微分算子的上循环和莱夫谢茨数公式的集成
Integration of Cocycles and Lefschetz Number Formulae for Differential Operators
摘要：
设${\mathcal E}$是复流形$X$上的全纯向量丛，使得$\dim_{\mathbb C}X=n$ .给定形式全纯微分算子代数${\rm Diff}_n$的任何连续的基本Hochschild $2n$ -上圈$ψ_{2n}$，从${\mathcal E}$上的任何全纯微分算子${\mathcal D}$得到$2n$ -形式$f_{{\mathcal E},ψ_{2n}}(\mathcal D)$.我们应用我们以前的结果[J。2（2008），405-448; J. Noncommut. 3（2009），27-45]，以表明$\int_X f_{{\mathcal E},ψ_{2n}}({\mathcal D})$给出$\mathcal D$的莱夫谢茨数，直到独立于$X$和${\mathcal E}$的常数。此外，我们得到了一个“局部”结果，推广了上述声明。当$ψ_{2n}$是[Duke Math.J.127（2005），487-517]中的上循环时，我们得到了Engeli-Felder的莱夫谢茨数定理的一个新的证明和推广.我们还得到了关于B的一个类似的“局部”结果。复可平行流形上平凡向量丛微分算子的全纯非交换剩余的构造。这使我们能够给出B定义的$\mathcal D$的全纯非交换剩余的严格构造。当${\mathcal E}$是任意紧致复流形$X$上的任意向量丛时，本文证明了Shoikhet定理.我们的局部结果立即证明了[Geom. Funct. Anal. 11（2001），1096-1124]。
英文摘要：
Let ${\mathcal E}$ be a holomorphic vector bundle on a complex manifold $X$ such that $\dim_{\mathbb C}X=n$. Given any continuous, basic Hochschild $2n$-cocycle $ψ_{2n}$ of the algebra ${\rm Diff}_n$ of formal holomorphic differential operators, one obtains a $2n$-form $f_{{\mathcal E},ψ_{2n}}(\mathcal D)$ from any holomorphic differential operator ${\mathcal D}$ on ${\mathcal E}$. We apply our earlier results [J. Noncommut. Geom. 2 (2008), 405-448; J. Noncommut. Geom. 3 (2009), 27-45] to show that $\int_X f_{{\mathcal E},ψ_{2n}}({\mathcal D})$ gives the Lefschetz number of $\mathcal D$ upto a constant independent of $X$ and ${\mathcal E}$. In addition, we obtain a "local" result generalizing the above statement. When $ψ_{2n}$ is the cocycle from [Duke Math. J. 127 (2005), 487-517], we obtain a new proof as well as a generalization of the Lefschetz number theorem of Engeli-Felder. We also obtain an analogous "local" result pertaining to B. Shoikhet's construction of the holomorphic noncommutative residue of a differential operator for trivial vector bundles on complex parallelizable manifolds. This enables us to give a rigorous construction of the holomorphic noncommutative residue of $\mathcal D$ defined by B. Shoikhet when ${\mathcal E}$ is an arbitrary vector bundle on an arbitrary compact complex manifold $X$. Our local result immediately yields a proof of a generalization of Conjecture 3.3 of [Geom. Funct. Anal. 11 (2001), 1096-1124].
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。
