摘要翻译:
给定连通紧致复流形X上的全纯向量丛$\cale$,[FLS]在$\hh{2n}{\compl}$上构造$\compl$-线性泛函$i_{\cale}$。这是用拓扑量子力学在$\cale$上全纯微分算子簇的第0个完全Hochschild同调$\choch{0}{(\dif(\cale))}$上构造线性泛函完成的。它们表明,如果$\cale$具有非零欧拉特性,则该函数为$\int_x$。他们猜想这个函数是$\int_x$对于所有$\cale$。作者随后的工作[Ram]证明了线性泛函$i_{\cale}$与向量丛$\cale$无关。本文在[Ram]中的工作基础上,证明了任意连通紧致复流形X上的任意全纯向量丛$\cale$的$i_{\cale}=\int_x$。这是用一个从几何角度来看非常自然的参数完成的。这个参数使我们能够将[FLS]中的构造扩展到任意连通复流形Y上任意全纯向量丛$\cale$上$\text{H}^{2n}_{c}(Y,\compl)$的线性泛函$I_{\cale}$的构造,并证明$I_{\cale}=\int_y$。我们还推广了[Ram]关于$I_{\cale}$的“循环同调类似物”的一个结果。
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英文标题:
《Integration over complex manifolds via Hochschild homology》
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作者:
Ajay C. Ramadoss
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最新提交年份:
2008
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分类信息:
一级分类:Mathematics 数学
二级分类:Algebraic Geometry 代数几何
分类描述:Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇,叠,束,格式,模空间,复几何,量子上同调
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一级分类:Mathematics 数学
二级分类:Quantum Algebra 量子代数
分类描述:Quantum groups, skein theories, operadic and diagrammatic algebra, quantum field theory
量子群,skein理论,运算代数和图解代数,量子场论
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英文摘要:
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 functional $I_{\cale}$ is independent of the vector bundle $\cale$. This note builds upon the work in [Ram] to prove that $I_{\cale}=\int_X$ for an arbitrary holomorphic vector bundle $\cale$ on an arbitrary connected compact complex manifold X. This is done using an argument that is very natural from the geometric point of view. This argument enables us to extend the construction in [FLS] to a construction of a linear functional $I_{\cale}$ on $\text{H}^{2n}_{c}(Y,\compl)$ for an arbitrary holomorphic vector bundle $\cale$ on an arbitrary connected complex manifold Y and prove that $I_{\cale} = \int_Y$. We also generalize a result of [Ram] pertaining to "cyclic homology analogs" of $I_{\cale}$.
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PDF链接:
https://arxiv.org/pdf/0707.4528