AI 中文总结
本文研究Ševera双复形，证明其谱序列第二页的微分在适当条件下赋予Batalin-Vilkovisky代数结构，并通过Cartan演算视角统一描述，推广了奇辛流形上的经典构造。
AI 中文摘要
Ševera双复形是一个双复形$(M,d_{0},d_{1})$，其微分$d_{i}$是关于$M$上给定的$A$-模结构的阶数不超过$i$的Grothendieck微分算子，其中$A$是一个分次交换代数。还要求$(M,d_{0})$具有一个收缩同伦$K$，它是与$d_{0}$同阶的微分算子。在适当的假设下，与$(M,d_{0},d_{1})$关联的谱序列第二页的微分$d_{2}$是$E_{2}$上的Batalin-Vilkovisky算子，因此当$E_{2}$是秩为1的自由$A$-模时，$A$的分次交换代数结构被增强为Gerstenhaber代数结构，并且选择$E_{2}$的一个$d_{2}$-闭基元素进一步将其增强为Batalin-Vilkovisky代数结构。这一构造的典型例子是Ševera对奇辛流形上光滑函数代数的Batalin-Vilkovisky代数结构的描述。在整篇文章中，我们通过Cartan演算的视角来审视Batalin-Vilkovisky代数。在结论部分简要讨论了向导出Cartan演算推广的迹象。
英文摘要
A Ševera bicomplex is a bicomplex $(M,d_{0},d_{1})$ whose differentials $d_{i}$ are Grothendieck differential operators of order $\leq i$ with respect to a given $A$-module structure on $M$, where $A$ is a graded commutative algebra. One also requires that $(M,d_{0})$ admits a contracting homotopy $K$ that is a differential operator of the same order as $d_{0}$. Under suitable assumptions, the differential $d_{2}$ of the second page of the spectral sequence associated with $(M,d_{0},d_{1})$ is a Batalin-Vilkovisky operator on $E_{2}$, so that when $E_{2}$ is a free rank 1 $A$-module, the graded commutative algebra structure of $A$ is enhanced to a Gerstenhaber algebra structure, and the choice of a $d_{2}$-closed basis element for $E_{2}$ further enhances this to a Batalin-Vilkovisky algebra structure. The prototypical example of this construction is Ševera's description of the Batalin--Vilkovisky algebra structure on the algebra of smooth functions of an odd symplectic manifold. Throughout the whole article we look at Batalin-Vilkovisky algebras through the lenses of Cartan calculus. Indications of a generalization to derived Cartan calculus are briefly discussed in the concluding section.
Comments30 pages
