guillemin-Sternberg conjecture, about the commutativity of geometric quantization and symplectic reduction, which was proved by Meinrenken and Tian-Zhang. Besides providing a new proof of this conjecture for the full non-abelian group action case, our methods lead to a generalisation for compact Lie group actions on manifolds that are not symplectic. Instead, these manifolds carry an invariant almost complex structure and an abstract moment map.

Riemann-Roch多项式的正性与超凯勒流形的Todd类

Positivity of Riemann-Roch polynomials and Todd classes of hyperkähler manifolds

本文证明了超凯勒流形的Riemann-Roch多项式系数全为正，证实了Cao-作者的猜想和Riess的严格单调性问题，并蕴含Kawamata有效非消失猜想；方法上利用Rozansky-Witten理论构造Todd类平方根的Lefschetz型分解。

The equivariant Riemann-Roch theorem and the graded Todd class

Let G be a torus and M a G-Hamiltonian manifold with Kostant line bundle L and proper moment map. Let P be the weight lattice of G. We consider a parameter k and the multiplicity $m(λ,k)$ of the quantized representation associated to M and the k-th power of L . We prove that the weighted sum $\sum m(λ,k) f(λ/k)$ of the value of a test function f on points of the lattice $P/k$ has an asymptotic development in terms of the twisted Duistermaat-Heckman distributions associated to the graded Todd class of M. When M is compact, and f polynomial, the asymptotic series is finite and exact.

A Riemann-Roch Theorem For One-Dimensional Complex Groupoids

We consider a smooth groupoid of the form Σ\rtimesΓwhere Σis a Riemann surface and Γa discrete pseudogroup acting on Σby local conformal diffeomorphisms. After defining a K-cycle on the crossed product C_0(Σ)\rtimesΓgeneralising the classical Dolbeault complex, we compute its Chern character in cyclic cohomology, using the index theorem of Connes and Moscovici. This involves in particular a generalisation of the Euler class constructed from the modular automorphism group of the von Neumann algebra L^{\infty}(Σ)\rtimesΓ.

谱不对称性、超对称性与等价黎曼-罗奇缺陷

Spectral asymmetry, supersymmetry and the equivariant Riemann-Roch defect

本文研究了复空间锥状奇点的等价黎曼-罗奇缺陷的两种解释，提出L²旋量空间的新三重划分，并探讨了超对称抵消与黎曼-罗奇缺陷的关系。

Riemann-Roch isometries in the non-compact orbifold setting

We generalize work of Deligne and Gillet-Soulé on a Riemann-Roch type isometry, to the case of the trivial sheaf on cusp compactifications of Riemann surfaces $Γ\backslash\mathbb{H}$, for $Γ\subset PSL_{2}(\mathbb{R})$ a fuchsian group of the first kind, equipped with the Poincaré metric. This metric is singular at cusps and elliptic fixed points, and the results of Deligne and Gillet-Soulé do not apply to this setting. Our theorem relates the determinant of cohomology of the trivial sheaf, with an explicit Quillen type metric in terms of the Selberg zeta function of $Γ$, to a metrized version…

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…