Riemann-Roch for quotients and Todd classes of simplicial toric varieties

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 relative Riemann-Roch theorem from Hochschild homology

This write up attempts to clarify a preprint by Markarian [2] which proves This paper attempts to clarify a preprint of Markarian [2]. The preprint by Markarian [2] proves the relative Riemann-Roch theorem using a result describing how the HKR map fails to respect comultiplication. This paper elaborates on the core computations in [2]. These computations show that the HKR map twisted by the square root of the Todd genus "almost preserves" the Mukai pairing. This settles a part of a conjecture of Caldararu[3]. The relative Riemann-Roch theorem follows from this and a result of Caldararu[4].

矩阵分解的典范配对与 Hirzebruch-Riemann-Roch 公式

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。

A generalized Hirzebruch Riemann-Roch theorem

This short note proves a generalization of the Hirzebruch Riemann-Roch theorem equivalent to the Cardy condition described in [1]. This is done using an earlier result [4] that explicitly describes what the Mukai pairing in [1] descends to in Hodge cohomology via the Hochschild-Kostant-Rosenberg map twisted by the root Todd genus.

几乎完美障碍理论的虚拟黎曼-罗赫定理

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。

Equivariant Grothendieck-Riemann-Roch theorem via formal deformation theory

We use the formalism of traces in higher categories to prove a common generalization of the holomorphic Atiyah-Bott fixed point formula and the Grothendieck-Riemann-Roch theorem. The proof is quite different from the original one proposed by Grothendieck et al.: it relies on the interplay between self dualities of quasi- and ind- coherent sheaves on $X$ and formal deformation theory of Gaitsgory-Rozenblyum. In particular, we give a description of the Todd class in terms of the difference of two formal group structures on the derived loop scheme $\mathcal LX$. The equivariant case is reduced to…

格罗滕迪克对希策布鲁赫-黎曼-罗赫定理的证明

Grothendieck's proof of Hirzebruch-Riemann-Roch theorem

本文针对代数闭域上光滑射影簇，自洽介绍并完成格罗滕迪克框架下希策布鲁赫-黎曼-罗赫定理的完整证明，验证其可还原经典情形。

关于一般组合团的切丛与除理理论

On the tangent bundle and the divisor theory of a general matroid

本文研究一般组合团的切丛与除理理论，构造K类T_M并推导其总示性类公式，引入假有效锥以表征大和拟正除子，定义β_S类以提供可计算的组合学拟正除子。

Hirzebruch classes and motivic Chern classes for singular (complex) algebraic varieties

In this paper we study some new theories of characteristic homology classes for singular complex algebraic varieties. First we introduce a natural transformation T_{y}: K_{0}(var/X) -> H_{*}(X,Q)[y] commuting with proper pushdown, which generalizes the corresponding Hirzebruch characteristic. Here K_{0}(var/X) is the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration. T_{y} is a homology class version of the motivic measure corresponding to a suitable specialization of the well known Hodge poly…

Characteristic classes of singular toric varieties

In this paper we compute the motivic Chern classes and homology Hirzebruch characteristic classes of (possibly singular) toric varieties, which in the complete case fit nicely with a generalized Hirzebruch-Riemann-Roch theorem. As special cases, we obtain new (or recover well-known) formulae for the Baum-Fulton-MacPherson Todd (or MacPherson-Chern) classes of toric varieties, as well as for the Thom-Milnor L-classes of simplicial projective toric varieties. We present two different perspectives for the computation of these characteristic classes of toric varieties. First, we take advantage of …

Genera of algebraic varieties and counting of lattice points

This paper announces results on the behavior of some important algebraic and topological invariants --- Euler characteristic, arithmetic genus, and their intersection homology analogues; the signature, etc. --- and their associated characteristic classes, under morphisms of projective algebraic varieties. The formulas obtained relate global invariants to singularities of general complex algebraic (or analytic) maps. These results, new even for complex manifolds, are applied to obtain a version of Grothendieck-Riemann-Roch, a calculation of Todd classes of toric varieties, and an explicit formu…

弗罗贝尼乌斯商的黎曼-罗赫定理

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。

等变环面几何与Euler-Maclaurin公式

Equivariant toric geometry and Euler-Maclaurin formulae

本文研究了拟射影环面簇的等变motivic Chern类和Hirzebruch特征类，通过等变K-理论和局部化方法推广了多个经典公式，并基于motivic特征类方法给出了全维格多面体的Euler-Maclaurin型公式的代数几何证明及其加权与局部推广。

Motivic Bivariant Characteristic Classes

Let K_0(V/X) be the relative Grothendieck group of varieties over X in obj(V), with V the category of (quasi-projective) algebraic (resp. compact complex analytic) varieties over a base field k. Then we constructed the motivic Hirzebruch class transformation in the algebraic context for k of characteristic zero and in the compact complex analytic context. It unifies the well-known three characteristic class transformations of singular varieties: MacPherson's Chern class, Baum-Fulton-MacPherson's Todd class and the L-class of Goresky-MacPherson and Cappell-Shaneson. In this paper we construct a…

等变环面几何与欧拉-麦克劳林公式——综述

Equivariant toric geometry and Euler-Maclaurin formulae -- an overview

本文综述了射影环面簇上等变 motivic Chern 类与 Hirzebruch 示性类的最新进展，利用局部化和等变 Hirzebruch-Riemann-Roch 形式体系建立加权及一般凝聚层系数的欧拉-麦克劳林型公式，并推广到面的内部和 Minkowski 加项。