AI 中文总结
本文证明光滑真微分Z/2-分次范畴的周期循环同调上的Getzler-Gauss-Manin联络具有正则奇点和拟幺单幺正单值，证实了KKP猜想，并通过p-曲率的乘法性质给出Jordan块大小的上界。
AI 中文摘要
设 $\mathcal{C}$ 是复数域 $\mathbb{C}$ 上的微分 $\mathbb{Z}/2$-分次范畴。其周期循环同调 $HH^{per}_*(\mathcal{C})$，当被视为形式穿孔圆盘上的向量丛时，配备了一个典范联络 $\nabla^{\mathcal{C}}_{\partial_t}$，称为 $t$ 方向上的Getzler-Gauss-Manin联络（或范畴 $t$-联络）。我们的主要结果是：当 $\mathcal{C}$ 光滑且真（proper）时，该联络在 $t=0$ 处具有正则奇点且拟幺单（quasi-unipotent）幺正单值，从而证实了Katzarkov-Kontsevich-Pantev的一个猜想 \cite{KKP}。我们的证明采用模 $p$ 约化论证，使用了Toën的展开（spreading out）技术 \cite{To} 和Katz的正则性判据 \cite{Ka1}。主要新颖之处在于通过双色Kontsevich-Soibelman operad的解释，证明了 $\nabla^{\mathcal{C}}_{\partial_t}$ 的 $p$-曲率的乘法性质。随后我们探讨了主要结果的两个应用。首先，我们给出（在额外假设下）光滑真 $d(\mathbb{Z}/2)$g 范畴的周期循环同调上的非交换Hodge滤过的显式描述，遵循Shklyarov的构造 \cite{Shk}。第二个应用，特别适用于我们特定的方法或证明，是 $\nabla^{\mathcal{C}}_{\partial_t}$ 的幺正单值的Jordan块大小的上界，这同时推广了Scherk关于孤立超曲面奇点的局部幺正单值定理 \cite{Sche} 以及（部分地）Pomerleano-Seidel关于闭单调辛流形的量子联络的近期结果 \cite{PS2}。作为一个特化，我们证明这些Jordan块的大小以 $\mathcal{C}$ 的对角维数加一为上界。
英文摘要
Let $\mathcal{C}$ be a differential $\mathbb{Z}/2$-graded category over $\mathbb{C}$. Its periodic cyclic homology $HH^{per}_*(\mathcal{C})$, when viewed as a vector bundle over the formal punctured disk, is equipped with a canonical connection $\nabla^{\mathcal{C}}_{\partial_t}$ called the Getzler-Gauss-Manin connection in the $t$-direction (or the categorical $t$-connection). Our main result is that when $\mathcal{C}$ is smooth and proper, this connection has a regular singularity at $t=0$ and quasi-unipotent monodromy, affirming a conjecture of Katzarkov-Kontsevich-Pantev \cite{KKP}. Our proof follows a reduction mod $p$ argument using a spreading out technique of Toën \cite{To} and a regularity criterion of Katz \cite{Ka1}. The main novelty is the proof of a multiplicative property of the $p$-curvature of $\nabla^{\mathcal{C}}_{\partial_t}$ through an interpretation in terms of the two-colored Kontsevich-Soibelman operad. We then explore two applications of the main result. First, we give an explicit description (under additional assumptions) of the non-commutative Hodge filtration on the periodic cyclic homology of a smooth proper d$(\mathbb{Z}/2)$g category, following a construction of Shklyarov \cite{Shk}. The second application, which is special to our particular method or proof, is an upper bound on the sizes of Jordan blocks of the monodromy of $\nabla^{\mathcal{C}}_{\partial_t}$, which simultaneously generalizes Scherk's local monodromy theorem for isolated hypersurface singularities \cite{Sche} and (partially) a recent result of Pomerleano-Seidel on the quantum connection of a closed monotone symplectic manifold \cite{PS2}. As a specialization, we show that the sizes of these Jordan blocks are bounded above by the diagonal dimension of $\mathcal{C}$ plus one.
Comments47 pages, 14 figures
