形式变形及其范畴一般纤维

On étale fundamental groups of formal fibres of $p$-adic curves

We investigate a certain class of (geometric) finite (Galois) coverings of formal fibres of $p$-adic curves and the corresponding quotient of the (geometric) étale fundamental group. A key result in our investigation is that these (Galois) coverings can be compactified to finite (Galois) coverings of proper $p$-adic curves. We also prove that the maximal prime-to-$p$ quotient of the geometric étale fundamental group of a (geometrically connected) formal fibre of a $p$-adic curve is (pro-)prime-to-$p$ free of finite computable rank.

范畴沿子范畴的形式完备化

Formal completion of a category along a subcategory

根据Kontsevich的思想，我们引入并研究了紧生成（由一组对象）增强的三角范畴沿着一个完整的厚本质上小的三角子范畴的形式完成的概念。特别是，我们证明（回答一个问题的Kontsevich），使用范畴的正式完成，可以得到普通的正式完成诺特计划沿闭子概形。此外，我们表明，Beilinson-Parshin adeles也可以得到使用范畴形式完成。

Categorical formal punctured neighborhood of infinity, I

In this paper we introduce and study the formal punctured neighborhood of infinity, both in the algebro-geometric and in the DG categorical frameworks. For a smooth algebraic variety $X$ over a field of characteristic zero, one can take its smooth compactification $\bar{X}\supset X,$ and then take the DG category of perfect complexes on the formal punctured neighborhood of the infinity locus $\bar{X}-X.$ The result turns out to be independent of $\bar{X}$ (up to a quasi-equivalence) and we denote this DG category by $\operatorname{Perf}(\hat{X}_{\infty}).$ We show that this construction can be…

Fukaya A-infinity structure near infinity and the categorical formal completion

For a stopped Liouville manifold arising from a Liouville sector, we construct a symplectic analogue of the formal neighborhood of the stop on the level of Fukaya categories. This geometric construction is performed via Floer-theoretic methods by allowing wrappings in the negative direction. On the other hand, inspired by homological mirror symmetry for pairs, where the mirror is the formal neighborhood of a divisor in an ambient projective variety, there is a different approach by taking a `categorical formal completion' introduced by Efimov. Our main results establishes equivalence of these …

Categorical proof of Holomorphic Atiyah-Bott formula

Given a symmetric monoidal $(\infty,2)$-category $\mathscr E$ we promote the trace construction to a functor. We then apply this formalism to the case when $\mathscr{E}$ is the $(\infty,2)$-category of $k$-linear presentable categories which in combination of various calculations in the setting of derived algebraic geometry gives a categorical proof of the classical Atiyah-Bott formula (also known as the Holomorphic Lefschetz fixed point formula).

（更高）范畴迹的应用 II：Deligne-Lusztig理论

Applications of (higher) categorical trace II: Deligne-Lusztig theory

利用AGCat范畴理论和更高范畴迹操作，重新推导Deligne-Lusztig理论中的若干关键结果。