Temperate holomorphic solutions and regularity of holonomic D-modules on curves

Regular and irregular holonomic D-modules

This is a survey paper based on a series of lectures given at the IHES in February/March 2015. In a first part, we recall the main results on the tempered holomorphic solutions of D-modules in the language of indsheaves and, as an application, the Riemann-Hilbert correspondence for regular holonomic modules. In a second part, we present the enhanced version of the first part, treating along the same lines the irregular holonomic case.

Bivariate hypergeometric D-modules

We undertake the study of bivariate Horn systems for generic parameters. We prove that these hypergeometric systems are holonomic, and we provide an explicit formula for their holonomic rank as well as bases of their spaces of complex holomorphic solutions. We also obtain analogous results for the generalized hypergeometric systems arising from lattices of any rank.

Curve test for enhanced ind-sheaves and holonomic D-modules

Recently, the Riemann-Hilbert correspondence was generalized in the context of general holonomic D-modules by A. D'Agnolo and M. Kashiwara. Namely, they proved that their enhanced de Rham functor gives a fully faithfully embedding of the derived category of cohomologically holonomic complexes of D-modules into the derived category of complexes of real constructible enhanced ind-sheaves. In this paper, we study a condition when a complex of real constructible enhanced ind-sheaves $K$ is induced by a cohomologically holonomic complex of D-modules. We characterize such $K$ in terms of the restric…

相对全纯 D-模的正则化

Regularization of relative holonomic D-modules

本文利用无穷阶相对微分算子层，将 Kashiwara-Kawai 经典定理推广到相对情形，函子性地构造相对全纯 D-模的正则化，并证明其与张量积形式同构。

一维全纯D-模傅里叶变换的Morse理论方法

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。

全纯D-模的中度与快速衰减邻近循环

Moderate and rapid-decay nearby cycles for holonomic D-modules

针对全纯D-模引入与全纯函数相关的中度、快速衰减邻近循环概念，证明其为特殊纤维与圆周乘积上的ℝ-可构造复形，验证对偶性质并明确其与非正则复形的关系，论证主要依托Mochizuki的相关成果。

Microlocal study of ind-sheaves I: micro-support and regularity

We define the notions of micro-support and regularity for ind-sheaves, and prove their invariance by contact transformations. We apply the results to the ind-sheaves of temperate holomorphic solutions of D-modules. We prove that the micro-support of such an ind-sheaf is the characteristic variety of the corresponding D-module and that the ind-sheaf is regular if the D-module is regular holonomic.

Tempered solutions of $\mathcal D$-modules on complex curves and formal invariants

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 $\mathcal M$.

Sheaves on subanalytic sites

In Asterisque 271 the authors introduced the notion of ind-sheaf, and defined the six Grothendieck operations in this framework. They defined subanalytic sheaves and they obtained the formalism of the six Grothendieck operations by including subanalytic sheaves into the category of ind-sheaves. The aim of this paper is to give a direct construction of the six Grothendieck operations in the framework of subanalytic sites avoiding the heavy theory of ind-sheaves. As an application we show how to recover the subanalytic sheaves of temperate and Whitney holomorphic functions.

An existence theorem for tempered solutions of D-modules on complex curves

Let X be a complex curve, $X_{sa}$ the subanalytic site associated to X, M a holonomic $D_X$-module. Let $O^t$ be the sheaf on $X_{sa}$ of tempered holomorphic functions, Sol(M) (resp. $Sol^t$(M)) the complex of holomorphic (resp. tempered holomorphic) solutions of M. We prove that the natural morphism from $H^1 Sol^t$(M) to $H^1$Sol(M) is an isomorphism of sheaves on $X_{sa}$. As a consequence, we prove that $Sol^t$(M) is R-constructible in the sense of sheaves on $X_{sa}$. Such a result is conjectured by M. Kashiwara and P. Schapira in Astérisque 284 in any dimension.

Functorial properties of the microsupport and regularity for ind-sheaves

The notion of microsupport and regularity for ind-sheaves was introduced by M. Kashiwara and P. Schapira in "Microlocal study of ind-sheaves I: microsupport and regularity". In this paper we study the behaviour of the microsupport under several functorial operations and characterize "microlocally" the ind-sheaves that are regular along involutive manifolds. As an application we prove that if a coherent D-module M is regular along an involutive manifold V (in the sense of "Systems of differential equations with regular singularities and their boundary value problems") then the ind-sheaf of temp…

A Topological Reconstruction Theorem for $D^{\infty}$-Modules

In this paper, we prove that any perfect complex of $D^{\infty}$-modules may be reconstructed from its holomorphic solution complex provided that we keep track of the natural topology of this last complex. This is to be compared with the reconstruction theorem for regular holonomic $D$-modules which follows from the well-known Riemann-Hilbert correspondence. To obtain our result, we consider sheaves of holomorphic functions as sheaves with values in the category of ind-Banach spaces and study some of their homological properties. In particular, we prove that a Künneth formula holds for them an…

A microlocal approach to the enhanced Fourier-Sato transform in dimension one

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.

通过增强 ind-层实现邻近循环的适度增长与快速衰减

Moderate Growth and Rapid Decay Nearby Cycles via Enhanced Ind-Sheaves

本文定义增强 ind-sheaf 的适度增长与快速衰减对象，证明其恢复非正则 Riemann–Hilbert 对应中的 de Rham 复形，并解决 Sabbah 对偶猜想。