Paul Bressler, Ryszard Nest, Boris Tsygan
We prove a Riemann-Roch formula for deformation quantization of complex manifolds and its corollary, an index theorem for elliptic pairs conjectured by Schapira and Schneiders.
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We prove a Hirzebruch-Riemann-Roch type formula for global matrix factorizations. This is established by an explicit realization of the abstract Hirzebruch-Riemann-Roch type formula of Shklarov. We also show a Grothendieck-Riemann-Roch type theorem.
Equivariant Riemann-Roch theorem for the complex variety under the action of complex linear reductive algebraic group.
In this lecture we review apprearance of the Riemann-Roch Theorem in classical function theory, Algebraic topology, in theory of pseudo-differential operators and finally in noncommutative geometry. We show also it usefulness in many problem of quantization and harmonic analysis on Lie groups.
We give a simple proof of the Riemann-Roch theorem for Deligne-Mumford stacks using the equivariant Riemann-Roch theorem and the localization theorem in equivariant K-theory together with some basic commutative algebra of Artin rings.
We prove two refinements of the higher Riemann-Roch without denominators: a statement for regular closed immersions between arbitrary finite dimensional noetherian schemes, with no smoothness assumptions, and a statement for the relative cohomology of a proper morphism.
In this paper, we formulate axioms of certain graded cohomology theory for which Chern class maps from higher K-theory are defined, following the method of Gillet [Gi1]. We will not include homotopy invariance nor purity in our axioms. It will turn out that the Grothendieck-Riemann-Roch theorem and the Riemann-Roch theorem without denominators hold for our higher Chern classes. We will also give two applications of our Riemann-Roch results.
We prove the Riemann-Roch theorem for homotopy invariant $K$-theory and projective local complete intersection morphisms between finite dimensional noetherian schemes, without smoothness assumptions. We also prove a new Riemann-Roch theorem for the relative cohomology of a morphism. In order to do so, we construct and characterize Gysin morphisms for regular immersions between cohomologies represented by spectra (examples include homotopy invariant $K$-theory, motivic cohomology, their arithmetic counterparts, real absolute Hodge and Deligne-Beilinson cohomology, rigid syntomic cohomology, mix…
This is the integral text of my thesis. The first part is an expanded version of "Riemann-Roch theorems for Deligne-Mumford stacks", where I deal with Artin stacks over general bases. In the second part, I prove some Riemann-Roch statment for D-modules on Deligne-Mumford stacks, and I also consider the problem of algebraization of analytic stacks.
In this article, we show that the combination of the constructions done in SGA 6 and the A^1-homotopy theory naturally leads to results on higher algebraic K-theory. This applies to the operations on algebraic K-theory, Chern characters and Riemann-Roch theorems.
For a $G$-scheme $X$ with a given equivariant perfect obstruction theory, we prove a virtual equivariant Grothendieck-Riemann-Roch formula, this is an extension of a result of Fantechi-Göttsche to the equivariant context. We also prove a virtual non-abelian localization theorem for schemes over $\mathbb{C}$ with proper actions.
We give a direct proof that the Freed-Lott differential analytic index is well defined and a condensed proof of the differential Grothendieck-Riemann-Roch theorem. As a byproduct we also obtain a direct proof that the R/Z analytic index is well defined and a condensed proof of the R/Z Grothendieck-Riemann-Roch theorem.
Local index theory and the Riemann-Roch-Grothendieck theorem for complex flat vector bundles II
该研究在微分形式层面证明了复平坦向量丛的Riemann-Roch-Grothendieck定理的实部，推进了相关局部指标理论研究。
For a separable $C^*$-algebra $A$, we introduce an exact $C^*$-category called the Paschke Category of $A$, which is completely functorial in $A$, and show that its K-theory groups are isomorphic to the topological K-homology groups of the $C^*$-algebra $A$. Then we use the Dolbeault complex and ideas from the classical methods in Kasparov K-theory to construct an acyclic chain complex in this category, which in turn, induces a Riemann-Roch transformation in the homotopy category of spectra, from the algebraic K-theory spectrum of a complex manifold $X$, to its topological K-homology spectrum.
In this paper we prove a categorification of the Grothendieck-Riemann-Roch theorem. Our result implies in particular a Grothendieck-Riemann-Roch theorem for Toën and Vezzosi's secondary Chern character. As a main application, we establish a comparison between the Toën-Vezzosi Chern character and the classical Chern character, and show that the categorified Chern character recovers the classical de Rham realization.
V. Lunts has recently established Lefschetz fixed point theorems for Fourier-Mukai functors and dg algebras. In the same vein, D. Shklyarov introduced the noncommutative analogue of the Hirzebruch-Riemann-Roch theorem. In this short article, we see how these constructions and computations formally stem from their motivic counterparts.
In this paper we give a simplified proof of the flat Grothendieck-Riemann-Roch theorem. The proof makes use of the local family index theorem and basic computations of the Chern-Simons form. In particular, it does not involve any adiabatic limit computation of the reduced eta-invariant.
Quadratic Riemann-Roch formulas
本文为非经典定向及辛定向的上同调理论建立了二次 Grothendieck-Riemann-Roch 公式，并展示了 Todd 类的计算方法，最后通过 Borel 特征进行了应用说明。
We prove that, for smooth quasi-projective varieties over a field, the $K$-theory $K(X)$ of vector bundles is the universal cohomology theory where $c_1(L\otimes \bar L)=c_1(L)+c_1(\bar L)-c_1(L)c_1(\bar L)$. Then, we show that Grothendieck's Riemann-Roch theorem is a direct consequence of this universal property, as well as the universal property of the graded $K$-theory $GK^\bullet (X)\otimes \mathbb{Q}$.
We define bivariant algebraic K-theory and bivariant derived Chow on the homotopy category of derived schemes over a smooth base. The orientation on the latter corresponds to virtual Gysin homomorphisms. We then provide a morphism between these two bivariant theories and compare the two orientations. This comparison then yields a homological and cohomological Grothendieck-Riemann-Roch formula for virtual classes.
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。
In this paper, we establish the Riemann-Roch theorem in twisted K-theory extending our earlier results. As an application, we prove a twisted index formula and show that D-brane charges in Type I and Type II string theory are classified by twisted KO-theory and twisted K-theory respectively in the presence of B-fields as proposed by Witten.
Canonical pairing and Hirzebruch-Riemann-Roch formula for matrix factorizations
本文研究矩阵分解的典范配对，并借助扭曲 de Rham 值 Todd 类，在周期循环同调中建立 Hirzebruch-Riemann-Roch 公式。
Let $S$ be a finite dimensional noetherian scheme. For any proper morphism between smooth $S$-schemes, we prove a Riemann-Roch formula relating higher algebraic $K$-theory and motivic cohomology, thus with no projective hypothesis neither on the schemes nor on the morphism. We also prove, without projective assumptions, an arithmetic Riemann-Roch theorem involving Arakelov's higher $K$-theory and motivic cohomology as well as an analogue result for the relative cohomology of a morphism. These results are obtained as corollaries of a motivic statement that is valid for morphisms between oriente…
The purpose of this paper is to prove an equivariant Riemann-Roch theorem for schemes or algebraic spaces with an action of a linear algebraic group $G$. For a $G$-space $X$, this theorem gives an isomorphism between a completion of the equivariant Grothendieck group and a completion of equivariant equivariant Chow groups. The key to proving this isomorphism is a geometric description of completions of the equivariant Grothendieck group. Besides Riemann-Roch, this result has some purely $K$-theoretic applications. In particular, we prove a conjecture of Köck (in the case of regular schemes) an…
We show a Riemann-Roch theorem for group ring bundles over an arithmetic surface; this is expressed using the higher adeles of Beilinson-Parshin and the tame symbol via a theory of adelic equivariant Chow groups and Chern classes. The theorem is obtained by combining a group ring coefficient version of the local Riemann-Roch formula as in Kapranov-Vasserot with results on K-groups of group rings and an explicit description of group ring bundles over P^1. Our set-up provides an extension of several aspects of the classical Fr"ohlich theory of the Galois module structure of rings of integers of …
The goal of this series of papers is to give a new non-commutative approach to problems about the density of reductions such as the conjecture of Joshi-Rajan, and the generalization of the conjecture of Serre. In this paper, we prove integral Grothendieck Riemann-Roch which was proved by Papas in the case ch$(k)=0$. As a corollary we prove an integral analogue of Kontsevich's comparison theorem, and we show that if a smooth projective variety $X$ has a full exceptional collection then there is an explicit formula of the motive of $X$ up to bounded torsion.
Mapping a locally free module to its l-th tensor power gives rise to a natural map from the Grothendieck group of all locally free modules to the Grothendieck group of all locally free representations of the l-th symmetric group. In this paper, we prove some formulas of Riemann-Roch type for the behaviour of this tensor power operation with respect to the push-forward homomorphism associated with a projective morphism between schemes. We furthermore establish analogous formulas for higher K-groups.
Smooth and proper dg-algebras have an Euler class valued in the Hochschild homology of the algebra. This Euler class is worthy of this name since it satisfies many familiar properties including compatibility with the familiar pairing on the Hochschild homology of the algebra and that of its opposite. This compatibility is the Riemann-Roch theorems of Shklyarov and Petit. In this paper we prove a broad generalization of these Riemann-Roch theorems. We generalize from the bicategory of dg-algebras and their bimodules to monoidal bicategories and from Euler class to traces of non identity maps. O…
A Motivic Riemann-Roch Theorem for Deligne-Mumford Stacks
本文为光滑分离 Deligne-Mumford 栈发展了在 Voevodsky 三角化动机范畴中可表示的动形上同调理论，证明其高阶 Chow 群与高阶 $K$-理论典范同构，从而将 Grothendieck-Riemann-Roch 定理推广至该范畴。
We prove the classical Riemann-Roch theorems for the Adams operations $\,ψ^j\,$ on $K$-theory: a statement with coefficients on $\mathbb{Z}[j^{-1}]$, that holds for arbitrary projective morphisms, as well as another one with integral coefficients, that is valid for closed immersions. In presence of rational coefficients, we also analyze the relation between the corresponding Riemann-Roch formula for one Adams operation and the analogous formula for the Chern character. To do so, we complete the elementary exposition of the work of Panin-Smirnov that was initiated by the first author in a previ…
We prove a localization formula in equivariant algebraic $K$-theory for an arbitrary complex algebraic group acting with finite stabilizer on a smooth algebraic space. This extends to non-diagonalizable groups the localization formulas H.A. Nielsen in equivariant $K$-theory of vector bundles and R.W. Thomason for higher $K$-theory of equivariant coherent sheaves. As an application we give a Riemann-Roch formula for quotients of smooth algebraic spaces by proper group actions. This formula extends previous work of B. Toen for stacks with quasi-projective moduli spaces and the authors for quotie…
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.
Deligne-Riemann-Roch and intersection bundles
本文为推进Deligne提出的拓展Grothendieck-Riemann-Roch定理余维1部分的研究计划，建立了取值于线丛的相对函子性相交理论，推广了Deligne与Elkik的相关构造。
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.
The Hirzebuch-Riemann-Roch (HRR) and Lefschetz type formulas for finite dimensional elementary algebras of finite global dimension are explicitly given. They have cohomological, homological, Hochschild cohomological and Hochschild homological four versions, and module, bimodule, module complex and bimodule complex four levels. For this, the dimension matrix of a bimodule (complex) and the trace matrix of a bimodule (complex) endomorphism are introduced. It is shown that Shklyarov pairing, Chern character and Hattori-Stallings trace can be concretely expressed by Cartan matrix, dimension vector…
This is the second of series of papers on the study of foliations in the setting of derived algebraic geometry based on the central notion of derived foliation. We introduce sheaf-like coefficients for derived foliations, called quasi-coherent crystals, and construct a certain sheaf of dg-algebras of differential operators along a given derived foliation, with the property that quasi-coherent crystals can be interpreted as modules over this sheaf of differential operators. We use this interpretation in order to introduce the notion of good filtrations on quasi-coherent crystals, and define the…
For an arbitrary proper DG algebra A (i.e. DG algebra with finite dimensional total cohomology) we introduce a pairing on the Hochschild homology of A and present an explicit formula for a Chern-type character of an arbitrary perfect A-module (the Chern characters take values in the Hochschild homology of A). The Hirzebruch-Riemann-Roch formula in this context expresses the Euler characteristic of the Hom-complex between two perfect A-modules in terms of the pairing of their Chern characters. We mention two examples of proper DG algebras and the HRR formulas for them. The first example is Ring…
A Riemann-Roch theorem for Frobenius quotients
该研究构造两类弗罗贝尼乌斯环作为光滑完全簇$K$-环和周环的模型，证明其满足黎曼-罗赫定理的组合类比，给出相关同构及应用。
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