Dimension Formulas for the Hyperfunction Solutions to Holonomic D-Modules

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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].

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. For a point x0AX let us set

XþN w F : 1 j dimC Hj F N 5 x0 ð Þ ¼ ðÀ Þ ð Þx0 o þ ð Þ j¼ÀN as an integer and call it the (local) Euler–Poincare´ index of F at x0AX: The following beautiful theorem is due to Kashiwara [10].

Theorem 2.2 (Kashiwara [10], see also Kashiwara and Schapira [11]). Let A b A y F DRÀcðXÞ and x0 X: Choosing a local coordinate system x ¼ðx1; x2; ; xnÞ of y - X such that x0 ¼ð0; 0; ; 0ÞP; let us consider the Morse function f : X R deﬁned by y / 2 n 2 x ¼ðx1; x2; ; xnÞ jxj ¼ j¼1 xj s.t. fðx0Þ¼0 and consider the associated ARTICLE IN PRESS

138 K. Takeuchi / Advances in Mathematics 180 (2003) 134–145

Lagrangean submanifold

n Lf :¼fðx; grad fðxÞÞjxAXgCT X: ð6Þ

Then the local Euler–Poincare´ index of F at x0AX is given by the formula

wx0 ðFÞ¼½CCðFÞ Á ½Lf; ð7Þ where the right-hand side is the intersection number between two Lagrangean Borel– Moore homology cycles in T nX:

Note that the intersection number appearing in the above theorem is deﬁned via some operations in the derived categories and it is in fact very hard to compute it explicitly (apart from the trivial case of transversal intersections). In the next section, we will see that the following remarkable theorem due to Schmid and Vilonen [16] will enable us to compute it very neatly.

Theorem 2.3 (Schmid and Vilonen [16]). Let X be a real analytic manifold and UCX an open subanalytic subset. Assume that there exists a real valued subanalytic CN- function f (that is, aCN-function f whose graph is a subanalytic set) s.t. f ðxÞ40 for any xAU and f is identically zero on Z ¼ X\U: Then for the open embedding j : U-X A b and F DRÀcðXÞ we have

À1 CCðRj j ðFÞÞ ¼ lim ½CCðFÞj n þ tdlogð f Þ * - T U t þ0 df ¼ lim CCðFÞjT nU þ t ; ð8Þ t-þ0 f where the meaning of the right-hand side is as follows. Consider the family of cycles n fStgtAð0;1 :¼fCCðFÞjT nU þ tdlogð f ÞgtAð0;1 in T X and its total space Sð0;1 :¼ n TtAð0;1½St ÂftgCT X Âð0; 1: Then the right-hand side of (8) is the boundary (limit)

n S0 ¼ lim ½St :¼fthe closure of Sð0;1 in T X Â Rg\Sð0;1 ð9Þ t-þ0 of this family.

This theorem is sometimes called the open embedding theorem and has many applications in the representation theory.

3. Dimension formula of solutions

In this section, we assume M is a real analytic manifold of dimension n and X is a complexiﬁcation of M: We will consider the problem of calculating the dimension of ARTICLE IN PRESS

K. Takeuchi / Advances in Mathematics 180 (2003) 134–145 139

the hyperfunction solutions HomDX ðM; BM Þ to holonomic DX -modules. Since the problem is always local, we may assume M ¼ Rn and X ¼ Cn: Then we have the following result.

n Theorem 3.1. Let MiCM ¼ R ði ¼ 1; 2; y; NÞ be linear subspaces of M passing n through the origin 0 and denote by XiCX ¼ C their complexiﬁcations in X: Then for any holonomic DX -module whose characteristic variety charðMÞ satisﬁes the condition

[N char C T n X; 10 ðMÞ Xi ð Þ i¼1 we have the formula

XN C n dim HomDX ðM; BM Þ0 ¼ multT X ðMÞ; ð11Þ Xi i¼1

n where multT n X ðMÞAZX0 is the multiplicity of the DX -module M along TX X: Xi i

A b Proof. Set F ¼ RHomDX ðM; BM Þ DRÀcðXÞ and SolðMÞ¼RHomDX ðM; A b OX Þ DRÀcðXÞðF ¼ RGM ðSolðMÞÞ½nÞ: Then it follows from a theorem of Lebeau [13] (see also [6] for another proof of it) that we have the vanishing of the cohomologies

xtj ; 0 for any jX1; 12 E DX ðM BM Þ0 ¼ ð Þ which implies

C dim HomDX ðM; BM Þ0 ¼ w0ðFÞ: ð13Þ

Hence it is enough to calculate the (local) Euler–Poincare´ index w0ðFÞ of F at 0AMCX: Next consider the following standard distinguished triangle:

F ¼ RGM ðSolðMÞÞ½n-SolðMÞ½n-RGXÀM ðSolðMÞÞ½n- þ 1: ð14Þ

Then we get

w0ðFÞ¼w0ðSolðMÞ½nÞ À w0ðRGXÀM ðSolðMÞÞ½nÞ

n ¼ðÀ1Þ ½w0ðSolðMÞÞ À w0ðRGXÀM ðSolðMÞÞÞ: ð15Þ

If we denote by di the complex codimension of Xi in X; then by virtue of an index theorem of Kashiwara [9] we have

XN di w0ðSolðMÞÞ ¼ ðÀ1Þ multT n X ðMÞ: ð16Þ Xi i¼1 ARTICLE IN PRESS

140 K. Takeuchi / Advances in Mathematics 180 (2003) 134–145

Therefore it remains to compute the term w0ðRGXÀM ðSolðMÞÞÞ: For this purpose, let us rewrite the complex RGXÀM ðSolðMÞÞ by the open embedding j : U ¼ \ - C À1 X M X as RGXÀM ðSolðMÞÞ Rj * j SolðMÞ: By taking a Morse function f : X-R s.t. fð0Þ¼0 as in Theorem 2.2, we get

À1 À1 w0ðRj * j SolðMÞÞ ¼ ½CCðRj * j SolðMÞÞ Á ½Lfð17Þ

À1 by Theorem 2.2. Now we have to know the cycle CCðRj * j SolðMÞÞ: Until some years ago, it had been very difficult in general to calculate the topological cycle À1 CCðRj * j SolðMÞÞ from the information of CCðSolðMÞÞ: However, now we can overcome this serious obstruction thanks to the open embedding theorem (Theorem 2.3) of Schmid and Vilonen [16]. Let us take a local coordinate system z ¼ x þ iy; x ¼ðx1; x2; y; xnÞ; y ¼ y ; y ; y; y X n M y n X: CN ð 1 2 nÞ of ¼ Cz s.t. ¼f ¼ 0g¼Rx inP Then the subanalytic - n- 2 n 2 function f : X ¼ Cz R defined by f ðzÞ¼jyj ¼ j¼1 yj satisfies the properties: \ ff jM ¼ 0 and f 40onU ¼ X Mg: Therefore we get by Theorem 2.3 the following equality:

À1 CCðRj j SolðMÞÞ ¼ lim ½CCðSolðMÞÞjT nU þ tdlogð f Þ; ð18Þ * t-þ0 where the right-hand side stands for the boundary (limit) S0 of the family fSt ¼

CCðSolðMÞÞjT nU þ tdlogð f ÞgtAð0;1 of the Borel–Moore homology cycles. There- fore to calculate the index w0ðRGXÀM ðSolðMÞÞÞ it sufﬁces to determine the intersection number ½S0Á½Lf between this limit cycle S0 and the Lagrangean cycle Lf: But usually, the limit cycle S0 has a very bad singularity at the intersection points with Lf: Hence, we cannot explicitly compute the intersection number ½S0Á½Lf: The next lemma, which is essentially contained in [2], enables us to resolve this problem.

Lemma 3.2. Let fStgtAð0;1 be a family of Borel–Moore homology cycles in a real analytic manifold X such that the total space Sð0;1 ¼ TtAð0;1½St ÂftgCX Âð0; 1 is a ﬁbre bundle over the interval ð0; 1: Also set S0 ¼ limt-þ0½St :¼ fthe closure of Sð0;1 in X Â Rg\Sð0;1 and let L be another Borel–Moore homology cycle in X: Finally, assume that the intersection number m ¼½StÁ½L is a constant integer for any tAð0; 1 and all intersection points are transversal ones. Then we have

½S0Á½L¼ lim ð½StÁ½LÞ ¼ m: ð19Þ t-þ0

The situation treated in this lemma is a special case of the ‘‘specialization of Borel– Moore homology’’, which is a notion introduced by Fulton and MacPherson [2]. For a review of that paper and the proof of this lemma, see also for example ARTICLE IN PRESS

K. Takeuchi / Advances in Mathematics 180 (2003) 134–145 141

Section 2.6.30. (Proposition 2.6.39.) of the book [1] of Chriss–Ginzburg. The proof is based on the interpretation of the operations of Borel–Moore homology cycles into the language of local cohomologies. With this lemma at hand, we may argue as

w0½RGXÀM ðSolðMÞÞ ¼ lim St Á½Lf t-þ0

¼ lim ð½StÁ½LfÞ ð20Þ t-þ0 if the family fStgtAð0;1 ¼fSt ¼ CCðSolðMÞÞjT nU þ tdlogð f ÞgtAð0;1 of cycles and L ¼ Lf satisfy the assumption of Lemma 3.2. The next lemma shows that we can choose a good Morse function f : X-R justifying the above argument.

n Lemma 3.3. For a positive deﬁnite symmetric n Â n matrix A ¼ðaijÞi;j¼1 let us deﬁne a N n- C -function fA : X ¼ Cz R by

2 t n fAðzÞ :¼jxj þ yAy for z ¼ x þ iyAX ¼ Cz : ð21Þ

n Then for generic matrices A’s, the associated Lagrangean cycles Lf in T X satisfy the condition:

fthe intersection number m ¼½StÁ½LfA is a constant

integer for any tAð0; 1 and all intersection points are transversal onesg:

Moreover, we can choose the positive deﬁnite symmetric matrix A so that we have in addition

L T n X tdlog f L T n X tdlog f | 22 ½ fA -ð Xi þ ð ÞÞ-½ fA -ð Xj þ ð ÞÞ ¼ ð Þ for any pair iaj and 8tAð0; 1:

Proof. To begin with, let us look at closely how the deformed cycle T n X ð Xi þ n tdlogð f ÞÞ intersects with the Lagrangean cycle LfA in T X: We choose an index i n and set Y ¼ XiCX: By a rotation of M ¼ Rx we may assume Y ¼fz1 ¼ z2 ¼ ? ¼ n 0 00 zd ¼ 0gCX ¼ Cz : Also, for the sake of simplicity, set z ¼ðz1; z2; y; zd Þ; z ¼ 0 00 0 0 0 ðzdþ1; zdþ2; y; znÞ (hence z ¼ðz ; z Þ and z ¼ x þ iy ; etc.). Here, we use a coordinate system ðz; zÞ¼ðz; ReðzÞ dx þ ImðzÞ dyÞ for the cotangent bundle T nðX RÞ of the underlying real manifold X R of X: Note there is a canonical isomorphism T nXCT nðX RÞ (we prefer to use T nðX RÞ for applying the Morse theory in the real domain). In this situation we have ( n 00 0 n C n R TY X ¼ fðð0; z Þ; ðz ; 0ÞÞgCT X T ðX Þ;

2ty1 2tyn n ð23Þ fThe graph of tdlogð f Þg ¼ fððz ¼ x þ iyÞ; 2 dy1 þ ? þ 2 dynÞgCT X: jyj jyj ARTICLE IN PRESS

142 K. Takeuchi / Advances in Mathematics 180 (2003) 134–145

Summing them up in T nX we get () !!

n 00 0 2tydþ1 2tyn n TY X þ tdlogð f Þ¼ ð0; z Þ; z ; dydþ1 þ ? þ dyn CT X; ð24Þ jy00j2 jy00j2

n where we used the fact that y1 ¼ y2 ¼ ? ¼ yd ¼ 0onTY X: On the other hand, for 2 t n the Morse function fAðzÞ :¼jxj þ yAy for A ¼ðaijÞi;j¼1 the associated Lagrangean cycle is () !! Xn Xn Xn

LfA ¼ðx þ iyÞ; 2xj dxj þ ðaij þ ajiÞyi dyj : ð25Þ j¼1 j¼1 i¼1

n A n A in T X: So each intersection point p ¼ðz; zÞ LfA -ðTY X þ tdlogð f ÞÞ ðt ð0; 1Þ must satisfy the conditions 8 > z0 ¼ 0; x00 ¼ 0; < P pffiffiffiffiffiffiffi n y zj ¼ i¼dþ1ðaij þ ajiÞ À1yi ð j ¼ 1; 2; ; dÞðAÞ ð26Þ > P : 2tyj n 2 ¼ ðaij þ ajiÞyi ð j ¼ d þ 1; y; nÞðBÞ jy00j i¼dþ1

00 n n Now consider the part A ¼ðaijÞi;j¼dþ1 of the matrix A ¼ðaijÞi;j¼1; which is still a positive deﬁnite symmetric ðn À dÞÂðn À dÞ matrix. If we diagonalize A00 ¼ n ðaijÞi;j¼dþ1 by an orthogonal change of coordinate with respect to the variables y 00 n ydþ1; yd þ 2; ; yn; we may assume A ¼ðlidijÞi;j¼dþ1 from the ﬁrst. Here the real 00 numbers ldþ1; ldþ2; y; ln40 are the eigenvalues of A : In this case Eq. (B) is reduced to the simpler one

2tyj ¼ 2ljyj ð j ¼ d þ 1; y; nÞðCÞð27Þ jy00j2

If moreover the positive eigenvalues ldþ1; ldþ; y; ln40 are all distinct each other, the solutions to Eq. ðCÞ must be the 2ðn À dÞ points sffiffiffiffi ! t ðydþ1; ydþ2; y; ynÞ¼ 0; y; 0; 7 ; 0; y; 0 ð j ¼ d þ 1; y; nÞ; ð28Þ lj

qffiffiffi where the number 7 t is placed in the jth entry ð j ¼ d þ 1; y; nÞ: Hence the lj A n intersection points p ¼ðz; zÞ LfA -ðTY X þ tdlogð f ÞÞ are the following 2ðn À dÞ ARTICLE IN PRESS

K. Takeuchi / Advances in Mathematics 180 (2003) 134–145 143 points for any tAð0; 1: !! pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2ty00 p ¼ðz; zÞ¼ ð0; À1y00Þ; z0; À1 jy00j2 sffiffiffiffi !! pffiffiffiffiffiffiffi t ¼ 0; 0; y; 0; 7 À1 ; 0; y; 0 ; lj sffiffiffiffi sffiffiffiffi pffiffiffiffiffiffiffi t pffiffiffiffiffiffiffi t z1 ¼ 7 À1ðaj1 þ a1jÞ ; y; zd ¼ 7 À1ðajd þ adjÞ ; lj lj !! pffiffiffiffiffiffiffipffiffiffiffiffiffi 00 z ¼ð0; y; 0; 72 À1 tlj; 0; y; 0Þ ð j ¼ d þ 1; y; nÞ: ð29Þ

It is clear that all intersection points are transversal ones and the intersection number n A ½LfA Á½ðTY X þ tdlogð f ÞÞ does not depend on the parameter t ð0; 1: We can also see from this explicit calculation that the set of matrices A’s satisfying the conditions for Y ¼ Xi (here we fix an index i) 8 > i the intersection L T n X tdlog f is transversal and the > ð Þ fA -ð Y þ ð ÞÞ > > intersection number L T n X tdlog f does not <> ½ fA Á½ð Y þ ð ÞÞ depend on tAð0; 1: ð30Þ > > n n | > ðiiÞ½LfA -ðTY X þ tdlogð f ÞÞ-½LfA -ðTX X þ tdlogð f ÞÞ ¼ > j : C C for any jai s:t: dim XjXdim Y and 8tAð0; 1: is open dense in the space PDM of positive definite symmetric n Â n matrices (we can perturb the matrix A so that condition (ii) is fulfilled by using the relation (A)). Let us denote this open dense subspace of PDM by OiCPDM: Then the assertion of Lemma 3.3 follows if we take a matrix A such that AAO1-O2-?-ON a|: It completes the proof of Lemma 3.3. &

Now let us return to the proof of Theorem 3.1. In order to compute the intersection number limt-þ0ð½StÁ½LfÞ we have to keep in mind the orientations of the cycles St and Lf: We will compute this topological number with the help of an algebraic method. Taking a good Morse function f ¼ fA as in Lemma 3.3, we have the following chain of equalities for any tAð0; 1:

½StÁ½Lf

XN di n ¼ ðÀ1Þ multT n X ðMÞÂ½TX X þ tdlogð f Þ Á ½Lf Xi i i¼1 ARTICLE IN PRESS

144 K. Takeuchi / Advances in Mathematics 180 (2003) 134–145 N " ¼ CC Rj jÀ1 " C mi ½Àd Á½L * Xi i f i¼1 N " ¼ w Rj jÀ1 " C mi ½Àd ; ð31Þ 0 * Xi i i¼1 where mi :¼ multT n X ðMÞ and we have used the transversality of the intersection at Xi nÀd À1 t40: Since the difference set Xi\M is homotopic to the ðn À di À 1Þ sphere S i ; it follows from a basic fact of elementary algebraic topology that j jÀ1 nÀdi w0½R * ðCXi Þ ¼ 1 ÀðÀ1Þ : ð32Þ

Combining these results together, we ﬁnally obtain

C dim HomDX ðM; BM Þ0

¼ w0½RHomDX ðM; BM Þ "# XN XN n di nÀdi di ¼ðÀ1Þ ðÀ1Þ mi À f1 ÀðÀ1Þ gðÀ1Þ mi i¼1 i¼1 XN ¼ multT n X ðMÞ: ð33Þ Xi i¼1

Hence it completes the proof of Theorem 3.1. &

Remark 3.4. If in the proof of Theorem 3.1 we choose the standard Morse function f z x 2 y 2; then the intersection sets L T n X tdlog f are not discrete ð Þ¼j j þj j f-ð Xi þ ð ÞÞ sets. They are in general higher dimensional spheres. This is the reason why we employed a perturbed Morse function fA associated to a generic positive deﬁnite symmetric matrix.

n Remark 3.5. In Theorem 3.1 we assumed that Mi’s are linear subspaces of M ¼ R ; but it also seems to be possible to extend our result to the cases where Mi’s are real analytic submanifolds passing through the origin. Even in such cases, the Lebeau’s theorem [13] asserts

C dim HomDX ðM; BM Þ0 ¼ w0½RHomDX ðM; BM Þ: ð34Þ

Perhaps, to get a good Morse function as in the proof above, we should impose the condition that each pair Mi and Mj ðiajÞ has a clean intersection at the origin. It would be also interesting to study the dimension formulas for hyperfunction solutions as in Theorem 3.1 for the holonomic DX -modules s. t. equality (34) does not hold. To treat such holonomic DX -modules, it might be necessary to return to the proof of Theorem 2.3 [16] modify it for our purpose. ARTICLE IN PRESS

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