Localization of the Riemann-Roch character

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。

Modified Futaki invariant and equivariant Riemann-Roch formula

In this paper, we give a new version of the modified Futaki invariant for a test configuration associated to the soliton action on a Fano manifold. Our version will naturally come from toric test configurations defined by Donaldson for toric manifolds. As an application, we show that the modified $K$-energy is proper for toric invariant Kähler potentials on a toric Fano manifold.

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' …

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…

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.

李群胚 Riemann-Roch-Hirzebruch 定理及应用

Lie groupoid Riemann-Roch-Hirzebruch theorem and applications

本文引入李代数胚指标理论并研究其 Dolbeault 算子，结合正则叶状流形上的 Connes 指标理论建立广义 Riemann-Roch 定理，并提出 Kodaira 消灭定理以通过拓扑信息验证正线丛判据。

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.

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…

Riemann-Roch for real varieties

If E is a C^\infty complex vector bundle on an oriented C^\infty manifold Σ, diffeomorphic to a circle, then the space of sections of E has a canonical polarization in the sense of Pressley and Segal and so one has its determinantal gerbe with lien C^*, the group of nonzero complex numbers. If q:Σ-->B is a smooth family of circles as above and E is a vector bundle on Σ, then the smooth direct image q_*(E) is an infinite-dimensional bundle with fibers as above and so we have its determinantal gerbe on B with lien being the sheaf of invertible complex valued C^\infty functions, it gives a class …

陈类不等式与 Riemann–Roch 型不等式

The inequalities of Chern classes and Riemann-Roch type inequalities

本文借助伴随丛有效极丰饶性与 Khovanskii–Teissier 对数凹性，建立射影流形上单项式陈类关于丰线丛和典范丛的通用不等式，并恢复及推广相关 Riemann–Roch 型结果。

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…