C. Roucairol
We compute formal invariants associated with the cohomology sheaves of the direct image of holonomic D-modules of exponential type. We also prove that every formal C[[t]]<\partial_t>-modules is isomorphic, after a ramification, to a germ of formalized direct image of analytic D-module of exponential type.
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Let $X$ be a complex analytic curve. In this paper we prove that the subanalytic sheaf of tempered holomorphic solutions of $\mathcal D_X$-modules induces a fully faithful functor on a subcategory of germs of formal holonomic $\mathcal D_X$-modules. Further, given a germ $\mathcal M$ of holonomic $\mathcal D_X$-module, we obtain some results linking the subanalytic sheaf of tempered solutions of $\mathcal M$ and the classical formal and analytic invariants of $\mathcal M$.
In this expository paper, we give a complete proof of van den Essen's theorem that the de Rham cohomology spaces of a holonomic D-module are finite-dimensional in the case of a formal power series ring over a field of characteristic zero. This proof requires results from at least five of van den Essen's papers as well as his unpublished thesis, and until now has not been available in a self-contained document.
We explain a formalism of regular holonomic $D$-modules for algebraic geometers using the distinguished triangles associated with algebraic local cohomology together with meromorphic Deligne extensions of local systems as well as the dual functor.
We present an algorithm to construct a basis of k-th extension group of a D-module M in ring of the formal power series Ext_D^k(M,O).
Fourier transforms of irregular holonomic D-modules, singularities at infinity of meromorphic functions and irregular characteristic cycles
本文利用不规则Riemann-Hilbert对应和Fourier-Sato变换研究不规则全纯D-模的傅里叶变换，借助亚纯消失循环函子精确分析奇点，并用不规则特征循环刻画高维情形下由指数因子不定点处线性扰动引起的额外秩跳跃。
We describe the category of regular holonomic modules over the ring D[[h]] of linear differential operators with a formal parameter h. In particular, we establish the Riemann-Hilbert correspondence and discuss the additional t-structure related to h-torsion.
We introduce the notion of a holonomic D-module on a smooth (idealized) logarithmic scheme and show that Verdier duality can be extended to this context. In contrast to the classical case, the pushforward of a holonomic module along an open immersion is in general not holonomic. We introduce a "perverse" t-structure on the category of coherent logarithmic D-modules which makes the dualizing functor t-exact on holonomic modules. This allows us to transfer some of the formalism from the classical setting and in particular show that every holonomic module on an open subscheme can be extended to a…
Holonomic D-cap-modules on rigid analytic spaces
本文定义了刚性解析空间上的全微分D-cap模，并证明了其在五种操作下的稳定性，同时探讨了其与常规六 functor 形式的关系。
A Morse theoretical approach to Fourier transforms of holonomic D-modules in dimension one
本文用扭曲Morse理论描述一维全纯D-模傅里叶变换的增强解复合体，定义非正则特征环，并给出经典特征环公式与全纯解自然基的构造。
Three results on holonomic D-modules
本文通过局部方法研究了范霍诺尼克D-模的三个关键问题，包括de Rham复形的欧拉特征不变性、局部一般消失定理以及Stokes过滤构造可析sheaf的拉普拉斯变换。
Holonomic $\mathcal{D}$-modules of arithmetic type and middle convolution
本文建立将G-联络扩展为完整全纯D-模的框架并证明其导出范畴具有格罗滕迪克六函子形式体系，由此得到中间卷积关于全局逆半径的稳定性，进而证明刚性Fuchs系统上多种算术性质等价，肯定回答了André与Baldassarri的猜想。
Characteristic cycles for coadmissible D-modules on smooth rigid analytic curves
本文研究光滑刚解析曲线上的可允许D模的特征环，定义了特征环并探讨了子全纯性与积分连接的关系，证明了可允许D模的子全纯性等价于对应的可允许D_∞-模的子全纯性。
Given a holonomic D-module M on a 1-dimensional torus, one can consider its Mellin transform, which is a difference system on the affine line. In this note we prove a formal stationary phase formula, which shows that the formal behavior at infinity of the Mellin transform of M is determined by the local germs defined by M at its singular points.
Hypergeometric $\mathcal D$-modules and exponential sums for reductive groups
本文定义了与有限域上reductive群表示家族相关的超几何指数和，引入超几何l-adic层和D-模，证明其是紧致的并估计其秩，利用Wang发展的向量丛一般基上的傅里叶变换理论，展示超几何D-模如何控制超几何层的一般行为，并应用结果估计超几何指数和。
Let $\mathcal{M}$ be a holonomic algebraic $\mathcal{D}$-module on the affine line. Its exponential factors are Puiseux germs describing the growth of holomorphic solutions to $\mathcal{M}$ at irregular points. The stationary phase formula states that the exponential factors of the Fourier transform of $\mathcal{M}$ are obtained by Legendre transform from the exponential factors of $\mathcal{M}$. We give a microlocal proof of this fact, by translating it in terms of enhanced ind-sheaves through the Riemann-Hilbert correspondence.
Open Quantum Systems as Regular Holonomic $\mathcal{D}$-Modules: The Mixed Hodge Structure of Spectral Singularities
本文提出将开放量子系统视为正则霍洛尼克D-模，并通过混合Hodge结构描述其谱奇点特性。
Realizations of exponential sheaves and Fourier transform
本文构造了从指数层的相对范畴到全纯（不一定正则）D-模的实现函子，并证明了它们在心上的t-正合性和忠实性，同时发展了与经典傅里叶变换交换的“通用”傅里叶变换，并证明其保持纯度。
Suppose we are given complex manifolds $X$ and $Y$ together with substacks $\mathcal{S}$ and $\mathcal{S}'$ of modules over algebras of formal deformation $\mathcal{A}$ on $X$ and $\mathcal{A}'$ on $Y$, respectively. Suppose also we are given a functor $Φ$ from the category of open subsets of $X$ to the category of open subsets of $Y$ together with a functor $F$ of prestacks from $\mathcal{S}$ to $\mathcal{S}'\circΦ$. Then we give conditions for the existence of a canonical functor, extension of $F$ to the category of coherent ${\sha}$-modules such that the cohomology associated to the action …
We introduce a category of possibly irregular holonomic D-modules which can be endowed in a canonical way with an irregular Hodge filtration. Mixed Hodge modules with their Hodge filtration naturally belong to this category, as well as their twist by $\expφ$ for any meromorphic function $φ$. This category is stable by various standard functors, which produce many more filtered objects. The irregular Hodge filtration satisfies the $E_1$-degeneration property by a projective morphism. This generalizes some results proved by Esnault-Sabbah-Yu arxiv:1302.4537 and Sabbah-Yu arxiv:1406.1339. We also…
Fourier-Orbit Construction of GKZ-Type Systems for Commutative Linear Algebraic Groups
本文基于Hotta等变D-模框架，提出傅里叶-轨道构造方法，将经典环面GKZ系统扩展至混合环面-幂幺群设置，并引入符号工具进行秩分析与精确计算，揭示了非正则等新现象。
The support S of Sabbah's specialization complex is a simultaneous generalization of the set of eigenvalues of the monodromy on Deligne's nearby cycles complex, of the support of the Alexander modules of an algebraic knot, and of certain cohomology support loci. Moreover, it equals conjecturally the image under the exponential map of the zero locus of the Bernstein-Sato ideal. Sabbah showed that S is contained in a union of translated subtori of codimension one in a complex affine torus. Budur-Wang showed recently that S is a union of torsion-translated subtori. We show here that S is always a…
This paper is devoted to the study of the relation between `formal exponential maps,' the Atiyah class, and Kapranov $L_\infty[1]$ algebras associated with dg manifolds in the $C^\infty$ context. Given a dg manifold, we prove that a `formal exponential map' exists if and only if the Atiyah class vanishes. Inspired by Kapranov's construction of a homotopy Lie algebra associated with the holomorphic tangent bundle of a complex manifold, we prove that the space of vector fields on a dg manifold admits an $L_\infty[1]$ algebra structure, unique up to isomorphism, whose unary bracket is the Lie der…
This memoir is devoted to the study of formal-analytic arithmetic surfaces. These are arithmetic counterparts, in the context of Arakelov geometry, of germs of smooth complex-analytic surfaces along a projective complex curve. Formal-analytic surfaces provide a natural framework for arithmetic algebraization theorems, old and new. Formal-analytic arithmetic surfaces admit a rich geometry which parallels the geometry of complex analytic surfaces. Notably the dichotomy between pseudoconvexity and pseudoconcavity plays a central role in their geometry. Our study of formal-analytic arithmetic surf…
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