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Dimension Formulas for the Hyperfunction Solutions to Holonomic D-Modules

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Dimension Formulas for the Hyperfunction Solutions to Holonomic D-Modules View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector ARTICLE IN PRESS Advances in Mathematics 180 (2003) 134–145 http://www.elsevier.com/locate/aim Dimension formulas for the hyperfunction solutions to holonomic D-modules Kiyoshi Takeuchi Institute of Mathematics, University of Tsukuba, 1-1-1, Tennodai, Tsukuba, Ibaraki 305-8571, Japan Received 15 July 2002; accepted 19 November 2002 Communicated by Takahiro Kawai Dedicated to Professor Mitsuo Morimoto on the Occasion of his 60th Birthday Abstract In this paper, we will prove an explicit dimension formula for the hyperfunction solutions to a class of holonomic D-modules. This dimension formula can be considered as a higher dimensional analogue of a beautiful theorem on ordinary differential equations due to Kashiwara (Master’s Thesis, University of Tokyo, 1970) and Komatsu (J. Fac. Sci. Univ. Tokyo Math. Sect. IA 18 (1971) 379). In the course of the proof, we will make use of a recent innovation of Schmid–Vilonen (Invent. Math. 124 (1996) 451) in the representation theory (in the theory of index theorems for constructible sheaves). r 2003 Elsevier Science (USA). All rights reserved. MSC: 32C38; 35A27 Keywords: D-module; Holonomic system; Index theorem 1. Introduction The aim of this paper is to prove a useful formula to compute the exact dimensions of the solutions to holonomic D-modules. The study of the dimension formula for the generalized function solutions to ordinary differential equations started only recently. In the 1970s, a beautiful theorem was obtained by Kashiwara [7] and Komatsu [12] using the hyperfunction theory. They showed that the dimension of the hyperfunction solutions to an ordinary differential equation could be expressed as a E-mail address: [email protected]. 0001-8708/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0001-8708(02)00099-3 ARTICLE IN PRESS K. Takeuchi / Advances in Mathematics 180 (2003) 134–145 135 sum of its multiplicities (this is an algebraic invariant of the equation). To explain the concrete meaning of their result, let M ¼ Rx and X ¼ Cz ð0AMÞ and denote by BM the sheaf of hyperfunctions (of one variable) on M ¼ Rx: Then for an a b ordinary differential operator P ¼ z @z þðlower order termsÞADX ; their theorem asserts C A C dim fu ðBM Þ0jPu ¼ 0g¼dim HomDX ðDX =DX P; BM Þ0 ¼ a þ b: For example, if we take an operator P ¼ z@z À lADX for a complex number lAC; the space of the local hyperfunction solutions to Pu ¼ðz@z À lÞu ¼ 0 at the origin 0 l A is two dimensional and spanned over C by the special functions x7 ðBM Þ0: Hence the theorem of Kashiwara [7] and Komatsu [12] can be considered as a vast generalization of this simple fact. In this paper, we would like to seek for a theorem of Kashiwara–Komatsu type in higher dimensional cases. In such cases, the special systems of partial differential equations called holonomic systems (or holonomic D-modules) are the most interesting objects, since the solutions to these systems shares the finite dimensionality with those to ordinary differential equations (even in higher dimensional situations). In fact, Kashiwara [8] proved that the space of the hyperfunction solutions to a holonomic D-modules is always finite dimensional. This decisive result was established by using the constructibility of the holomorphic solution complexes to holonomic systems, whose proof also required a deep argument with the aid of even functional analysis. However, to the best of our knowledge, it seems that the general theory of the dimension formulas for the generalized function solutions to holonomic D-modules have not yet been fully developed (although the finite dimensionality is always guaranteed by Kashiwara’s theorem [8]). Such formulas were discretely studied in several branches of mathematics in some special situations. The people in the representation theory studied the distribution characters on real Lie groups, the solutions to the Harish- Chandra’s system. Also the dimension of invariant distributions was investigated by the researchers in the theory of prehomogeneous vector spaces. For example, see the important contributions in [5,14] etc. in this direction. Here in the present paper, we will prove a general formula which resembles more the Kashiwara–Komatsu’s theorem. For this purpose, we employ a very recent innovation in the representation theory (index theorem theory) due to Schmid and Vilonen [16]. In order to state our main theorem, let us set M ¼ Rn and X ¼ Cn (a complexification of M). Denote also by BM (resp. DX ) the sheaf of hyperfunctions on M (resp. the sheaf of partial differential operators with holomorphic coefficients on X) and regard as usual coherent DX -modules M with systems of linear partial differential equations by Sato’s philosophy. Then our result is as follows. n Theorem. Let MiCM ¼ R ði ¼ 1; 2; y; NÞ be linear subspaces passing through the n origin 0 and denote by XiCX ¼ C their complexifications in X: Then for any holonomic DX -module whose characteristic variety charðMÞ satisfies the ARTICLE IN PRESS 136 K. Takeuchi / Advances in Mathematics 180 (2003) 134–145 geometric condition [N char C T n X; 1 ðMÞ Xi ð Þ i¼1 we have the formula XN C n dim HomDX ðM; BM Þ0 ¼ multT X ðMÞð2Þ Xi i¼1 for the dimension of the hyperfunction solutions to M at 0, where multT n X ðMÞAZX0 Xi stands for the multiplicity of the -module along T n X (see [9] for the precise DX M Xi definition). Note that condition (1) is always satisfied when n ¼ 1; i.e. in the case of ordinary differential equations. So this theorem is a higher dimensional analogue of Kashiwara–Komatsu’s formula. The proof of this theorem utilize the theory of the index theorems for constructible sheaves developed by Kashiwara [9,10], Kashiwara and Schapira [11] and Schmid–Vilonen’s recent result in [16] on the behaviour of the characteristic cycles of constructible sheaves under the push- forwards by open embeddings. The main point in the proof is the calculation of the intersection numbers between Borel–Moore homology cycles (i.e. homology cycles with infinite supports) via a Schmid–Vilonen’s theorem. There are many examples of systems satisfying the assumption (1). For example, let us mention that the moduli space of the D-modules satisfying condition (1) was recently studied by Nitsure [15] in the case when the complex subspaces Xi’s are normally crossing, and the corresponding perverse sheaves were classified by Galligo et al. [3]. However, even under this strong condition of normal crossing, our result would not follow from their (categorical) classification. Also Prof N. Takayama1 kindly informed us of the fact that the holonomic system called A-hypergeometric equation introduced by Gelfand et al. [4] satisfies our condition (1) at every point. 2. Review on index theorems In this section, we will briefly recall the theory of the index theorems for R- constructible sheaves. The main reference is Chapter IX of Kashiwara–Schapira [11] and we will also follow the terminology in it throughout this paper. Let X be a real analytic manifold and denote by DbðXÞ the derived category of the complexes of sheaves of C-vector spaces on X with bounded cohomologies. We also denote by b b DRÀcðXÞ the full subcategory of D ðXÞ consisting of objects with R-constructible cohomology sheaves. In 1985, Kashiwara [10] defined for each R-constructible object A b F DRÀcðXÞ its characteristic cycle CCðFÞ as a (real Lagrangean) Borel–Moore 1 The author would like to thank him on this occasion. ARTICLE IN PRESS K. Takeuchi / Advances in Mathematics 180 (2003) 134–145 137 homology cycle in the cotangent bundle T nX of X: This topological cycle CCðFÞ plays an crucial role in calculating the Euler–Poincare´ indices of R-constructible A b sheaves F DRÀcðXÞ: Let us explain this notion more precisely. For each A b F DRÀcðXÞ we can always take a Whitney stratification X ¼ TaAAXa of X by subanalytic submanifolds Xa’s in X so that the micro-support SSðFÞ of F satisfies the condition n SSðFÞC T TX X: ð3Þ aAA a Then the characteristic cycle CCðFÞ of F can be expressed as a formal sum of the Borel–Moore chains T n X aAA as follows. As a real Lagrangean cycle in T nX ½ Xa ð Þ X CC F m T n X ; 4 ð Þ¼ a½ Xa ð Þ aAA where m is an integer called the multiplicity of F along T n X: For example, for a a Xa n closed submanifold Y of X we have CCðCY Þ¼½TY X: Moreover, if X is a complex A b manifold and F DRÀcðXÞ is the holomorphic solution complex RHomDX ðM; OX Þ of a holonomic DX -module, then SSðFÞ¼chðMPÞ and the absolute value jCCðFÞj of CC F defined by the equation CC F : m T n X coincides with the ð Þ j ð Þj ¼ aAA j aj½ Xa characteristic cycle CCðMÞ of the DX -module M (see [9] for the definition). In this case, if we denote by da the complex codimension of the complex strata Xa in X; then da we have in fact jmaj¼ðÀ1Þ ma as in the example below. n Example 2.1. Let YCX ¼ Cz be a complex submanifold of complex codimension d in X and assume the holonomic DX -module M is the sheaf BYjX of the holomorphic C hyperfunctions along Y: Then F ¼ RHomDX ðM; OX Þ CY ½Àd and CCðFÞ¼ d n n ðÀ1Þ ½TY X; whereas the characteristic cycle CCðMÞ of the DX -module M is ½TY X: A b Now let us quickly review how the characteristic cycle CCðFÞ of F DRÀcðXÞ encodes the cohomological data of the sheaf F into a topological one.
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