首页/论文/2003.09457

arXiv 2003.094572020-03math.AG

Compactly supported $\mathbb{A}^{1}$-Euler characteristic and the Hochschild complex

Niny Arcila-Maya, Candace Bethea, Morgan Opie, Kirsten Wickelgren, Inna Zakharevich

备注：: 24 pages. Accepted for publication in Topology and its Applications

英文摘要：

We show the $\mathbb{A}^{1}$-Euler characteristic of a smooth, projective scheme over a characteristic $0$ field is represented by its Hochschild complex together with a canonical bilinear form, and give an exposition of the compactly supported $\mathbb{A}^{1}$-Euler characteristic $χ^{c}_{\mathbb{A}^{1}}: K_0(\mathbf{Var}_{k}) \to \text{GW}(k)$ from the Grothendieck group of varieties to the Grothendieck--Witt group of bilinear forms. We also provide example computations.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Microlocal Euler classes and Hochschild homology

We define the notion of a trace kernel on a manifold M. Roughly speaking, it is a sheaf on M x M for which the formalism of Hochschild homology applies. We associate a microlocal Euler class to such a kernel, a cohomology class with values in the relative dualizing complex of the cotangent bundle over M and we prove that this class is functorial with respect to the composition of kernels. This generalizes, unifies and simplifies various results of (relative) index theorems for constructible sheaves, D-modules and elliptic pairs.

2013-05-01math.AG

曲线模空间权二紧支撑欧拉示性数

The weight two compactly supported Euler characteristic of moduli spaces of curves

本文通过图复形与操作子方法推导出曲线模空间权二紧支撑等变欧拉示性数的生成函数公式。

2021-12-24math.AGmath.ATmath.QA

$\mathbb{A}^1$-欧拉特征数的低对称幂与分裂的托里纤维品种

$\mathbb{A}^1$-Euler Characteristic of Low Symmetric Powers and Split Toric Varieties

本文计算了低对称幂和分裂托里纤维品种的$\mathbb{A}^1$-欧拉特征数，并展示了其与Pajwani-Pál构造的幂结构预测的一致性。

2026-01-09math.AG

紧支撑的$\mathbb{A}^1$-欧拉特征数的对称幂细胞状变种

Compactly supported $\mathbb{A}^1$-Euler characteristics of symmetric powers of cellular varieties

University of Bristol（布里斯托大学）；Heilbronn Institute for Mathematical Research（希尔布龙数学研究所）；Universität Duisburg-Essen（杜伊斯堡-埃森大学）；Leibniz Universität Hannover（汉诺威莱布尼茨大学）

本文研究了满足特定幂结构的变种类$\mathrm{Sym}_k$，并计算了格拉姆斯曼流形和某些德尔皮佐表面的对称幂的$\mathbb{A}^1$-欧拉特征数。

2026-01-12math.AGmath.KT

对数光滑对的对称学的函子性

Functoriality of logarithmic Hochschild homology of log smooth pairs

Budapest University of Technology and Economics（布达佩斯技术与经济大学）；Leiden University（莱顿大学）；Virginia Tech（弗吉尼亚理工大学）

本文提出基于对数傅里叶-穆卡变换的对应理论方法，证明对数Hochschild同调在诱导变换下的函子性，并构建了对数对应双范畴及相关的对数Chern字符和欧拉配对。

2026-10-14math.AG

Riemann-Roch theorems in monoidal 2-categories

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…

2023-01-08math.AGmath.ATmath.CT

更多相关论文收起更多论文

曲线对称幂的二次 Euler 示性数

Quadratic Euler Characteristic of Symmetric Powers of Curves

本文借助 Levine–Raksit 动机 Gauss–Bonnet 定理计算非特征二域上光滑射影曲线对称幂的二次 Euler 示性数，并证明 Pajwani–Pál 幂结构在特征零下给出相应紧支撑 $\mathbb{A}^1$-Euler 示性数。

2024-06-17math.AG

Euler characteristic of the space of real multivariate irreducible polynomials

We compute the compactly supported Euler characteristic of the space of degree $d$ irreducible polynomials in $n$ variables with real coefficients and show that the values are given by the digits in the so-called balanced binary expansion of the number of variables $n$.

2020-11-10math.AGmath.ATmath.NT

Integration over complex manifolds via Hochschild homology

Given a holomorphic vector bundle $\cale$ on a connected compact complex manifold X, [FLS] construct a $\compl$-linear functional $I_{\cale}$ on $\hh{2n}{\compl}$. This is done by constructing a linear functional on the 0-th completed Hochschild homology $\choch{0}{(\dif(\cale))}$ of the sheaf of holomorphic differential operators on $\cale$ using topological quantum mechanics. They show that this functional is $\int_X$ if $\cale$ has non zero Euler characteristic. They conjecture that this functional is $\int_X$ for all $\cale$. A subsequent work [Ram] by the author proved that the linear fun…

2008-06-05math.AGmath.QA

关于 $\mathcal{H}_{g, n}$ 的权零紧支上同调

On the weight zero compactly supported cohomology of $\mathcal{H}_{g, n}$

本文研究超椭圆曲线模空间 $\mathcal{H}_{g,n}$ 的权零紧支上同调，通过可容许覆盖构造对偶复形并定义图复形，给出等变欧拉示性数的图求和公式，并推广至阿贝尔覆盖模空间。

2024-11-05math.AGmath.CO

Topological Hochschild homology and Zeta-values

Using work of Antieau and Bhatt-Morrow-Scholze, we define a filtration on topological Hochschild homology and its variants $TP$ and $TC^-$ of quasi-lci rings with bounded torsion, which recovers the BMS-filtration after $p$-adic completion. Then we compute the graded pieces of this filtration in terms of Hodge completed derived de Rham cohomology relative to the base ring $\mathbb{Z}$. We denote the cofiber of the canonical map from $\mathrm{gr}^{n}TC^-(-)$ to $\mathrm{gr}^{n}TP(-)$ by $LΩ^{<n}_{-/\mathbb{S}}[2n]$. Let $\mathcal{X}$ be a regular connected scheme of dimension $d$ proper over $\…

2021-07-09math.AGmath.NT

$\ell$-adic局部系统在$\mathcal{A}_n$上的欧拉特征

The Euler characteristic of $\ell$-adic local systems on $\mathcal{A}_n$

研究$\ell$-adic局部系统在$\mathcal{A}_n$上的欧拉特征，证明了与自守表示相关的$\mathrm{GSpin}$-valued伽罗瓦表示的存在性，并给出了紧支撑欧拉特征的显式公式。

2026-01-12math.AGmath.NTmath.RT

虚拟Hodge数的$\mathcal{M}_{g, n}(\mathbb{P}^r, d)$：稳定性和计算

Virtual Hodge numbers of $\mathcal{M}_{g, n}(\mathbb{P}^r, d)$: stability and calculations

Massachusetts Institute of Technology（麻省理工学院）；University of Cambridge（剑桥大学）

本文研究了模空间$\mathcal{M}_{g, n}(\mathbb{P}^r, d)$的动机不变量，并给出了其 Serre 特征式的公式，特别在 genus 一和二时确定了该特征式。

2026-01-26math.AGmath.CO