数理解析研究所講究録
第 750 巻 1991 年 58-79

58

The Fourier-Sato Transformation of Pure Sheaves

SYOITI NINOMIYA
Dept. of Math., Faculty of Science, University of Tokyo\S 0. INTRODUCTION.
Kashiwara-Schapira introduced the notion of pure sheaves in [K-S] inorder to calculate the shifts which appear when contact transformations
are applied to sheaves. The purity of a sheaf describes the obstruction forthe prolongment of its sections across critical points of Morse functions
and played an important role in studying R-constructible sheaves and, inparticular, their index theorems (see M. Kashiwara [K] and P. Schapira
and N. Tose [S-T]). Under the assumption of purity, the obstruction isexpressed as cohomology groups, which can be calculated with two microlocal data, the Lagrangian variety associated to the Morse function
and that of the micro-support of the sheaf. Then we use the inertia index
of three Lagrangian planes. Kashiwara-Schapira studied the functorialproperties of pure sheaves by several fundamental operators in [K-S].
The Fourier-Sato transformation is a geometric counterpart of Fouriertransformation, which is introduced by Sato et al. [S-K-K] when they
constructed the sheaf of microfunctions. The Fourier-Sato transforma.tion of a conic sheaf on a real vector bundle $E$ is a conic object on the. In the category of , this transformation is closely
dual bundle
related with the Gauss sum, etc.
In this paper, the author calculates the Fourier-Sato transformationof pure sheaves. In \S 4 we have the result and the proof. In \S 5 as acorollary of this result, we obtain another proof of the proposition by
Kashiwara-Schapira [K-S2] which asserts that the Fourier-Sato transform of a perverse sheaf is also perverse. J. L. Brylinski proved analogous
propositions in the algebraic category [ , corollaire 7.23] and in the category of [ , corollaire 9.11]. The important point of the present paper
is that we use only techniques purely in the real domain. Thus the proof
is independent of the monodromy structure of perverse sheaves.
$E^{*}$$F_{q}$$B$$F_{q}$$B$

The author would like to express his sincere thanks to Prof. T. Oshima for his warm
guidance, to Prof. N. Tose for his advice on the direction of study and to Dr. N. Honda
for his advice and encouragement.

59\S 1. NOTATION AND CONVENTIONS.
The following notation is taken from [K-S].

1.0. Throughout this paper, let be a commutative unitary ring withfinite global dimension, $Sh(X)$ the abelian category of sheaves of Amodules on a topological space $X,$ $D(X)$ the derived category of $Sh(X)$ .
We denote by $D^{+}(X)$ the full subcategory of $D(X)$ consisting of complexes with cohomology bounded from below and by $D^{b}(X)$ the fullsubcategory of $D(X)$ consisting of complexes with bounded cohomolthe object obtained
ogy. For an object of $D(X)$ , we denote byand.
by k-shifts; that is to say$X$
Sheaves on are identified with complexes of $D(X)$ which are concentrated in degree . We use usual notation of derived categories and sheaf,
cohomology. Refer to [K-S] for functors,,, orientation sheaf
relative orientation.
and constant sheaf
sheaf
$A$

$\mathcal{F}[k]$$\mathcal{F}$$H^{J}(\mathcal{F}[k])=H^{i+k}(\mathcal{F})$$d_{\mathcal{F}[k]}^{n}=(-1)^{k}d_{\mathcal{F}}^{n+k}$$0$$\underline{Hom}(\cdot, \cdot),$$|z,$$\cdot\otimes\cdot,$$(\cdot)_{Z},$$\Gamma_{Z}(\cdot, \cdot),$$\Gamma_{Z}(\cdot)$$\cdot\otimes\cdot,$$f_{*},$$f^{-1},$$f_{!},$$f^{!}$$\underline{or}_{X}$$\underline{M}_{X}$$\underline{or}_{Y/X}$

$ooo$

o

T$X,$ $T^{*}X,$ $T_{Y}X,$
1.1. $TX,$ $T^{*}X,$ $T_{Y}X,$-real manifold $X,$ $TX$ (resp. $T^{*}X$ ) denotes the tangent
For a(resp. cotangent) bundle to $X$ . If is a submanifold of $X,$ $T_{Y}X$ (resp.
$T_{Y}^{*}X$$T_{Y}^{*}X,$$c\infty$$Y$takes for the normal (resp. conormal) budle to Y.

$T_{Y}X)$$T_{Y}X\circ$and$T_{Y}^{*}X\circ$$T^{*}X\circ$,are defined by

T $X=TX\backslash T_{X}X$,$T^{*}X\circ=T^{*}X\backslash T_{X}^{*}X$

T $YX=T_{Y}X\backslash T_{Y}Y$,$T_{Y}^{*}X\circ=T_{Y}^{*}X\backslash T_{Y}^{*}$

1.2.
-map between
For adefined by$\varpi_{f},$$TX\circ,$

Y.

$\rho_{f}$$C^{\infty}$$C^{\infty}$-real manifolds$f$: $Yarrow X,$$\varpi_{f}$and$\rho_{f}$are$Y\cross T^{*}XX$$\prime^{f}\rho$$\varpi\backslash ^{J}$$T^{*}X$$T^{*}Y$

1.3.
For a vector bundle $Earrow Z,$
is the image ofsubset of $E,$.

$a$

$G^{a}$$a$$G$

is an antipodal map in $E$ . If
by thi map.
$s$1.4. Micro-support
We recall the definition of micro-support for sheaves.
2

$G$

is a

60
-real manifold and
DEFINITION ([K-S]). Let $X$ be aan object, is a $su$ bset ofof $D^{+}(X)$ . Then the micro-support of , denoted$T^{*}X$ defined as follows.
Let
be an open $su$ bset of $T^{*}X$ . Then
$C^{\infty}$$\mathcal{F}$$SS(\mathcal{F})$$\mathcal{F}$$U$$U\cap SS(\mathcal{F})=\emptyset$$\Leftrightarrow\{\begin{array}{l}foranyreaIC^{\infty}- function\phi onX(x_{1}.\cdot d\phi(x_{1}))\in Uimp1ies(R\Gamma_{\{\phi\langle x)\geqq\phi(x_{1})\}}(\mathcal{F}))_{x_{1}}=0\end{array}$

1.5. $D^{+}(X;\Omega)$be a subset of $T^{*}X$ .
Consider the same situation as above. Let
Then
is the set of arrows in $D^{+}(X)$ , given as follows. :if there exists a distinguished triangle
belongs to$\Omega$$S(\Omega)$$f$$\mathcal{F}arrow \mathcal{G}$$S(\Omega)$$?t$$+\swarrow^{1,}$$\backslash$$arrow^{f}$$\mathcal{F}$$\mathcal{G}$which satisfies$SS(H)\cap\Omega=\emptyset$.

is a multiplicative system of $D^{+}(X)$ . Then $D^{+}(X;\Omega)$ is
The set.
defined as the localization of $D^{+}(X)$ with respect to$S(\Omega)$$S(\Omega)$\S 2. THE FOURIER-SATO TRANSFORMATION.
2.1. We recall the definition of the Fourier-Sato transformaion from $[K-$$S]$ . The notion of Fourier-Sato transformation is due to Sato-KashiwaraKawai ([S-K-K]) although they defined it for sphere bundles. Let$Earrow rZ$ be a real vector budle with finite fibre dimension
over a locally compact topological space and $D_{conic}^{+}(E)$ be the full subcategory
of $D^{+}(E)$ consisting of complexes whose cohomology groups are locallybe a dual vector bundle
constant on any half-line of $E$ . Letof $E$ . Set$Z$$E^{*}arrow\pi Z$$D^{+}=\{(x,y)\in E\cross E^{*}|Z<x,y>\geqq 0\}$,$D^{-}=\{(x,y)\in E\cross ZE^{*}|<x,y>\leqq 0\}$.

Consider the diagram3

61$E\cross E^{*}Z$$\prime^{2}p$$\backslash ^{1}p$$J$$D^{\pm}$$E^{*}$For an objectof by$\mathcal{F}$$E$.

of $D_{conic}^{+}(E)$ , we define the Fourier-Sato transform$\mathcal{F}^{\wedge}$$\mathcal{F}$$\mathcal{F}^{\wedge}=Rp_{2*}R\Gamma_{D+}(p_{1}^{-1}\mathcal{F})=Rp_{2!}(p_{1}^{-1}\mathcal{F})_{D}-$.

2.2. micro-support of
Let $(z)$ be a coordinate system of $Z,$ $(z, x)$ that of $E$ and $(z, x;\zeta, \xi)$ theassociated coordinate system of $T^{*}E$ . Let $(z,y)$ be a coordinate system
for whichofand $(z, y;\zeta, \eta)$ the associated coordinate system ofis given by
the canonical pairing between $E$ and$\mathcal{F}^{A}$$T^{*}E^{*}$$E^{*}$$E^{*}$$<x,y>= \sum_{i}x_{i}y$ ;and for which the canonical l-forms of $T^{*}E$ andtively by$T^{*}E^{*}$$\omega_{E}=<\zeta,dz>+<\xi,dx>$

and$\omega_{E}\cdot=<\zeta,$$dz>+<\eta,dy>$ .

Then the canonical isomorphism$\Phi_{E}$:$T^{*}Earrow^{\underline\simeq}T^{*}E^{*}$

is defined by$(z, x;\zeta,\xi)(z,\xi;\zeta, -x)$

Under the above situation, we have
THEOREM 2.2.1 ([K-S, THEOREM 5.1.4]).
$SS(\mathcal{F}^{\wedge})=\Phi_{E}(SS(\mathcal{F}))$2.3. Another proposition from [K-S].

4

.

.

are given respec-

62-manifold and $E$ a real vector
PROPOSITION 2.3.1. Let $Y$ be a real$te$space wiih fini dimension. Let be a dosed convex cone (not necessarily proper) in $E$ with $O\in G$ . Set $X=Y\cross E$ and $X_{G}=Y\cross E_{G}$ . Hereis the space $E$ endowed with G-topology (see $\int K-S]$ for defnition).
Let be the natural continuous map$c\infty$$G$$E_{G}$$\phi$$\phi:Xarrow X_{G}$.

Then following daims hold.
$(a)$ For $\mathcal{F}\in Ob(D^{+}(X)),$
is contain $ed$ in $T^{*}Y\cross(E\cross G^{oa})$ if
is an isomorphism.
and only if the morphism$(b)$ For $\mathcal{F}\in Ob(D^{+}(X))$ , we have$SS(\mathcal{F})$$\phi^{-1}R\phi_{*}\mathcal{F}arrow \mathcal{F}$in$\phi^{-1}R\phi_{*}\mathcal{F}\cong \mathcal{F}$$D^{+}$( $X;T^{*}Y\cross(E\cross$ Int$G^{oa}$)).

\S 3. PURE SHEAVES.
We recall the definition of pure sheaves from [K-S].
3.1. Inertia index
be a real symplectic finite dimensional vector space; i.e. isLet
a non-degenerate skew symmetric bilinear form on the finite dimensional
R-vector space $E$ . Let be a linear subspace of $E$ . Set$\tau(\lambda_{1}, \lambda_{2}, \lambda_{3})$$(E, \sigma)$$\sigma$$’\rho$$\rho^{\perp}=$

Then
if

$\rho${ $x\in E|\sigma(x,y)=0$ foris called Lagrangian if$\rho^{\perp}\supset\rho$$\rho^{\perp}=\rho$.

DEFINITION 3.1.1([K-S]). Let
Here the quadratic form on$Q$$\lambda_{1},$$\forall y\in\rho$, involutive if$\lambda_{2},$$\lambda_{3}$$\lambda_{1}\oplus\lambda_{2}\oplus\lambda_{3}$}.

$\rho^{\perp}\subset\rho$

and isotropic

be Lagrangian planes of $E$ .
is defined by$Q(x_{1}, x_{2},x_{3})=\sigma(x_{1}, x_{2})+\sigma(x_{2}, x_{3})+\sigma(x_{3}, x_{1})$for$(x_{1}, x_{2}, x_{3})\in\lambda_{1}\oplus\lambda_{2}\oplus\lambda_{3}$. Then the index,$\tau_{E}(\lambda_{1}, \lambda_{2}, \lambda_{3})$is definedas the signat $ure$ of , that is the difference of the um $ber$ of positive
eigenval $ues$ and that of nega tive eigenvalues of .
$Q$

$n$

$Q$

3.2. Properties of the inertia index
if there is no fearhthe following part of this paper, we write foranda subset of $E$ .
isotropic
subspaceofbeanconfusion.
Letofis defined by
Then$\tau$$\tau_{E}$$E$$\rho$$\lambda^{\rho}$.

$\lambda^{\rho}=((\lambda\cap p^{\perp})+\rho)/\rho$5$\lambda$

63
PROPOSITION 3.2.1 ([K-S, PROPOSITION 7.1.2]). Let$\lambda_{i}$be Lagrangianplanes of E. Then we have following statements.
,(i) For all$s\in \mathfrak{S}_{3}$$\tau(\lambda_{1}, \lambda_{2}, \lambda_{3})=sgn(s)\tau(\lambda_{s(1)}, \lambda_{s(2)}, \lambda_{s(3)})$holds.
(ii) If$\rho$

is a $su$ bspace and satisfies$\rho\subset(\lambda_{1}\cap\lambda_{2})+(\lambda_{2}\cap\lambda_{3})+(\lambda_{3}\cap\lambda_{1})$then we have,.

$\tau_{E}(\lambda_{1}, \lambda_{2}, \lambda_{3})=\tau_{E^{\rho}}(\lambda_{1}^{\rho}, \lambda_{2}^{\rho}, \lambda_{3}^{\rho})$

In particular if$\lambda_{1}\cap(\lambda_{2}+\lambda_{2})\subset(\lambda_{1}\cap\lambda_{2})+(\lambda_{1}\cap\lambda_{3})$holds, we have$\tau(\lambda_{1}, \lambda_{2}, \lambda_{3})=0$.

3.3. Definition of pure sheaves
Let $X$ be a
the projection $T^{*}Xarrow X,$-real manifold,aa real function on $X$ and
Lagrangian submanifold of $T^{*}X,$$\{(x, d\phi(x));x\in X\}$ . For any point
in $T^{*}X,$ $T_{p}T^{*}X$ has a canonicalspace.
symplecticofvectorThen three Lagrangian planes instructure$T_{p}T^{*}X$ are defined by$c\infty$$\Lambda$$\pi$$Y_{\phi}=$$\phi$$p$$\lambda_{0}(p)=T_{p}(\pi^{-1}\pi(p))$,$\lambda_{\Lambda}(p)=T_{p}\Lambda$

and.
DEFINITION 3.3.1. Under the above situation, we say that is transver$sai$ toat if $\phi(\pi(p))=0$ and if
and interse transversally at .
LEMMA 3.3.2. Let be a Lagrangian submanifold of $T^{*}X,$ a poinof A and an object of $D^{+}(X)$ . Assume that in a neighborhood of ,holds. Let be a real function on $X$ and transversal toat . Let be a number which satisfies$\lambda_{\phi}(p)=T_{p}Y_{\phi}$$\phi$$\Lambda$$p$$Y_{\phi}$$ct$$\Lambda$$p$$\Lambda$$P$$\mathcal{F}$$p$$SS(\mathcal{F})\subset\Lambda$

$p$

$t$

$\Lambda$$\phi$$j$$j \equiv\frac{1}{2}(\dim X+\dim(\lambda_{0}(p)\cap\lambda_{\Lambda}(p)))$mod Z.

Then the cohomology group$H_{\{x|(x)\geqq 0\}}^{j+\frac{1}{\phi 2}\tau_{\phi}(p)}(\mathcal{F})_{\pi(p)}$does $not$ depend on$\phi$

where.
After these preparations, we can define pure sheaves.
$\tau_{\phi}(p)=\tau(\lambda_{0}(p), \lambda_{\Lambda}(p),$6$\lambda_{\phi}(p))$

$4
DEFINITION 3.3.3([K-S]). Let A $be$ a Lagrangian submanifold of $T^{*}X$ ,, and $\mathcal{F}\in Ob(D^{+}(X))$ . We
in a neigh\’oorhoodof . If we have, for a real function
tran sversai to A at and A$M$
maodule ,$assumeSS(\mathcal{F})\subset\Lambda$$p\in\Lambda$$\phi$$p$$p$$H_{\{x|\phi(x)\geqq 0\}}^{j}(\mathcal{F})_{\pi(p)}=\{\begin{array}{l}M,forj=-d+\frac{1}{2}d_{\mathring{1}}mX+\frac{1}{2}\tau_{\phi}(p)\circ\}0,otherwise\end{array}$withtype $M$ with shift$\tau_{\phi}(p)=\tau(\lambda_{0}(p), \lambda_{\Lambda}(p),$$d$$\lambda_{\phi}(p))$

along A at .

, then we say that$\mathcal{F}$is pure of$p$

3.4. Properties of pure sheaves
We recall properties of pure sheaves from [K-S].

PROPOSITION 3.4.1 ([K-S, PROPOSITION 7.2.8, 7.2.9]).
$p\circ$ a point of A(i) Le A be a Lagrangian submanifold of $T^{*}X,$
is pure of type $M$ with shift
an object of $D^{b}(X)$ . Assume that
$Ext^{j}(M,A)=0(j\neq0)$ . Thenisalong A at and thatat .
pure of type $Hom(M, A)$ with $shift-d$ along(ii) Letbe a Lagrangian submanifold ofa point ofis pure of typeandan object of $D^{+}(X_{j})$ . Assume thatwithat $p_{j}(j=1,2)$ . Let
be the j-th projection ona long$8I2d\mathcal{F}$

$f$

$d$

$\mathcal{F}$$R\underline{Hom}(\mathcal{F},\underline{A}_{X}\grave{)}$$p$$\Lambda^{a}$$p^{a}$$T^{*}X_{j},$$\Lambda_{j}$$M_{j}$$\mathcal{F}_{j}$$\mathcal{F}_{j}$$sb_{\check{I}}ftd_{j}$$\Lambda_{j}$$X_{1}\cross X_{2}$$q_{j}$.

then
If $Tor_{j}(M_{1}, M_{2})=0$ for
typewith $shiftd_{1}+d_{2}$ alongat$(b)$ If $Ext^{j}(M_{1}, M_{2})=0$ for, thenpure of $tp^{r}peHom(M_{1}, M_{2})$ with shift $d_{2}-d_{1}$ along$(a)$$q_{1}^{-1}\mathcal{F}_{1}\otimes Lq_{2}^{-1}\mathcal{F}_{2}$$\forall j\neq 0_{f}$$M_{1}\otimes M_{2}$$f$: $Yarrow X$ be a$C^{\infty}$-map between

is ure of$p$$(p_{1},p_{2})$$\Lambda_{1}\cross\Lambda_{2}$$\forall j\neq 0$Let$A_{j}$$p_{j}$.

$R\underline{Hom}(q_{1}^{-1}\mathcal{F}_{1}, q_{2}^{-1}\mathcal{F}_{2})$$C^{\infty}$$\Lambda_{1}^{a}\cross\Lambda_{2}$atis$(p_{1}^{a},p_{2})$.

-manifolds.

THEOREM 3.4.2 ([K-S, THEOREM 7.3.1]). Let
manifold of $T^{*}Y,$ a point of $Y\cross T^{*}XX$ and$p$$\Lambda$$\mathcal{G}$be a Lagrangian suban object of $D^{+}(Y)$ .

$Assume$ :(i) is proper over supp ,is transversal to at and(ii)
is isomorph ic to a submanifold
of $T^{*}X$ ,
,
(iii)
$M$
(iv) is $pure$ of typewith $shiftd$ along A at.
is a Lagrangian submanifold andThen
is pure of type $M$at
where
Wtth shift along$f$$(\mathcal{G})$$\Lambda$$\rho_{f}$$\varpi_{f}\rho_{f}^{-1}(\Lambda)$$p$$\Lambda_{0}$$\rho_{f}^{-1}(SS(\mathcal{G}))\cap\varpi_{f}^{-1}\varpi_{f}(p)\subset\{p\}$$\rho_{f}(p)$$\mathcal{G}$$Rf_{*}(\mathcal{G})$$\Lambda_{0}$$d’$$\Lambda_{0}$$\varpi_{f}(p)$$d’-d= \frac{1}{2}(\dim X-\dim Y)-\frac{1}{2}\tau(\lambda_{0}(\rho_{f}(p)), \lambda_{\Lambda}(\rho_{f}(p)),$7$\rho_{f}\varpi_{f}^{-1}(\lambda_{0}(\varpi_{f}(p))))$.

65
are the same as
THEOREM 3.4.3 ([K-S, THEOREM 7.3.3]). $f,$ $X,$
those of Theorem 3.4.2. Let A be a Lagrangian submanifold of $T^{*}X,$a point of
A and an object of $D^{+}(X)$ . Assume:
$Y$

$p$

$\mathcal{F}$$Y\cross X$(i) is non-characteristic for ,(ii)
is tran sversal to A at andof $T^{*}Y$ ,submanifold$f$$\mathcal{F}$$p$$\varpi_{f}$is isomorphic to a$\rho_{f}\varpi_{f}^{-1}(\Lambda)$$\Lambda_{0}$,(iv) is pure of type $M$ with shift a longThen
is a Lagrangian submanifold anat.
with shift a long$(j\ddot{u})\varpi_{f}^{-1}(SS(\mathcal{F}))\cap\rho_{f}^{-1}\rho_{f}(p)\subset\{p\}$$d$$\mathcal{F}$$\Lambda$

at$\varpi_{f}(p)$$df^{-1}(\mathcal{F})$$\Lambda_{0}$$d$.

is pure of type $M$$\rho_{f}(p)$$\Lambda_{0}$THEOREM 3.4.4 ([K-S, COROLLARY 7.3.4]). Let $X$ and be-real$X\crossY$$X$
manifolds, and the projections fromto and respectively
the proje ctions from $T^{*}(X\cross Y)=T^{*}X\cross T^{*}Y$ to $T^{*}X$andandand $T^{*}Y$ respectively. Set $p_{j}^{a}=p_{j}oa(j=1,2)$. Let A be a Lagrangiana Lagrangian submanifold of $T^{*}Y$ andsubmanifold of $T^{*}(X\cross Y),$a point of A. Set $p_{Y}=p_{2}(p)$ and $px=p_{1}^{a}(p)$ . Let
be an object of’$D^{b}(X\cross Y)$ an$D^{+}(Y)$. Assume:
an object of(i)
at and(Ay) A is isomorph to ais transversal tosubmanifold $\Lambda_{X}ofT^{*}X$ by ,(ii) is pure of type $M$ with shift a long at ,is pure of type $N$ with shift along
at ,
,
(iv)
is proper over,(v)(vi) $(SS(\mathcal{K})\cross SS(\mathcal{F}))\cap(T_{X}^{*}X\cross T^{*}Y)\subset T_{X}^{*}X\cross T_{Y}^{*}Y$ holds in
$Y$

$Y$

$q_{2}$$q_{1}$$c\infty$$p_{2}$$p_{1}$$\Lambda_{Y}$$\mathcal{K}$$p$$d\mathcal{F}$$\Lambda_{Y}$$p_{2}|_{\Lambda}$$p_{2}^{-1}$

$p$

$ic$

$\cap$$p_{1}^{a}$

rc

$d$

$d’$

$(i\ddot{n})\mathcal{F}$$\Lambda$$p$$\Lambda_{Y}$$p_{Y}$$supp(\mathcal{K})\cap q_{2}^{-1}(supp(\mathcal{F}))$$q_{1}$$(p_{1}^{a})^{-1}(p_{x})\cap SS(\mathcal{K})\subset\{p\}$$a$$T^{*}Y$.
neighborhood of(vii) $Ext^{j}(M, N)=0$ for
Then
at
where
along
$\pi_{X}(p_{X})$$\forall j\neq 0$$Rq_{1*}R\underline{Hom}(\mathcal{K},q_{2}^{-1}\mathcal{F})$$\Lambda_{X}$.

is pure of type $Hom(M, N)wi$ th shift $d”$

$p_{X}$$d”=d’-d- \frac{1}{2}\dim Y+\frac{1}{2}\tau$

and$\tau=\tau(\lambda_{0}(p), \lambda_{\Lambda}(p),$ $\lambda_{0}(p_{X}^{a})\cross\lambda_{\Lambda_{Y}}(p_{Y}))$$=\tau(\lambda_{0}(p_{Y}),p_{2}(\lambda_{\Lambda}(p)\cap(p_{1}^{a})^{-1}(\lambda_{0}(p_{X}))),$$\lambda_{\Lambda_{Y}}(p_{Y}))$.

Thi proposition describes the contact transformation of pure sheaves.
$s$

8

66
3.5. Microlocal uniqueness of pure sheaves.
The following fact is important.([K-S]) Let
and be object of$M$$D^{+}(X)$ . Assume
is pure of typewith shift along at . Thenwithshiftd alongA atpifand only if
is pure of typeM$\mathcal{F}$$\mathcal{G}$$d$$\mathcal{F}$$s$$\Lambda$$p$$\mathcal{G}$in$\mathcal{F}\cong \mathcal{G}$$D^{+}(X;\{p\})$holds.

\S 4. THE $FoURIER- SATO$ TRANSFORMATION OF PURE SHEAVES.
4.1. The Main Theorem.

THEOREM 4.1.1. Let $Earrow Z$ be an R-vector bun $dle$ with finite fibre dimension over a $lo$ cally compac topologica 1 space and A a Lagrangian. Assume
submanifold of $T^{*}E$ . Let $\mathcal{F}\in Ob(D_{conic}^{+}(E))$ andis pure of type $M$ with shift along A at . Then the Fourier-Satoof is pure of type $M$ with shift along
wheretransform
$Z$

$t$

$n$

$p\in\Lambda$$d$$\mathcal{F}^{A}$

$p$

$d’$

$\mathcal{F}$$p^{*}=\Phi_{E}(p)$$\Lambda^{*}at$,$\Lambda^{*}=\Phi_{E}(\Lambda)$,

and$d’=d- \frac{n}{2}+\frac{1}{2}\tau(\lambda_{0}(p), \Phi_{E}^{-1}(\lambda_{0}(p^{*})),$PROOF;

Identify$Z$$\lambda_{\Lambda}(p))$

with the zero sections of $E$ and$Eo=E\backslash Z$,$S=E/R^{+}\circ$$E^{*}$,$S^{*}=E^{*}/R^{+}o$$D_{E}=\{(x, y)\in E\cross ZE^{*}|<x,y>\geqq 0\}$,

$D_{\Phi,E}=\{(x, y)\in E\cross E^{*}|ooZ<x,y>\geqq 0\}$,$D_{S}=\{(x, y)\in S\cross ZS^{*}|<x,y>\geqq 0\}$

$D_{0}=D_{E}\cross_{Z}(E^{*}x^{o}EEExEz^{E)}’$$D\circ=Ds_{S_{Z}}\cross(S^{*}x^{o}SE\cross s*z^{E)}$First we give three lemmas.

9

,

.

. Set

$E^{*}o=E^{*}\backslash Z$,$\mathcal{F}$$p^{*}$

67
LEMMA 4.1.2. Consider the diagram,$E^{*}\cross EooZ$$D_{0,E}$$p$$p\backslash ^{11}$$Eo$$E^{*}o$$\downarrow s_{3}$$\downarrow s_{1}$$\downarrow f_{2}p$$S^{*}x^{o}z^{E}$

,

$s*$

$D_{SE}o$$S$$\downarrow s_{2}$$\downarrow s_{4}$$ps\nearrow_{1}$$P^{\backslash _{32}}$$\underline{i_{S}}$$S^{*}\cross SZ$

Then for an arbitrary object$\mathcal{F}$of$D_{conic}^{+}(E)\circ$$D_{S}$.

, we have.

$\mathcal{F}^{\wedge 0}=f_{2}^{-1}((Rf_{1*}\mathcal{F})^{\wedge S})$

Here for $\mathcal{F}\in Ob(D_{conic}^{+}(E))0$ and $\mathcal{G}\in Ob(D^{+}(S)),$
defined by$\mathcal{F}^{Ao}$and$\mathcal{G}^{\wedge S}$

are$\mathcal{F}^{Ao}=Rp_{12_{*}}R\Gamma_{D_{o}}p_{11^{-1}}\mathcal{F}B$$\mathcal{G}^{AS}=Rp_{32*}R\Gamma_{D_{S}}p_{31^{-1}}\mathcal{G}$PROOF OF LEMMA 4.1.2: Since.

$R\Gamma_{D_{S}}(\cdot)=Ri_{3*}i_{3}^{!}(\cdot)$, we have(4.1.2.1)$f_{2}^{-1}((Rf_{1*}\mathcal{F})^{\wedge S})=f_{2}^{-1}Rp_{32_{*}}R\Gamma_{D_{S}}p_{31}^{-1}Rf_{1_{*}}.\mathcal{F}$$=f_{2}^{-1}Rp_{32*}Ri_{3*}i_{3}^{!}p_{31}^{-1}Rf_{1*}\mathcal{F}$.

is a topological submersion of codimension 1, we
From the fact that
Poincar\’e-Verdier
duality theorem,
get, by$s_{2}$(4.1.2.2).

$p_{31}^{-1}Rf_{1*}\mathcal{F}\otimes\underline{or}_{(s*x^{l}E)/(S\cross S)}[1]=Rs_{2*}s_{1}^{!}p_{31}^{-1}Rf_{1*}\mathcal{F}$10

68
Now we remark that the following part of the diagram in Lemma 4.1.2
is a Cartesian diagram.
$S^{*}\cross Eoarrow^{:_{2}}D_{SE}\circ$$\downarrow^{Z_{s_{2}}}$$\downarrow s_{4}$$S^{*}\cross Szarrow^{i_{3}}$$D_{S}$

From (4.1.2.1),(4.1.2.2) and this fact, we deduce(4.1.2.3)$f_{2}^{-1}((Rf_{1*}\mathcal{F})^{\wedge S})\otimes\underline{or}_{(Sx^{o}E)/(SxS)}[1]$$=f_{2}^{-1}Rp_{32*}Ri_{3*}i_{3}^{!}Rs_{2*}s_{2}^{!}p_{31}^{-1}Rf_{1*}\mathcal{F}$$=f_{2}^{-1}Rp_{32*}Ri_{3*}Rs_{4*}i_{2}^{!}s_{2}^{!}p_{31}^{-1}Rf_{1*}\mathcal{F}$$=f_{2}^{-1}Rp_{32*}Rs_{2*}Ri_{2*}i_{2}^{!}s_{2}^{!}p_{31}^{-1}Rf_{1*}\mathcal{F}$$=f_{2}^{-1}Rp_{22*}Ri_{2*}i_{2}^{!}s_{2}^{!}p_{31}^{-1}Rf_{1*}\mathcal{F}$Since$p_{31}$.

is a topological submersion of codimension $(n-1)$ ,(4.1.2.4)$p_{31}^{-1}Rf_{1*}\mathcal{F}\otimes\underline{or}_{\langle SxS)/S}[n-1]=p_{31}^{!}Rf_{1*}\mathcal{F}$Remarking that$s_{2}$.

is a topological submersion and that the diagram$s*\downarrow^{x_{s_{2}}^{o}}z^{E}arrow^{p_{21}}E_{f_{1}}\downarrow\circ$$S^{*}\cross Szarrow^{p_{\theta 1}}S$

is a Cartesian diagram, we have, by (4.1.2.3) and (4.1.2.4),(4.1.2.5)$f_{2}^{-1}((Rf_{1*}\mathcal{F})^{AS})\otimes\underline{or}_{(SxE)/s^{[n]}}$$=f_{2}^{-1}Rp_{22*}Ri_{2*}i_{2}^{!}s_{2}^{!}p_{31}^{!}Rf_{1*}\mathcal{F}$$=f_{2}^{-1}Rp_{22*}Ri_{2*}i_{2}^{!}s_{2}^{!}Rs_{2*}p_{21}^{!}\mathcal{F}$$=f_{2}^{-1}Rp_{22*}Ri_{2s}i_{2}^{!}p_{21}^{!}\mathcal{F}\otimes\underline{or}_{(Sx^{o}E)/(S^{*}xS)}[1]$

11

.

69
Now

$s_{1}$

is a opological submersion of codimension 1 and the diagram$t$$E\cross E^{*}o0arrow^{:_{1}}D_{0}$$s\epsilon\downarrow^{E}$$s_{1}\downarrow Z$$S^{*}x^{o}z^{E}arrow^{i_{2}}D_{SE}\circ$is Cartesian. Thus we have from (4.1.2.5),$f_{2}^{-1}((Rf_{1*}\mathcal{F})^{AS})\otimes\underline{or}_{(ExE)/s^{[n]}}ooz$$=f_{2}^{-1}Rp_{22*}Ri_{2*}i_{2}^{!}Rs_{1*}s_{1}^{!}p_{21}^{!}\mathcal{F}$$=f_{2}^{-1}Rp_{22*}Ri_{2*}Rs_{3*}i_{1}’s_{1}^{!}p_{21}^{!}\mathcal{F}$$=f_{2}^{-1}Rp_{22*}Rs_{1*}Ri_{1*}i_{1}^{!}s_{1}^{!}p_{21}^{!}\mathcal{F}$$=f_{2}^{-1}Rf_{2*}Rp_{12*}Ri_{1*}i_{1}^{!}s_{1}^{!}p_{21}^{!}\mathcal{F}$$=Rp_{12*}Ri_{1*}i_{1}^{!}p_{11}^{!}\mathcal{F}=(*)$

Since$p_{11}$.

is a topological submersion of codimension , we have, more$n$

over,$(*)\underline{or}_{(E^{*}\cross E)/E}[n]$$=RR\Gamma_{D_{0}}\mathcal{F}\otimes oooBz$$=\mathcal{F}^{\wedge 0}\otimes\underline{or}_{t_{z}^{ooo}E^{*}xE)/E}[n]$.

Then we have$f_{2}^{-1}((Rf_{1*}\mathcal{F})^{AS})=\mathcal{F}^{Ao}$

1
12

.

70
LEMMA 4.1.3. Consider th diagraln$e$,$E^{*}\cross EooZ$$D_{E}\circ$$p$$E^{*}o$$p\backslash ^{11}$$E\circ$$]|1$$\}2p$,$E^{*}$$)|s$$E^{*}\cross EZ^{\circ}$$D_{EE}o$$E$$1^{1}\cdot 2$$1^{i_{4}}$$p\nearrow_{S1}$$P32\backslash$$\underline{ja}$$E^{*}\cross Ez$

Then for an arbitrary object$\mathcal{F}$$D_{E}$.

of $D_{conic}^{+}(E)$ ,,$(g_{1}^{-1}\mathcal{F})^{Ao}$$+$$\backslash$$g_{2}^{-1}q_{2}^{-1}Rq_{1_{*}}R\Gamma_{Z}\mathcal{F}$$arrow$

is a distinguished triangle where
and the $nat$ ural projection$q_{1}$$g_{2}^{-1}(\mathcal{F}^{A})$is the $nat$ ural projection $Earrow^{q_{1}}Z$$E^{*}arrow^{q_{2}}Z$$q_{2}$.

PROOF OF LEMMA 4.1.3: Remark that if is an open inclusion,. Now
are open inclusions.
andcoincides wi th
Taking into account of the fac that diagrams$i^{!}$$i$$i^{-1}$$g_{1},$$g_{2},$$i_{1},$$i_{2},$$i_{3}$$t$$E^{*}\circ\cross Eoarrow^{j_{1}}$$D_{\mathscr{O}}$$i_{\theta}\downarrow^{E}$$i_{1}\downarrow^{Z}$$E^{*}x^{o}z^{E}arrow^{j_{2}}D_{EE}\circ$13$i_{4}$

71$E^{*}oarrow^{p_{12}}E^{*}\cross Eoo$$i_{1}\downarrow^{Z}$$g_{2}\downarrow$$E^{*}arrow^{p_{22}}E^{*}x^{o}z^{E}$

and$E^{*}E:_{2}\downarrow^{x^{o}}zarrow^{j_{2}}D_{E^{o}}:_{4}\downarrow^{E}$$E^{*}\cross EZarrow^{j_{\theta}}D_{E}$are Cartesian diagrams, we have by Poincar\’e-Verdier duali $ty$ theorem(4.1.3.1)$(g_{1}^{-1}\mathcal{F})^{Ao}=Rp_{12*}R\Gamma_{D_{o}}p_{11}^{-1}g_{1}^{-1}\mathcal{F}B$$=Rp_{12*}Rj_{1_{*}}j_{i^{p_{11}^{-1}g_{1}^{-1}\mathcal{F}}}^{l}$$=Rp_{12*}Rj_{1_{*}}j_{i^{i_{1}^{-1}p_{21}^{-1}g_{1}^{-1}\mathcal{F}}}^{1}$$=Rp_{12*}Rj_{1*}i_{3}^{!}j_{2}^{!}p_{21}^{-1}g_{1}^{-1}\mathcal{F}$$=Rp_{12*}i_{1}^{!}Rj_{2*}j_{2}^{!}p_{21}^{-1}g_{1}^{-1}\mathcal{F}$$=g_{2}^{!}Rp_{22_{*}}Rj_{2*}j_{2}^{!}p_{21}^{-1}g_{1}^{-1}\mathcal{F}$$=g_{2}^{!}Rp_{22*}Rj_{2*}j_{2}^{!}i_{2}^{-1}p_{31}^{-1}\mathcal{F}$$=g_{2}^{!}Rp_{22_{*}}Rj_{2_{*}}j_{2}^{!}i_{2}^{!}p_{31}^{-1}\mathcal{F}$$=g_{2}^{!}Rp_{22_{*}}Rj_{2*}i_{4}^{!}j_{3}^{!}p_{31}^{-1}\mathcal{F}$$=g_{2}^{-1}Rp_{22*}i_{2}^{!}Rj_{3*}j_{3}^{!}p_{31}^{-1}\mathcal{F}$$=g_{2}^{-1}Rp_{22*}i_{2}’R\Gamma_{D_{B}}p_{31}^{-1}\mathcal{F}$$=g_{2}^{-1}Rp_{32_{*}}Ri_{2*}i_{2}^{!}R\Gamma_{D_{B}}p_{31}^{-1}\mathcal{F}$$=g_{2}^{-1}Rp_{32*}R\Gamma$.

$oR\Gamma_{D_{B}}p_{31}^{-1}\mathcal{F}$$ExEz$

14

72
Apply the functor

to the distinguished triangle$g_{2}^{-1}Rp_{32*}R\Gamma_{D_{B}}$RF,$op_{31}^{-1}\mathcal{F}$$E^{*}xEz$$+$$\backslash$$arrow$$R\Gamma_{E\cross\{0\}}p_{31}^{-1}\mathcal{F}$$p_{31}^{-1}\mathcal{F}$.

Then combinig this with (4.1.3.1), we get the distinguished triangle,$(g_{1}^{-1}\mathcal{F})^{Ao}$$+$$\backslash$$g_{2}^{-1}q_{2}^{-1}Rq_{1*}R\Gamma_{Z}\mathcal{F}$$arrow$$g_{2}^{-1}(\mathcal{F}^{\wedge})$.

1
LEMMA 4.1.4. Let $(z)$ be a coordin ate system of $Z,$ $(x)$ that of and$(y)$ that of: $S^{*}\cross ZSarrow S^{*}\cross S$ and. Define two inclusions$S$$S^{*}$$i_{2}$$i_{1}$: $D_{S}arrow S^{*}\cross S$ as they embed$Z$in to the diagonal set of$S^{*}\cross SZ\ni(z;x,y)\underline{i_{1}}((z;x), (z;y))\in S^{*}\cross S$$D_{S}\ni(z;x,y)((z;x),(z;y))\in S^{*}\cross S\underline{i_{2}}$.

Consider the diagram$D_{S}$$S^{*}\cross SZ$$\downarrow i_{1}$$s*$$s*xS$$arrow^{p_{2}}$Then for an arbitrary object$\mathcal{F}$$arrow^{p_{1}}$$S$.

of $D^{+}(S)$ , we have.

$Rp_{2_{*}}R\Gamma_{D_{S}}p_{1}^{-1}\mathcal{F}\otimes\underline{or}_{(S\cross S)/\langle SxS)}[-\dim Z]\sim\sim.$$=Rp_{2*}R\Gamma_{D_{S}}p_{1}^{-1}\mathcal{F}$15$Z\cross Z$; i.e.

73$(i.e. \mathcal{F}^{AS}\otimes\underline{or}_{(SxS)/(S^{*}\cross S)}[-\dim Z]=Rp_{2*}R\Gamma_{D_{S}}p_{1}^{-1}\mathcal{F})$.

is a topological submersion of
PROOF OF LEMMA 4.1.4: Remark that
codimension $(n-1)$ and that a topological submersion of codimension$(n-1+\dim Z)$ . Then by Poincar\’e-Verdier duality theorem, we get$p_{1}\sim$$p_{1}$.

$Rp_{2*}R\Gamma_{D_{S}}p_{1}^{-1}\mathcal{F}\otimes\underline{or}_{(S\cross S)/S}[n-1]\sim\sim$$=Rp_{2*}R\Gamma_{D_{S}}p_{1}^{!}\mathcal{F}\sim\sim$$=Rp_{2*}Ri_{3*}i_{3}^{!}i_{1}^{!}p_{1}^{!}\mathcal{F}\sim$.

$=Rp_{2*}Ri_{3*}i_{3}^{!}i_{1}^{!}p_{1}^{-1}\mathcal{F}\otimes\underline{or}_{(S\cross S)/S}[n-1\sim.+\dim Z]$

From this, we deduce$R\tilde{p}_{2*}Rr_{D_{S}p_{1}^{-1}\mathcal{F}\otimes\underline{or}_{(SxS)/\langle S^{*}xS)z}[-\dim Z]}^{\sim}$$=R\tilde{p}_{2*}Ri_{3*}i_{3}^{!}i_{1}^{!}p_{1}^{-1}\mathcal{F}$$=Rp_{2*}Ri_{1*}Ri_{3*}i_{2}^{!}p_{1}^{-1}\mathcal{F}$$=Rp_{2*}Ri_{2*}i_{2}^{!}p_{1}^{-1}\mathcal{F}$$=Rp_{2_{*}}R\Gamma_{D_{S}}p_{1}^{-1}\mathcal{F}$.

I
Now we enter into the Proof of Theorem 4.1.1.
.
1. Proof in the case thatcase, we have an isomorphism
this
in
Since
$p\in\Lambda\cap T^{*}Eoo$$p^{*}\not\in SS(q_{2}^{-1}Rq_{1*}R\Gamma_{Z}\mathcal{F})$(4.1.1.1)$(g_{1}^{-1}\mathcal{F})^{\wedge 0}\cong g_{2}^{-1}(\mathcal{F}^{A})$in$D^{+}(E;\circ\{p^{*}\})$by Lemma 4.1.3.
Define$ps=\varpi_{f}1\rho_{f_{1}}^{-1}(p)$$\Lambda_{S}=\varpi_{f_{1}}\rho_{f_{1}}^{-1}(\Lambda)$,

using the notation in Lemma 4.1.2. It is known by [K-S, Proposition
5.1.1], that an object of $D^{+}(E)$ belongs to $Ob(D_{conic}^{+}(E))$ if and only
if $SS(F)$ is contained inwhere
is the characteristic variety of$E$
the Euler vector field on ; i.e. $S_{E}=\{(z,x;\zeta,\xi)\in T^{*}E|<x,\xi>=0\}$
where $(z, x)$ is a coordinate system of with the fibre coordinate system$(x)$ and its dual coordinate system. Then we may regard A as a,ofsubmanifold
andthiscoincides with. Since is
Lagrangian
in a neighborhood of
in $T^{*}E,$ coincides with. Consequently
is pure with shift along A at .
$\mathcal{F}$$S_{E}$$S_{E}$$E$$(\zeta, \xi)$$T^{*}S$$\mathcal{F}$$g_{1}^{-1}\mathcal{F}$$\Lambda_{S}$$\pi(p)$$g_{1}^{-1}\mathcal{F}$

$d$

$p$

16

$p$

74
CLAIM 1.

$Rf_{1*}g_{1}^{-1}\mathcal{F}$is pure of type $M$ with shift

along

$d$

1: Let $\mathcal{G}\in Ob(D^{+}(S))$ , and assume
along A at . We have an isomorphism

PROOF OF CLAIM

with shift$d$$\Lambda_{S}$at$f_{1}^{-1}\mathcal{G}$$p_{S}$.

is pure$p$$f_{1}^{-1}\mathcal{G}\cong g_{1}^{-1}\mathcal{F}$in$D^{+}(E;\{p\})0$.

Then by Proposition 2.3.1 we have an isomorphism$\mathcal{G}\cong Rf_{1*}g_{1}^{-1}\mathcal{F}$in$D^{+}(S;\{p_{S}\})$.

Remark that pure heaves are micro-locally unique (see 3.5). Therefore itis enough to show that if $\mathcal{G}\in Ob(D^{+}(S))$ is pure with shift along
is pure with shift alongat . Sinceis a
atis smooth,projection, is non-characteristic for . Now sinceat the point. Remark thatis transversal tois. Now we havean injection and
$s$

$d$

$p_{S},$$d$$f_{1}^{-1}\mathcal{G}$$E\circarrow^{f_{1}}S$$\Lambda$$f_{1}$$\Lambda_{S}$$p$$\mathcal{G}$$\varpi_{f_{1}}$$\varpi_{f_{1}}$$p_{0}=\rho_{f_{1}}^{-1}(p)$$\rho_{f_{1}}^{-1}(\Lambda)$$\rho_{f_{1}}$$\varpi_{f_{1}}(p_{0})=ps$$\rho_{f_{1}}\varpi_{f_{1}}^{-1}(\Lambda_{S})\cong\Lambda$

and$\varpi_{f_{1}}^{-1},(\dot{S}S(\mathcal{G}))\cap\rho_{f_{1}}^{-1}\rho_{f_{1}}(p_{0})\subset\{p_{0}\}$.

Then it follows from Theorem 3.4.3 (the theorem of inverse image ofis pure with shift along A at . I
pure sheaves) that$d$$f_{1}^{-1}\mathcal{G}$$p$

CLAIM 2. If $\mathcal{G}\in Ob(D^{+}(S))$ is pure of type $M$ wi th shift along
is pure of type $M$ with shift alongatwhereat ps, then
$d$

$d’$

$\mathcal{G}^{AS}$$\Lambda_{S}$$p_{S}^{*}$$\Lambda_{S}^{*}$$p_{S}^{*}=\Phi_{S}(p_{S})$$\Lambda_{S}^{*}=\Phi_{S}(\Lambda_{S})$and$d’=d- \frac{n}{2}+\frac{1}{2}\tau(\lambda_{0}(p), \Phi_{E}^{-1}(\lambda_{0}(p^{*})),$ $\lambda_{\Lambda}(p))$.

This claim is essential. The proof depends on the fact that consideredon , Fourier-Sato transformation is a contact transformation. This
claim is proved by use of Proposition 3.4.4 (the proposition of contact
transformation of pure sheaves).
PROOF OF CLAIM 2: Consider the following diagram and that ofLemma 4.1.4.
$S$

17

75

$T^{*}(S^{*}\cross S)\circ$$\prime^{2}r$$\backslash ^{1}r$$Ji$$T^{*}S^{*}o$$T^{*}So$$\Lambda_{K}=T_{D_{S}}^{*}(S^{*}o\cross S)$

Here
is embedded into
in the same way as Lemma 4.1.4; i.e.
he base space ofis embedded into the diagonal ofin.
Set$S^{*}\cross S$$D_{S}$$Z$$t$$Z\cross Z$$D_{S}$$D_{S}o=\{(x,y)\in S^{*}\cross S|Z<x, y>=0\}$

Here $(x)$ is the fibre coordinate system ofsystem. Remark that

$S$

$S^{*}\cross S$.

and $(y)$ its dual coordinate$T_{D_{S}}^{*}(S^{*}\circ\cross S)=T_{D_{S}}^{*_{l}}(S^{*}o\cross S)$

and$Rp_{2*}R\Gamma_{D_{S}p_{1}^{-1}}\mathcal{G}=Rp_{2*}R\underline{Hom}(\underline{A}_{D_{S}}\circ,p_{1}^{-1}\mathcal{G})$.

In the following part, we show that the conditions in Proposition 3.4.4
are satisfied. Let $(z)$ be a coordinate.system of $Z,$
its associatedcoordinate system of $T^{*}Z,$ $(z,x)$ a homogeneous coordinate system of ,$(z, x;\zeta, \xi)$ its associated homogeneous coordinate system of $T^{*}S,$ $(z, y)$
the dual of $(z, x)$ and $(z, y;\zeta,\eta)$ the dual of $(z,x;\zeta, \xi)$ . We have$(z, \zeta)$$S$$(z, x;\zeta,\xi)\in T^{*}So\Leftrightarrow|x|=1,$$|\xi|\neq 0,$$<x,$ $\xi>=0$$(z,y;\zeta, \eta)\in T^{*}S^{*}\circ\Leftrightarrow|y|=1,$ $|\eta|\neq 0,$$<y,$ $\eta>=0$and$(z_{1}, z_{2},x,y;\zeta_{1}, \zeta_{2},\xi, \eta)\in T_{o}^{*}(S^{*}\cross S)=\Lambda_{K}0_{D_{S}}$$\Leftrightarrow\{\begin{array}{l}z_{1}=z_{2},|x|=|y|=l,\zeta_{1}+\zeta_{2}=0\exists t\in R\backslash \{0\}s.t.\xi=ty,\eta=tx\end{array}$Since Claim 2 is a local statement, we may take a neighborhood, Ag totoof $ps$ inand may restrictandto. From now on, we work in the situation under thi restriction. Thus we have the equivalence$\Omega_{S}$$T^{*}So$$\Lambda_{S}$$\Omega_{S}$$\Phi_{S}(\Omega_{S})^{a}\cross\Omega_{S}$$\Phi_{S}(\Omega_{S})$$\Lambda_{K}$$s$$(z_{1}, z_{2},x,y;\zeta_{1}, \zeta_{2}, \xi,\eta)\in\Lambda_{K}$

18

76

$\Leftrightarrow\{\begin{array}{l}z_{1}=z_{2},|x|=|y|=1,\zeta_{l}+\zeta_{2}=0\exists t\in R^{+}s.t.\xi=ty,\eta=tx\end{array}$Then$r_{1}|_{\Lambda_{K}}$:$\Lambda_{K}arrow\Omega_{S}$is a diffeomorphism. In fact$r_{1}|_{\Lambda_{K}}((z, z, x, y;\zeta, -\zeta,ty, tx))=(z, x;\zeta,ty)$and$(r_{1}|_{\Lambda_{K}})^{-1}((z, x; \zeta, \xi))=(z, z, x, \frac{\xi}{|\xi|};\zeta, -\zeta, \xi, |\xi|x)$.

The map$r_{2}^{a}|_{\Lambda_{K}}$is also diffeomorphic. Let
at $pss$ since
sal tohave$\Lambda_{S}$:$\Lambda_{K}arrow\Phi_{S}(\Omega_{S})$$r_{1}|_{\Lambda_{K}}$and$r_{2}^{a}|_{\Lambda_{K}}$. Thenis transverare diffeomorphic. Moreover we$p_{SS}=(r_{1}|_{\Lambda_{K}})^{-1}(p_{S})$$r_{1}|_{\Lambda_{K}}$$r_{2}^{a}|_{\Lambda_{K}}((r_{1}|_{\Lambda_{K}})^{-1}(\Lambda_{S}))=\Phi_{S}(\Lambda_{S})$

and
that

$\underline{A}_{o,D_{S}}$$r_{2}$

is pure with shift

is proper on$\frac{1}{2}co\dim_{S^{*}xS^{\circ}}D_{S}$along$supp(\underline{A}_{\mathring{D}_{S}})\cap r_{1}^{-1}(supp(\mathcal{G}))$

Proposition 3.4.4 to this situation. Thenpure with shift $d”$ along$\Lambda_{S}^{*}$

at$p_{S}^{*}$$\Lambda_{K}$. Thus we can apply$Rp_{2*}R\underline{Hom}(\underline{A}_{o,D_{S}},p_{1}^{-1}\mathcal{G})$where$d”=d- \frac{1}{2}co\dim_{S^{*}xS}D_{S}-\frac{1}{2}\dim So+\frac{1}{2}\tau$$=d- \frac{1}{2}n-\dim Z+\frac{1}{2}\tau$

and$\tau=\tau(\lambda_{0}(p_{S}), \Phi_{S}^{1}(\lambda_{0}(p_{S}^{*})),$$\lambda_{\Lambda_{S}}(p_{S}))$

By Proposition 3.2.1 (ii), we have$\tau=\tau(\lambda_{0}(p), \Phi_{E}^{-1}(\lambda_{0}(p^{*})),$

19. It is clear$\lambda_{\Lambda}(p))$

.

.

is

77
Now, from Lemma 4.1.4, we deduce$\mathcal{G}^{AS}[-\dim Z]=Rp_{2*}R\Gamma_{D_{S}}p_{1}^{-1}\mathcal{G}$$=Rp_{2*}R\underline{Hom}(\underline{A}_{D_{S}}\circ,p_{1}^{-1}\mathcal{G})$.

Finally we have Claim 2. 1
From Claim 1 and Claim 2 we get the following claim.

$\{\begin{array}{l}(Rf_{1*}g_{1}^{-1}\mathcal{F})^{\wedge s}ispureoftypeMwithshiftd’=d-\frac{n}{2}+\frac{1}{2}\tau(\lambda_{0}(p),\Phi_{E}^{-1}(\lambda_{0}(p^{*})),\lambda_{\Lambda}(p))along\Lambda_{S}^{*}atp_{S}^{*}\end{array}$

From Lemma 4.1.2 we have$(g_{1}^{-1}\mathcal{F})^{\wedge 0}=f_{2}^{-1}(Rf_{1*}g_{1}^{-1}\mathcal{F})^{AS}$.

is a projection, we can apply Theorem 3.4.3. Then we find out
at . Considering theis pure of type $M$ with shift along
last statement, (4.1.1.1) and the microlocal uniqueness of pure sheaves
is pure of type $M$ with shift alongat .
(see 3.5), we findcoincides within a neighborhood of
becauseNow

Since$f_{2}$$d’$$(g_{1}^{-1}\mathcal{F})^{Ao}$$\Lambda^{*}$$p^{*}$$d’$$g_{2}^{-1}(\mathcal{F}^{A})$$\mathcal{F}^{\wedge}$

is pure of type $M$ with shift alongatThus the proof of the theorem is finished in the case of $p\in T^{*}Eoo$ .
.
2. Proof in the case that
Define $\mathcal{F}’\in Ob(D^{+}(E\cross R^{2}))Z$ as follows.

$p^{*}\in T^{*}E^{*}oo$$p^{*}$$\pi(p^{*})$$g_{2}^{-1}(\mathcal{F}^{\wedge})$. Therefore$\Lambda^{*}$$d’$$\mathcal{F}^{A}$$\Lambda^{*}$$p^{*}$.

$p\in T^{*}E\backslash T^{*}Eoo$$\mathcal{F}’=\mathcal{F}\otimes\underline{Z}_{\{0\}}\otimes\underline{Z}_{R}$Remark$SS(\mathcal{F}’)=SS(\mathcal{F})\cross T_{\{0\}}^{*}R\cross T_{R}^{*}R$.

Then, from Proposition 3.4.1 (ii), it follow thatis pure of type $M$
at $(p, (0;1),$ $(1;0))$ . Now we canalongwith shift
is pure with shift
apply the above result. Then we have$\mathcal{F}’$$s$$d+ \frac{1}{2}$$\Lambda\cross T_{\{0\}}^{*}R\cross T_{R}^{*}R$$\mathcal{F}^{\prime\wedge}$$d”=d+ \frac{1}{2}-\frac{n+2}{2}+\frac{1}{2}\tau_{ExR^{2}}(\lambda_{0}(p’), \Phi_{ExR^{2},z}^{-1}(\lambda_{0}(p’*)),$

along$\Lambda^{*}\cross T_{R}^{*}R\cross T_{\{0\}}^{*}R$

at $(p^{*}, (1;0),$ $(0;1))$ where$p’=(p, (0;1),$ $(1,0))$ ,$\Lambda’=\Lambda\cross T_{\{0\}}^{*}R\cross T_{R}^{*}R$

and

20

,

$\lambda_{\Lambda’}(p’))$78$p’*=\Phi_{E\cross R^{2}}(p’)$.

By Proposition 3.2.1 (ii), we have$\lambda_{\Lambda’}(p’))$$\tau_{ExR^{2},z}(\lambda_{0}(p’),\Phi_{E\cross R^{2},z}^{-1}(\lambda_{0}(p’*)),$$=\tau_{E}(\lambda_{0}(p), \Phi_{E}^{-1}(\lambda_{0}(p^{*})),$$\lambda_{\Lambda}(p))$.

Then$d”=d+ \frac{1}{2}-\frac{n+2}{2}+\frac{1}{2}\tau_{E}(\lambda_{0}(p), \Phi_{E}^{-1}(\lambda_{0}(p^{*})),$

Since
that

$\mathcal{F}^{\prime A}=\mathcal{F}^{A}\otimes\underline{Z}_{R}\otimes\underline{Z}_{\{0\}}[-1]$$\mathcal{F}^{A}$$\Lambda^{*}$.

, it follows from Proposition 3.4.1 (ii)is pure with shift$d- \frac{n}{2}+\frac{1}{2}\tau(\lambda_{0}(p), \Phi_{E}^{-1}(\lambda_{0}(p^{*})),$

along$\lambda_{\Lambda}(p))$

at$p^{*}$$\lambda_{\Lambda}(p))$. Thus the proof of Theorem 4.1.1 is completed. 1\S 5. APPLICATION.

As a corollary of Theorem 4.1.1, we prove that the Fourier-Sato transformation of perverse sheaves with n-shifts are also perverse.
5.1 Perverse Sheaves.
We do not recall here the definition of stratification, constructible.sheaves and perverse sheaves. Refer to [K-S] and [G-M] for the se definitions. We denote by
the subcategory of $D^{b}(X)$ consistingof C-constructible complexes. For a complex manifold $X$ , we denote by, the real underlying manifold of $X$ .
$D_{\mathbb{C}-c}^{b}(X)$$X^{R}$.

5.2 Perverse Sheaves and Pure heave
First we have the following theorem from [K-S] which describes therelation between perverse sheaves and pure sheaves.
$S$

$s$

THEOREM 5.2.1 ([K-S, THEOREM 9.5.2]). Let $X$ be a complex $m$ anifold,be an object of $D_{C-c}^{b}(X)$ and. Then the following
conditions are $eq$ uivalent.
is a perverse sheaf$(b)$ At any point of the non-singul ar $loc$ usofis pure withshift .
$\Lambda=SS(\mathcal{F})$$\mathcal{F}$$(a)\mathcal{F}$$\Lambda,$$\Lambda_{reg}$$\mathcal{F}$$0$

5.3 Fourier-Sato transformation of perverse sheaves.
Let $X$ be a complex vector budle with finite fibre dimension . When
we apply the Fourier-Sato transformation to object of $D^{+}(X)$ , we reand the object as tho se of $D^{+}(X^{R})$ .
gard $X$ as
$n$

$s$

$X^{R}$

$s$

21

79
THEOREM 5.3.1. For an arbitrary object
if and only if
is perverse.

$\mathcal{F}$of $D_{C-c}^{b}(X),$$\mathcal{F}$is perverse$\mathcal{F}^{A}[n]$This proposition was proved by Kashiwara-Schapira [K-S2] and analogou proposition in the algebraic category and in the category ofwere proved by J. L. Brylinsky in [ $B$ , corollaire 7.23 and corollaire 9.11].
We give a different proof by use of Theorem 4.1.1.
is perverse. Set A. Let be a point of
PROOF: Assume. By Theorem 5.2.1, is pure with shift along A at . Since isperverse, we can regard as
where is a smooth submanifold of$X$ ([K-S]). Thus we deduce from Theorem 4.1.1 thatis pure withatwhere
shift along
$s$

$s$

$F_{q}$$=SS(\mathcal{F})$$\mathcal{F}$$\Lambda$

$p$

$0$

$\mathcal{F}$$\Lambda_{reg}$$p$$\mathcal{F}$$Y$$T_{Y}^{*}X$$\mathcal{F}^{A}$$d$$\Lambda^{*}=\Phi_{X^{\blacksquare}}(T_{Y}^{*}X)$$p^{*}=\Phi_{X^{\bullet}}(p)$$d=- \frac{2n}{2}+\frac{1}{2}\tau(\lambda_{0}(p), \Phi_{X^{\bullet}}^{-1}(\lambda_{0}(p^{*})),$$\lambda_{T_{Y}X}(p))$.

By Proposition 3.2.1 (ii), we have$\tau(\lambda_{0}(p), \Phi_{X^{\bullet}}^{-1}(\lambda_{0}(p^{*})),$$\lambda_{T_{Y}X}(p))=0$

From Theorem 5.2.1 again, it follows that$\mathcal{F}^{A}[n]$.

is perverse. vice versa. 1

REFERENCES[B]. J. L. Brylinski, Transformation canoniques, Dualit\’e projective, Th\’eorie deLefschetz, Tra nsformations de Fourier et sommes trigonom\’etriq ues, Ast\’erisque140-141 (1986), 3-134.
[G-M]. M. Goresky and R. MacPherson, Intersection homology II, Inv. Math. 71(1983), 77-129.
[K]. M. Kashiwara, Index theorem for constructible sheaves, Ast\’erisque 130 (1985),
193-209.
[K-S]. M. Kashiwara and P. Schapira, Microlocal study of sheaves, Ast\’erisque 128(1985).
[K-S2]. M. Kashiwara and P. Schapira, “Sheaves on Manifolds,“ Springer, 1990.
[S-K-K]. M. Sato, M. Kashiwara and T. Kawai, Microfunctions and pseudo-differential
equations, in “Springer Lecture Notes in Math. 287,“ Springer, 1973, pp. 265-529.
[S-T]. P. Schapira and N. Tose, Morse inequalities for $R$ -constructible sheaves, Advances in Math. (to appear).

Hongo 7-3-1, Bunkyo-ku, Tokyo 113 Japan

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