Reduction theory for connections over the formal punctured disc

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…

关于穿孔对数映射粘合的评注

Remarks on gluing punctured logarithmic maps

本文研究穿孔稳定对数映射的粘合形式体系，在良好情形下建立对数 Gromov–Witten 不变量粘合公式，并将其应用于 Gross–Siebert 的 K3 曲面典范墙结构。

Derived Geometry and Non-Linear Differential Equations on the Punctured Disc

We study non-linear differential equations on the punctured formal disc by considering the natural derived enhancements of their spaces of solutions. In particular, by appealing to results of the inverse theory in the calculus of variations, we show that a variational formulation of a differential equation is \emph{equivalent} to the residue pairing inducing a (-1)-symplectic form on the derived space of solutions equipped with a certain decoration of its tangent complex.

Calculation of local formal Mellin transforms

Much recent work has been done on the local Fourier transforms for connections on the punctured formal disk. Specifically, the local Fourier transforms have been introduced, shown to induce certain equivalences of categories, and explicit formulas have been found to calculate them. In this paper, we prove analogous results in a similar situation, the local Mellin transforms for connections on the punctured formal disk. Specifically, we introduce the local Mellin transforms and show that they induce equivalences between certain categories of vector spaces with connection and vector spaces with …

Calculation of local formal Fourier transforms

We calculate the local Fourier transforms for connections on the formal punctured disk, corroborating the results of J. Fang and C. Sabbah using a different method. Our method is similar to Fang's, but more direct.

A Relative Lubin-Tate Theorem via Meromorphic Formal Geometry

We formulate a theory of punctured affine formal schemes, suitable for certain problems within algebraic topology. As an application, we show that the Morava K-theoretic localizations of Morava E-theory corepresent a version of the Lubin-Tate moduli problem in this framework.

伴随、环绕与无穷远处的态射

Adjoints, wrapping, and morphisms at infinity

本文证明光滑真范畴局部化中无穷远邻域的态射可由局部化函子左右伴随间的自然锥计算，从而统一恢复Ganatra–Gao–Venkatesh关于环绕Fukaya范畴的Rabinowitz环绕结果。

Any flat bundle on a punctured disc has an oper structure

We prove that any flat G-bundle, where G is a complex connected reductive algebraic group, on the punctured disc admits the structure of an oper. This result is important in the local geometric Langlands correspondence proposed in arXiv:math/0508382. Our proof uses certain deformations of the affine Springer fibers which could be of independent interest. As a byproduct, we construct representations of affine Weyl groups on the homology of these deformations generalizing representations constructed by Lusztig.

形式 torsor 的模空间

Moduli of formal torsors

本文构造了特征 p > 0 下形式穿孔圆盘上由 p-群与驯顺循环群半直积给出的有限群 torsor 的模叠，并证明其为具有有限万有单射转移映射的分离 Deligne-Mumford 叠的极限。

A geometric analogue of a conjecture of Gross and Reeder

Let G be a simple complex algebraic group. We prove that the irregularity of the adjoint connection of an irregular flat G-bundle on the formal punctured disk is always greater than or equal to the rank of G. This can be considered as a geometric analogue of a conjecture of Gross and Reeder. We will also show that the irregular connections with minimum adjoint irregularity are precisely the (formal) Frenkel-Gross connections.

Balanced embedding of degenerating Abelian varieties

For a certain maximal unipotent family of Abelian varieties over the punctured disc, we show that after a base change, one can complete the family over a disc such that the whole degeneration can be simultaneously balanced embedded into a projective space by the theta functions. Then we study the relationship between the balanced filling-in and the Gromov-Hausdorff limit of flat Kähler metrics on the nearby fibers.

Log intermediate Jacobians

We study the degenerations of intermediate Jacobians by means of log geometry. We extend the family of intermediate Jacobians over a punctured disc to a "log intermediate Jacobian" over a disc.

Local monodromy in non-archimedean analytic geometry -- fifth release

We study the topology of the punctured disc defined over a non-archimedean field of characteristic zero. Chapter two includes a new proof of the so-called p-adic Riemann existence theorem. This release completes the study of breaks and break decompositions of the monodromy representation of a sheaf around the origin of the punctured disc.

Formal gluing along non-linear flags

In this paper we prove formal glueing along an arbitrary closed substack $Z$ of an arbitrary Artin stack $X$ (locally of finite type over a field $k$), for the stacks of (almost) perfect complexes , and of $G$-bundles on $X$ (for $G$ a smooth affine algebraic $k$-group scheme). By iterating this result, we get a decomposition of these stacks along an arbitrary nonlinear flag of closed substacks in $X$. By taking points over the base field, we deduce from this both a formal glueing, and a flag-related decomposition formula for the corresponding symmetric monoidal derived $\infty$-categories of …