Alberto Navarro
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.
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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 Singular Riemann-Roch Theorem
复线性约化代数群作用下复簇的等变Riemann-Roch定理。
Riemann-Roch for Deligne-Mumford stacks
利用等变K-理论中的等变Riemann-Roch定理和局部化定理，结合Artin环的一些基本交换代数，给出Deligne-Mumford栈的Riemann-Roch定理的一个简单证明.
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…
K-theory and cohomology of algebraic stacks: Riemann-Roch theorems, D-modules and GAGA theorems
这是我论文的组成部分。第一部分是“Deligne-Mumford栈的Riemann-Roch定理”的扩展版本，其中我处理一般基上的Artin栈。第二部分证明了Deligne-Mumford栈上D-模的一些Riemann-Roch定理，并考虑了解析栈的代数化问题。
Algebraic K-theory, A^1-homotopy and Riemann-Roch theorems
在这篇文章中，我们表明，在SGA 6和A^1-同伦理论的结合所做的建设，自然会导致更高的代数K理论的结果。这适用于代数K理论、陈特征标和Riemann-Roch定理的运算。
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 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 $GK^\bullet (X)\otimes \mathbb{Q}$.
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。
Riemann-Roch for equivariant Chow groups
本文证明了具有线性代数群$G$作用的概型或代数空间的等变Riemann-Roch定理。对于$G$ -空间$X$，该定理给出了等变Grothendieck群的完备化与等变等变Chow群的完备化之间的同构.证明这个同构的关键是等变Grothendieck群的完备化的几何描述。除了Riemann-Roch，这个结果还有一些纯$K$理论的应用。特别是，我们证明了一个猜想Köck（在情况下的定期计划）和扩展到任意特征的结果Segal表示环。
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.
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。
Nonabelian localization in equivariant K-theory and Riemann-Roch for quotients
证明了光滑代数空间上任意复代数群作用于有限稳定子的等变代数$K$ -理论中的一个局部化公式。这将局部化公式H. A. Nielsen在等变$K$ -向量丛理论和R.W.高阶$K$的基础-等变相干层理论。作为应用，我们给出了光滑代数空间中由真群作用生成的Riemann-Roch公式.这个公式推广了B. Toen关于具有拟投射模空间的栈和作者关于可对角化群的等价物。
Lefschetz and Hirzebruch-Riemann-Roch formulas via noncommutative motives
V. Lunts最近建立了Fourier-Mukai函子和dg代数的Lefschetz不动点定理。同样，D。Shklyarov介绍了Hirzebruch-Riemann-Roch定理的非交换类似物。在这篇简短的文章中，我们看到这些构造和计算是如何从它们的动机对应物中正式起源的。
Grothendieck-Riemann-Roch for derived schemes
在光滑基上导出格式的同伦范畴上定义了双变代数K理论和双变导出Chow。后者的方向对应于虚拟Gysin同态。然后，我们提供了这两个双变理论之间的态射，并比较两个方向。这种比较然后产生一个同调和上同调的Grothendieck-Riemann-Roch公式的虚类。
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。
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…
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.
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 …
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…
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…
Riemann-Roch via deformation quantization, II
证明了复流形形变量子化的Riemann-Roch公式及其推论，Schapira和Schneiders关于椭圆对的指标定理.
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。
Riemann-Roch for tensor powers
将一个局部自由模映射到它的第l个张量幂上，会产生一个从所有局部自由模的Grothendieck群到第l个对称群的所有局部自由表示的Grothendieck群的自然映射。本文证明了这种张量幂运算关于与格式间射影态射相关的前推同态的Riemann-Roch型性质。我们还建立了高阶K-群的类似公式。
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…
Hirzebruch-Riemann-Roch theorem for DG algebras
对任意的真DG代数A（即具有有限维全上同调的DG代数），在A的霍赫希尔德同调上引入一个配对，并给出了任意完备A-模的Chern型特征标的一个显式公式（Chern型特征标取值于A的霍赫希尔德同调中）. Hirzebruch-Riemann-Roch公式在此上下文中表示两个完全A-模之间的Hom-复形的欧拉示性数，表示为它们的陈特征标的配对。给出了真DG代数的两个例子及其HRR公式。第一个例子是林格尔的公式与关系的箭袋。第二个例子与V/G形式的轨道奇点有关，其中V是复向量空间，G是SL（V）的有限子群。进一步证明了当DG代数是光滑的时，霍赫希尔德同调上的上述配对是非退化的。我们还提出了一个猜想：对于Calabi-Yau DG代数A，它的配对与拓扑域论中的配对一致，并在Frobenius代数的情况下验证了这个猜想.
A condensed proof of the differential Grothendieck-Riemann-Roch theorem
本文给出了Freed-Lott微分解析指标定义良好的一个直接证明和微分Grothendieck-Riemann-Roch定理的一个简明证明。作为副产品，我们还得到了一个直接的证明，R/Z解析指数是定义良好的和一个压缩证明的R/Z Grothendieck-Riemann-Roch定理。
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 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。
Riemann-Roch Theorem and Index Theorem in Non-commutative Geometry
在这堂课中，我们回顾了黎曼-罗克定理在经典函数论、代数拓扑学、伪微分算子理论以及非交换几何中的认识。我们还证明了它在李群上量子化和调和分析的许多问题中的有用性。
Riemann-Roch and index formulae in twisted K-theory
本文建立了扭曲K理论中的Riemann-Roch定理，推广了我们以前的结果。作为应用，我们证明了一个扭曲指数公式，并证明了在Witten提出的B-场存在的情况下，I型弦理论和II型弦理论中的D-膜电荷分别可以用扭曲KO-理论和扭曲K理论进行分类.
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.
Localization of the Riemann-Roch character
给出了Meinrenken和Tian-Zhang证明的关于几何量子化和辛约化的交换性的Guillemin-Sternberg猜想的K-理论方法。除了提供了一个新的证明，这一猜想的充分非阿贝尔群行动的情况下，我们的方法导致了一个推广的紧凑李群行动的流形上，不是辛的。相反，这些流形携带不变的殆复结构和抽象的矩映射。
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…
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