Riemann-Roch, Stability and New Non-Abelian Zeta Functions for Number Fields
Lin WENG
本文引入了数域上的一个几何形式化的算术上同调。基于这样一个上同调，我们定义和研究新的真正的非阿贝尔zeta函数的数域，使用交叉稳定性。
In this paper, we introduce a geometrically stylized arithmetic cohomology for number fields. Based on such a cohomology, we define and study new yet genuine non-abelian zeta functions for number fields, using an intersection stability.
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继续探索
This is the first of a series of papers. Our final goal is to establish Deligne-Riemann-Roch isomorphisms in various settings. In this paper, we establish a uniqueness theorem for Deligne pairings and prove the degree $1$ part of the Deligne-Riemann-Roch theorem.
Riemann-Roch Theorems for Deligne-Mumford Stacks
本文的目的是证明Deligne-Mumford代数栈的Riemann-Roch型定理。为此，我们引入了一个“上同调系数表示”和陈特征，我们证明了一个Grothendieck-Riemann-Roch定理的Riemann-Roch变换定义。作为推论，我们得到了凝聚层的欧拉特征线的Hirzebruch-Riemann-Roch公式，以及复代数栈的不同拓扑欧拉示性数的公式.
Riemann-Roch For Equivariant K-Theory
本文的目的是证明具有线性代数群作用的簇的高等等变K-理论的Riemann-Roch同构。
Explicit Orbifold Riemann-Roch for quasismooth varieties
考虑拟光滑变元作为整体$\CC^*$商，通过商栈的一般Riemann-Roch公式，给出了一个Riemann-Roch公式.进一步，我们给出了具有轨道曲线和不同点的极化拟光滑投射Gorenstein代数簇的Hilbert级数的一个parcing公式，它是\cite{BRZ}中结果的推广.
Orbifold Riemann-Roch for threefolds with an application to Calabi-Yau geometry
本文证明了一个关于极化三重（X，D）的轨道Riemann-Roch公式.作为应用，我们构造了新的投射Calabi-Yau三重簇.
We establish a Hirzebruch-Riemann-Roch type theorem and Grothendieck-Riemann-Roch type theorem for matrix factorizations on quotient Deligne-Mumford stacks. For this we first construct a Hochschild-Kostant-Rosenberg type isomorphism explicit enough to yield a categorical Chern character formula. We next find an expression of the canonical pairing of Shklyarov under the isomorphism.
The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem
我们发展了一个形式主义，涉及Atiyah类层光滑流形上，Hochschild链和上链复形。作为应用，我们证明了一个版本的Riemann-Roch定理。
Grothendieck-Riemann-Roch and the moduli of Enriques surfaces
我们给出了Borcherds定理Enriques曲面的模空间是拟仿射的一个简短而经典的证明。在我们的证明中，Borcherds乘积的使用被Grothendieck-Riemann-Roch定理的应用所取代。
Riemann-Roch Theorems via deformation quanitzation I
从辛变形量子化的周期循环上圈的Riemann-Roch型定理出发，利用伴随符号复形的Chern特征标和流形的Todd类，导出了表示D-模的完全复形的微局部Euler类的Riemann-Roch型公式.后者的证明载于本文的续集。
We explain how the Riemann-Roch theorem for divisors on an abelian variety $A$ is related to the reduced norms of the Wedderburn components of $\operatorname{End}^0(A)$ the $\mathbb{Q}$-endomorphism algebra of $A$. We then describe consequences and examples.
A Lefschetz-Riemann-Roch theorem for singular schemes
本文提出一个适用于奇异方案的Lefschetz-Riemann-Roch定理，扩展了之前针对奇异流形的结果。
A Deligne-Riemann-Roch isomorphism I: Preliminaries on virtual categories
这是即将发表的一系列论文的第一篇。他们已经出现了通过试图回答开放的问题德利涅提出的“乐决定德拉cohomologie”，Contemp。数学67（1987）。它相当于Grothendieck-Riemann-Roch定理的函子化和度量化版本，以及Lefschetz-Riemann-Roch公式的形式，参见。“Lefschetz-Riemann-Roch定理和相干迹公式”Ann. Sci. EC.诺姆辅助核算18（1985），Theorem 3.5.在这篇文章中，我们对待各种初步结果的虚拟范畴，这是范畴的Deligne-Riemann-Roch定理最初制定。最后，我们比较了我们的结构的Franke对陈省身交函子和Chow范畴的建设。
On the Adams-Riemann-Roch theorem in positive characteristic
本文给出了光滑射影态射$X\to Y$的Adams-Riemann-Roch定理的一个新的证明，其中$Y$是正则概型，它在$\mF_p$上是拟投射的.我们还部分地回答了B的一个问题.科克。
We show that the space of theta functions on tropical tori is identified with a convex polyhedron. We also show a Riemann-Roch inequality for tropical abelian surfaces by calculating the self-intersection numbers of divisors.
On Deligne's functorial Riemann-Roch theorem in positive characteristic
本文根据Pink和Rössler关于Adams-Riemann-Roch正特征定理的证明（见\cite{Pi}）中的思想，给出了函子Deligne-Riemann-Roch正特征定理的一个变体的证明. \cite{Pi}中给出的证明方法与经典的证明方法完全不同，它对任何正特征都有效，这将使我们更容易、更直接地证明函子Deligne-Riemann-Roch定理。我们的证明也部分兼容芒福德的同构。
On the Riemann-Roch formula: old and new
本文通过回答Matthew Baker和Serguei Norine于2007年提出的问题，提出了黎曼-罗奇公式证明的新方法。
Representations of finite groups on Riemann-Roch spaces, II
设G是射影曲线X的自同构群的有限子群，D是X上被G稳定的因子，在D非特殊的假设下，我们计算了G的自然表示在Riemann-Roch空间L（D）上的迹的一个简化公式.
We produce a Grothendieck transformation from bivariant operational $K$-theory to Chow, with a Riemann-Roch formula that generalizes classical Grothendieck-Verdier-Riemann-Roch. We also produce Grothendieck transformations and Riemann-Roch formulas that generalize the classical Adams-Riemann-Roch and equivariant localization theorems. As applications, we exhibit a projective toric variety $X$ whose equivariant $K$-theory of vector bundles does not surject onto its ordinary $K$-theory, and describe the operational $K$-theory of spherical varieties in terms of fixed-point data. In an appendix, V…
Integral Grothendieck-Riemann-Roch theorem
我们证明，在特征零点，明显的积分版本的Grothendieck-Riemann-Roch公式获得通过清除的Todd和Chern特征标的分解子是真实的（而不必划分周群由他们的扭子群）。证明介绍了一种替代Grothendieck的策略：我们使用奇点的分解和弱分解定理的双有理映射。
For a divisor $D$ on a tropical variety $X$, we define two amounts in order to estimate the value of $h^{0}(X,D)$, which are described by terms of global sections and computed more easily than $h^{0}(X,D)$. As an application of its estimation, we show that a Riemann-Roch inequality holds for smooth tropical toric surfaces.
Bases for Riemann--Roch spaces of linearized function fields with applications to generalized algebraic geometry codes
该研究通过给出线性化函数域 Riemann–Roch 空间的显式基，建立广义代数几何码最小距离下界，并据此构造参数良好的编码。
We introduce the most general to date version of the permutation-equivariant quantum K-theory, and express its total descendant potential in terms of cohomological Gromov-Witten invariants. This is the higher-genus analogue of adelic characterization from the paper [7] by Givental-Tonita, and is based on the application of Kawasaki's Riemann-Roch formula to moduli spaces of stable maps.
Orbifold Hirzebruch-Riemann-Roch
该研究针对商Deligne-Mumford栈的轨形HRR定理展开，分析其与有限群表示论的联系，借助轨形Mukai配对得到了新的轨形HRR公式，丰富了代数几何领域的相关理论。
A Riemann-Roch formula for traces of algebraic differential operators
得到了代数微分算子的一个迹公式，相应的解析结果已由M. Engeli和G. Felder
We introduce K-theoretic Gromov-Witten invariants of algebraic orbifold target spaces. Using the methods developed by Givental-Tonita we characterize Giventals Lagrangian cone of quantum K theory of orbifolds in terms of the cohomological cone.
Riemann-Roch for quotients and Todd classes of simplicial toric varieties
本文给出了可对角化群的光滑概形的等价奇异格式的Riemann-Roch映射的一个显式公式。作为应用，我们得到了单纯复曲面簇的Todd类的一个公式的简单证明。一个等变版本的这个公式以前获得了完整的单纯复曲面品种的Brion和Vergne使用不同的技术。
Using the arithmetic Schottky uniformization theory, we show the arithmeticity of $PSL_{2}({\mathbb C})$ Chern-Simons invariant. In terms of this invariant, we give an explicit formula of the Riemann-Roch isomorphism as Zograf-Mcintyre-Takhtajan's infinite product for families of algebraic curves. By this formula, we determine the unknown constant which appears in the holomorphic factorization formula of determinant of Laplacians on Riemann surfaces via the classical Liouville action. As an application, we show the rationality of Ruelle zeta values for Schottky uniformized $3$-manifolds.
The relative Riemann-Roch theorem from Hochschild homology
本文试图澄清Markarian [2]的一个预印本，它证明了本文试图澄清Markarian [2]的一个预印本。Markarian [2]的预印本使用描述HKR映射如何不尊重余乘的结果证明了相对Riemann-Roch定理。本文详细阐述了[2]中的核心计算。这些计算表明，由Todd亏格的平方根扭曲的HKR映射“几乎保持”Mukai配对。这一点，也是李嘉诚的一部分。从这一点和Caldararu[4]的一个结果导出了相对的Riemann-Roch定理。
Suppose $\mathcal{S}$ is a smooth, proper, and tame Deligne-Mumford stack. Toën's Grothendieck-Riemann-Roch theorem requires correction terms, involving components of the inertia stack, to the standard formula for schemes. We give a brief overview of Toën's Grothendieck-Riemann-Roch theorem, and explicitly compute the correction terms in the case of an orbifold surface with stabilizers of types ADE.
Riemann-Roch theorems and elliptic genus for virtually smooth Schemes
对于一个具有固定1-完全阻塞理论的适当方案X，我们定义了全纯欧拉示性数、χ y-亏格和椭圆亏格的虚形式，它们是变形不变的，并且推广了光滑情形下的通常定义.我们证明虚拟版本的Grothendieck-Riemann-Roch和Hirzebruch-Riemann-Roch定理。我们证明了虚chi y-亏格是一个多项式，并利用它定义了一个虚拓扑欧拉示性数.我们证明了虚拟椭圆亏格满足Jacobi模块化属性，我们的状态和证明的局部化定理的复曲面等变的情况下。我们展示了我们的一些结果如何适用于稳定层的模空间。
Group representations on Riemann-Roch spaces of some Hurwitz curves
设q>1表示与2，3，7互素的互素整数，G=PSL（2，q）是定义在C上的光滑射影曲线X的Hurwitz群.我们计算了Riemann-Roch空间L（D）的G-模结构，其中D是X上正次数的不变因子.这取决于计算的分歧模块，我们明确给出。特别地，我们得到了H^1（X，C）作为G-模的分解。
Let $ \; G \; $ be a group acting on a compact Riemann surface $ \; {\mathcal X} \; $ and $ \; D \; $ be a $ \; G$-invariant divisor on $\; {\mathcal X}. \; $ The action of $ \; G \; $ on $ \; {\mathcal X} \; $ induces a linear representation $ \; L_G(D) \; $ of $ \; G \; $ on the Riemann-Roch space associated to $ \; D.$ In this paper we give some results on the decomposition of $ \; L_G(D) \; $ as sum of complex irreducible representations of $ \; G, \; $ for $ \; D \; $ an effective non-special $\; G$-invariant divisor. In particular, we give explicit formulae for the multiplicity of each c…
In this paper, we prove the functorial Riemann-Roch theorem in positive characteristic for a smooth and projective morphism with any relative dimension. In the case of relative dimension $1$, we have given an analogue with Deligne's functorial Riemann Roch theorem in previous author's paper. For any relative dimension, our result can deduce an analogue to the Knudsen-Mumford extension. The present result is a generalization, which mainly originated from the extended Deligne pairing by S. Zhang and the Adams Riemann Roch theorem in positive characteristic by R. Pink and D. Rössler.
Towards boundedness of minimal log discrepancies by Riemann--Roch theorem
利用Riemann-Roch定理讨论了定维最小对数偏差的有界性问题。在将其归结为Gorenstein终端奇点的情形后，首先证明了当重数或嵌入维数有界时，其最小对数偏差有界。其次，我们恢复的Gorenstein终端三重奇异里德的特性，其最小的日志差异的精确边界Markushevich，没有明确的分类。最后给出了一般超平面截面具有一个终端段的四重奇点的精确边界。
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.
Ice cream and orbifold Riemann-Roch
在X是射影Gorenstein的且只有孤立的轨道点的假设下，给出了拟光滑极化n重X，D的Hilbert级数的封闭形式的轨道Riemann-Roch公式.我们的公式是一个总和的部分，其中每一个是积分和Gorenstein对称的相同的典型重量; orbifold部分被称为“冰淇淋功能”。这种形式的希尔伯特级数对计算机代数特别有用，我们用K3曲面和卡-丘3-折叠的例子来说明它。这些结果也适用于更高维的orbifold地层（见[A。巴克利和B. Szendroi，Orbifold Riemann-Roch for 3-folds with an application to Calabi-Yau geometry，J. Algebraic Geometry 14（2005）601- 622]和[Shengtian Zhou，Orbifold Riemann-Roch and Hilbert series，University of Warwick PhD thesis，March 2011，91+vii pp.]，尽管正确的陈述相当棘手。我们希望在今后的出版物中再次讨论这一问题。
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。
An arithmetic Riemann-Roch theorem in higher degrees
我们证明了一个类似的Arakelov几何的Grothendieck-Riemann-Roch定理。
In this work we study the generalized Weierstrass semigroup $\widehat{H} (\mathbf{P}_m)$ at an $m$-tuple $\mathbf{P}_m = (P_{1}, \ldots , P_{m})$ of rational points on certain curves admitting a plane model of the form $f(y) = g(x)$ over $\mathbb{F}_{q}$, where ${f(T),g(T)\in \mathbb{F}_q[T]}$. In particular, we compute the generating set $\widehatΓ(\mathbf{P}_m)$ of $\widehat{H} (\mathbf{P}_m)$ and, as a consequence, we explicit a basis for Riemann-Roch spaces of divisors with support in $\{P_{1}, \ldots , P_{m}\}$ on these curves, generalizing results of Maharaj, Matthews, and Pirsic.
In this thesis we study toric rank functions for chip firing games and prove special cases of a conjectural Riemann-Roch. The original motivation for an investigation into this area of study came for the adaptation (due to Matt Baker) of Riemann-Roch into a graph theoretic analogue through the use of chip-firing games. Here, we collect known results and present new observations that indicate Riemann--Roch holds for trees and polygons. We also prove an asymptotic case of Riemann--Roch (i.e.~Riemann--Roch for divisors of large degree). Finally, we also provide magma code and computational eviden…
The Deligne-Riemann-Roch isomorphism
本文通过发展新的陈类范畴和相对交理论，确立了德利尼计划中关于精细格罗滕迪克-里曼-罗奇公式的几何部分，并展示了其在Knudsen-Mumford展开式和格里菲斯丛中的应用。
Riemann-Roch Polynomials of the known Hyperkähler Manifolds
本文为O'Grady的两个例外超凯勒流形计算线束欧拉示性数显式公式，并在附录中用于计算10维例子的陈数。
A Riemann-Roch theorem in tropical geometry
最近，Baker和Norine证明了有限图的Riemann-Roch定理。我们将他们的结果推广到度量图，从而建立了（抽象）热带曲线上因子的Riemann-Roch定理。
A Riemann-Roch theorem for flat bundles, with values in the algebraic Chern-Simons theory
设$f: X \to S$是代数闭域$k$上的平坦态射，其相对法交叉因子为$Y\subset X$，$(E, \nabla)$是一个丛，其对数极点沿$Y$连接，曲率值在$f^*Ω^2_{k(S)}$中.然后Gaueline-Manin层$R^if_*(Ω^*_{X/S}({\rm log} Y)\otimes E)$携带Gaueline-Manin连接$GM^i(\nabla)$ .建立了联系$\nabla$、$GM^i(\nabla)$和$Ω^1_{X/S}({\rm log}Y)$的顶Chern类的代数Chern-Simons不变量的Riemann-Roch公式.
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.
The Gauss map and a noncompact Riemann-Roch formula for constructible sheaves on semiabelian varieties
对于代数群G中的一个不可约子簇Z，我们定义一个非负整数gdeg（Z）作为Z的Gauss映射在一定意义下的次数.它可以看作是Z的余法丛与T^*G的零截面的交指标的替代，即使G可能是非紧的。对于半交换簇G（特别是代数环面（C^*）^n），我们证明了G上可构造层的Riemann-Roch型公式，其中包含了我们对交指标的替换.作为推论，我们得到了这样一个G上的反常层具有非负的欧拉示性数，推广了Loeser-Sabbah的一个定理.
A remark on an algebraic Riemann-Roch formula for flat bundles
本文给出了特征为零的常数族上的移动平坦丛的代数Riemann-Roch公式，其值在代数微分特征标环上。该公式推广了复数域上Chow群中的Grothendieck-Riemann-Roch公式和复数域上具有$\C/\Z$系数的奇维Betti上同调的Bismut-Lott公式.
Virtual Riemann-Roch Theorems for Almost Perfect Obstruction Theories
本文在几乎完美障碍理论框架下证明虚拟黎曼-罗赫定理，涵盖非等变与等变情形及余截面局部化版本，推广并简化了既有结果，为K-理论不变量提供坚实基础。
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.
L^2-Riemann-Roch Inequalities for Covering Manifolds
本文研究了Moishezon流形中Zebraki开集的Galois覆盖上不变线丛的$L^2$全纯截面的存在性.我们证明了$L^2$全纯截面空间的von Neuman维数在合理的曲率条件下是有界的。我们还给出了一个具有孤立奇点的紧致复空间和一些相关的强伪凹流形是Moishezon的判别准则。然后用同样的方法研究它们的覆盖物。作为应用，我们给出弱Lefschetz定理使用的Napier-Ramachandran证明的Nori定理。
The arithmetic Grothendieck-Riemann-Roch theorem for general projective morphisms
经典的算术Grothendieck-Riemann-Roch定理只适用于在复数上光滑的射影态射。本文将算术Grothendieck-Riemann-Roch定理推广到正则算术簇之间的一般射影态射的情形。为此，我们依靠作者发展的广义解析挠理论。
In this article, we consider regular projective arithmetic schemes in the context of Arakelov geometry, any of which is endowed with an action of the diagonalisable group scheme associated to a finite cyclic group and with an equivariant very ample invertible sheaf. For any equivariant morphism between such arithmetic schemes, which is smooth over the generic fibre, we define a direct image map between corresponding higher equivariant arithmetic K-groups and we discuss its transitivity property. Then we use the localization sequence of higher arithmetic K-groups and the higher arithmetic conce…
A Riemann-Roch theorem for edge-weighted graphs
证明了实上边权图上实因子的Riemann-Roch定理，推广了Baker和Norine关于重边图上整因子的结果.
Riemann-Roch theorems via deformation quantization
本文证明了P. Schapira和J. P. Schneiders关于D-模的特征类。
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.
The Riemann-Roch theorem in a singular setting
本文将 Grothendieck–Riemann–Roch 定理推广到奇异情形，证明其对奇数特征 Noether 分离概形间的任意真平坦态射成立。
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…
Riemann-Roch theory for weighted graphs and tropical curves
我们定义了一个因子理论的图形和热带曲线赋予一个权函数的顶点，我们证明了黎曼-罗克定理在这两种情况下。我们将Baker的特化引理推广到加权图。
Arithmetic Riemann-Roch and Hilbert-Samuel formulae for pointed stable curves
证明了点稳定曲线的算术Riemann-Roch和Hilbert-Samuel型公式。我们给出了应用程序的积分尖点形式格的体积为0的稳定曲线。
We introduce twisted permutation-equivariant GW-invariants, and compute them in terms of untwisted ones. The computation is based on Grothendieck-like RR formula corresponding to Adams' operations from K-theory to itself, and the result can be understood as a "quantum" version of such Adams-RR. As in the case of cohomological quantum RR theorem [3], the result is applied to express the invariants of bundle and super-bundle spaces in terms of those of the base. The bonus feature of permutation-equivariant K-theory is that the twisting classes can be understood as the simpler kappa-classes of Ka…
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