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24
Hyperfunctions and microdifferential equations
 Lecture Notes in Math
, 1973
"... This is the third of the series of the papers dealing with holonomic systems(*}. A holonomic system is, by definition, a left coherent (fModule (or ^Modules)(*sS:) whose characteristic variety is Lagrangian. It shares the finiteness theorem with a linear ordinary differential equation, namely, al ..."
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Cited by 68 (10 self)
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This is the third of the series of the papers dealing with holonomic systems(*}. A holonomic system is, by definition, a left coherent (fModule (or ^Modules)(*sS:) whose characteristic variety is Lagrangian. It shares the finiteness theorem with a linear ordinary differential equation, namely, all the
On the Holonomic Systems of Linear Differential Equations, II
, 1978
"... In this paper we shall study the restriction of holonomic systems of differential equations. Let X be a complex manifold and Y a submanifold, and let (9 x and D x be the sheaf of the holomorphic functions and the sheaf of the differential operators of finite order, respectively. If a function u on X ..."
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Cited by 38 (1 self)
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In this paper we shall study the restriction of holonomic systems of differential equations. Let X be a complex manifold and Y a submanifold, and let (9 x and D x be the sheaf of the holomorphic functions and the sheaf of the differential operators of finite order, respectively. If a function u on X satisfies a system of differential equations, the restriction of u onto Y also satisfies the system of differential equations derived from the system on X. This leads to the following definition. Let JC { be a DxModule. The restriction of ~/onto Y is, by definition, (gr  J//. In [4] it is proved that if Jr is a coherent DxModule and if Y is noncharacteristic to Jg, then the restriction of Jg is also a coherent DrModule. However, if Y is characteristic, the restriction is no longer coherent in general. For examples, if X=II2 " and Y={x=(x 1...., x,)eX; xl=O} and Jg=Dx, the restriction JC[/xlJCl is a free DrModule generated by D~(m=0, 1,2,...) and is not coherent. We shall prove the following theorems in this paper.
Bfunctions and holonomic systems (Rationality of roots of bfunctions
 Invent. Math
, 1976
"... A bfunction of an analytic function f(x) is, by definition, a gcnerator of the ideal formed by the polynomials b(s) satisfying P(s, x, Dx) f (x) " + 1 = b(s) f(x) ~ for some differential operator P(s, x, Dx) which is a polynomial on s. Professor M.Sato introduced the notions of "afunctio ..."
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Cited by 31 (1 self)
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A bfunction of an analytic function f(x) is, by definition, a gcnerator of the ideal formed by the polynomials b(s) satisfying P(s, x, Dx) f (x) " + 1 = b(s) f(x) ~ for some differential operator P(s, x, Dx) which is a polynomial on s. Professor M.Sato introduced the notions of "afunction", "bfunction " and "'cfunction " for relative invariants on prehomogeneous vector spaces, when he studied the fourier transforms and ~functions associated with them (see [10, 12]). He defined, in the same time, bfunctions for arbitrary holomorphic functions and conjectured their existence and the rationality of their roots. Professor Bernstein introduced, independently of Prof. Sato, bfunctions and proved any polynomial has a non zero bfunction [1]. Professor Bj6rk extended this result o an arbitrary analytic functions by the same method [3]. The rationality of roots of bfunctions is closely related to the quasiunipotency of local monodromy. In fact, Professor Malgrange showed that the eigenvalues of local monodromy are exp (2 n l fZ~a) for a root c ~ of the bfunction when f has an isolated singularity [9]. In this paper, the proof of the existence of bfunctions and the rationality of their roots are given. The method employed here is to study the system of differential equations which satisfies f(x) ~. First, we will show that ~J ~ is a subholonomic system and prove the existence of bfunctions as its immediate consequence. Next, we study the rationality of roots of bfunctions by using the desingularization theorem due to Hironaka. So, the main result of this paper is the following two theorems. Theorem, The characteristic variety oJ'~J ~ is equal to W s. Wf is, by dffinition, the closure of {(x, ~); ~ = s grad log f(x) for some sol2} in the cotangent vector bundle.
Constructibility and duality for simple holonomic modules on complex symplectic manifolds
 2005, arXiv:math.QA/0512047. STAR PRODUCTS 15
"... Consider a complex symplectic manifold X and the algebroid WX of quantizationdeformation. For two regular holonomic WXmodules Li (i = 0, 1) supported by smooth Lagrangian manifolds, we prove that the complex RHom WX (L1, L0) is constructible and perverse and dual to the complex RHom WX (L0, L1). ..."
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Consider a complex symplectic manifold X and the algebroid WX of quantizationdeformation. For two regular holonomic WXmodules Li (i = 0, 1) supported by smooth Lagrangian manifolds, we prove that the complex RHom WX (L1, L0) is constructible and perverse and dual to the complex RHom WX (L0, L1).
An Introduction to DModules
"... The purpose of these notes is to introduce the reader to the algebraic theory of systems of partial differential equations on a complex analytic manifold. We ..."
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The purpose of these notes is to introduce the reader to the algebraic theory of systems of partial differential equations on a complex analytic manifold. We
Holonomic Dmodule with Betti structure
, 2010
"... This is an attempt to define a notion of Betti structure with nice functorial property for algebraic holonomic Dmodules which are not necessarily regular singular. ..."
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Cited by 4 (1 self)
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This is an attempt to define a notion of Betti structure with nice functorial property for algebraic holonomic Dmodules which are not necessarily regular singular.
A coherence criterion for Fréchet modules
 Astérisque
, 1994
"... In the literature, one nds essentially two general criteria to get the niteness of the cohomology groups of complexes of locally convex topological vector spaces. They are (a) If u : G −! F is a compact morphism of complexes of Frechet spaces then dimHk(F ) < +1 for any k 2 ZZ such that Hk(u) ..."
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Cited by 3 (0 self)
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In the literature, one nds essentially two general criteria to get the niteness of the cohomology groups of complexes of locally convex topological vector spaces. They are (a) If u : G −! F is a compact morphism of complexes of Frechet spaces then dimHk(F ) < +1 for any k 2 ZZ such that Hk(u) is surjective.
INTRODUCTION TO A THEORY OF bFUNCTIONS
, 2006
"... We give an introduction to a theory of bfunctions, i.e. BernsteinSato polynomials. After reviewing some facts from Dmodules, we introduce bfunctions including ..."
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We give an introduction to a theory of bfunctions, i.e. BernsteinSato polynomials. After reviewing some facts from Dmodules, we introduce bfunctions including