CC BY
13
УДК 517.55
Microlocal Study of Lefschetz Fixed Point Formulas
Yutaka Matsui*
Graduate School of Mathematical Sciences University of Tokyo, 3-8-1, Komaba Meguro-ku, Tokyo, 153-8914,
Japan
Kiyoshi Takeuchi^
Institute of Mathematics University of Tsukuba, 1-1-1, Tennodai Tsukuba, Ibaraki, 305-8571,
Japan
Received 1.10.2007, accepted 5.12.2007 The aim of this short paper is to introduce our recent study on Lefschetz fixed point formulas over singular varieties. In particular, we generalize Kashiwara’s theory of characteristic cycles by introducing new Lag-rangian cycles associated with endomorphisms of constructible sheaves. Some examples related with Schubert varieties and toric hypersurfaces will also be given.
Key words: singular varieties, Lefschetz fixed point formulas, Lagrangian cycles, Schubert varieties.
Introduction
In this paper, we introduce our recent study of Lefschetz fixed point formulas in [15]. First of all, let us consider the (classical) Lefschetz fixed point formulas for morphisms ^: X —► X of real analytic manifolds X. We denote the fixed point set of ^ by M = {x G X | ^(x) = x} C X (since here we mainly consider the case where the fixed point set is smooth, we used the symbol M) and let m = |j.ei Mi be the decomposition of M into connected components. Then it is well-known that if X and M are compact the global Lefschetz number of ^
tr(^) := ^(-1)jtr{Hj(X; Cx) Hj(X; Cx)} G C
j£ Z
is a sum of some numbers c(^)Mi G C associated with the fixed point components M.’s. The number c(<p)Mi is called the local contribution from M.. But if the fixed point component M. is "higher-dimensional" , it is a very hard task to compute the local contribution in general. In our paper [15], we studied more general problems for the Lefschetz numbers of hypercohomology groups of constructible sheaves and overcame this difficulty for smooth fixed point components Mi’s. In particular, we obtained some Lefschetz fixed point formulas over singular varieties (see e.g. Corollary 1). Moreover new Lagrangian cycles in the cotangent bundles T*M., which we call Lefschetz cycles, were introduced and their functorial properties were studied precisely. This result generalizes Kashiwara’s theory of characteristic cycles. For details, see [15].
1. Preliminary Notions and Results
In this note, we essentially follow the terminology in [11]. For example, for a topological space X, we denote by Db(X) the derived category of bounded complexes of sheaves of CX-modules on X. Since
* e-mail: you317@ms.u-tokyo.ac.jp te-mail: takemicro@nifty.com © Siberian Federal University. All rights reserved
we focus our attention on Lefschetz fixed point formulas for constructible sheaves in this paper, we treat here only real analytic manifolds and morphisms. Now let X be a real analytic manifold. We denote by Dr_c(X) the full subcategory of Db(X) consisting of the bounded complexes of sheaves whose cohomology sheaves are R-constructible (see [11, Chapter VIII] for the definition).
Denote also by шх — °rX [dimX] Є DRR_c(X) the dualizing complex of X. Let ф: X —u X
be an endomorphism of the real analytic manifold X. Then our initial data is a pair (F, Ф) of F Є DR-c(X) and a morphism Ф: ф-1ґ —u F in DR-c(X). If the support supp(F) of F is compact, Hj (X; F) is a finite-dimensional vector space over C for any j Є Z and we can define the following number from (F, Ф).
Definition 1. We set
tr(F, Ф) := £(-1)jtr{Hj (X; F) —U Hj(X; F)} Є C,
je Z
where the morphisms Hj (X; F) —U Hj (X; F) are induced by
F -u Дф*ф-1 F —U Дф*F..
We call tr(F, Ф) the global trace (Lefschetz number) of the pair (F, Ф).
Now consider the fixed point set of ф: X u X in X:
M := {x Є X | ф(х) = x} С X.
Since we mainly consider the case where the fixed point set is smooth, we use the symbol M to express it. Note that if a compact group G is acting on X and ф is the left action of an element of G, then the fixed point set is smooth by Palais’s theorem [17] (see also [6] for a survey on this subject). Also in this very general setting, Kashiwara [10] proved the following result.
Theorem 1 (Kashiwara [10]). If supp(F) is compact, then there exists a class C(F, Ф) Є Hs0upp(F)nM(X; ) — HdimXF)nm (X; orx)
such that the equality
tr(F, Ф) = j C(F, Ф) (1)
X
holds. Here
J: HcdimX (X; orx) -u C
X
is the morphism induced by the integral of differential (dimX)-forms with compact support.
Let M = LI іє/ Mi be the decomposition of M into connected components and
H0upp(F)nM (X;^X ) = (Biel HsUpp(F)nMi (X; ), C(F, Ф) = ФіЄ/ C(F, Ф)Мі the associated di-
rect sum decomposition.
Deñnition 2. When supp(F) П Mi is compact, we define a complex number c(F, Ф)м» by
c(F, Ф)Mi := У C(F, Ф)Mi
X
and call it the local contribution of the pair (F, Ф) from Mi.
With this notation, (1) is rewritten as
tr(F, Ф) = £ c(F, Ф)мі .
іЄ/
Therefore the next important problem in the Lefschetz fixed point formula for constructible sheaves is to describe these local contributions c(F, $)m* • However, the direct computation of local contributions is in general a very difficult task. Instead of directly considering the local contribution, let us first consider the following number tr(FM, ^|m*), which is much more easily computed. Let Mj be a compact fixed point component such that supp(F) fi Mj is compact.
Definition 3. Consider the morphism (the restriction of ft-1F —► F to Mj):
$|.M : F|mî — (ft-1F)|mî —^ FM
and set
tr(F|Mi, $|m4) := ^(-1)jtr{Hj(Mj; FM^ H(Mj; FM)}.
Then we can easily compute tr(F|Mi, $|Mi) € C as follows. Let Mj = |JaeA Mjja be a stratification of Mj by connected subanalytic manifolds Mjj0, such that (F)M a is a locally constant sheaf for any a € A and j € Z. Namely, we assume that the stratification Mj = |JaeA Mjja is adapted to F M .
Definition 4. For each a € A, we set
ca := ^(-1)jtr{Hj(F)xa ^} H(F)xa} € C,
j£Z
where xa is a reference point of Mjja.
Then we have the following very useful result due to Goresky-MacPherson.
Proposition 1 (Goresky-MacPherson [5]). We have
tr(F M , $|Mi ) = ^ Ca • Xc(Mja),
aeA
where xc is the Euler-Poincaré index with compact supports.
Therefore our main problem is:
Problem 1. When does the following equality hold?
c(F, $)mî = tr(FM, $|.m). (2)
For this problem, we have the following known results.
Example 1. (i) (Kashiwara-Schapira [11]) If X, ft and F are all complex analytic, Mj is a point
{pt} such that 1 is not an eigenvalue of the linear map ft' : TMX — CdlmX —> TMX — CdlmX, then (2) holds.
(ii) (Goresky-MacPherson [5]) If ft : X —> X is weakly hyperbolic (see [5] for the definition) along Mj, then (2) holds. In particular, if ft is of finite order, then (2) holds.
2. Localization Theorems and Their Applications
In this section, we give an answer to Problem 1 by partially generalizing the results of Kashiwara-Schapira [11] and Goresky-MacPherson [5] in Example 1. Since we always consider the same fixed point component Mj in this section, we denote Mj, c(F, $)m* etc. simply by M, c(F, $)m etc. respectively. Let us consider the natural morphism
ft : TMreg X -^ TMreg X
Definition 5. For x G Mreg, we set
Evx := {the eigenvalues of ftX : (?MregX)x —► (TMregX)x} C C.
Theorem 2. Assume the following conditions:
(i) supp(F) n M C Mreg is compact.
(ii) 1 G Evx for any x G supp(F) n M.
(iii) Evx n R>i = 0 for any x G supp(F) n M
(when X, ft and F are all complex analytic, (iii) is not necessary). Then (2) holds.
Proof. Consider the specialization
VMreg (F) G DR-C(TMreg X )
of F along Mreg and the natural morphism
$: ft^1^(F) VMreg(F).
Then our proof is similar to that of [11, Proposition 9.6.11 and 9.6.12]. We also use some results in Section . For details, see [15]. □
As a very special case of Theorem 2, we obtain the following Lefschetz fixed point formula over singular varieties.
Corollary 1. Let X be a complex manifold, ft: X —> X a holomorphic map and V C X a ft-invariant compact analytic subset. Assume that for the fixed point set M = {x G X | ft(x) = x} C X the following conditions are satisfied.
(i) V n m c Mreg,
(ii) 1 G Evx for any x G V n M.
Then we have
tr(ft|v) := ^(-1)jtr{Hj(V; Cy) (—1 * (V; Cy)} = x(V n M).
j£Z
Example 2. Let Gn = SLn(C) and let Bn C Gn be the Borel subgroup of Gn consisting of upper triangular matrices. Then the homogeneous space X = Gn/Bn is the flag manifold. Take an element
g = diag(Ai, • • • , Ai, A2, • • • , A2, • • • , A^, • • • , A^)
'----^----' '---^------' '--------^'
ni-times n2-times nk-times
(n = ni + • • • + nk) in Bn such that Aj = Aj for any i = j, where diag(^ • •) denotes a diagonal matrix. Let ft: X —► X be the left action : X —► X by g G Bn C Gn. Then it is easy to see that the fixed point set M of ft is a smooth complex submanifold of X. More precisely, M
n!
is isomorphic to the disjoint union of —-------------- copies of the product of smaller flag manifolds
ni! • • • nfc!
Gni/Bni • • xGnfc/Bnk . Therefore the assumptions of Corollary 1 are satisfied for any ft-invariant analytic subset V of X, if 1 G Evx for any x G M (we expect this is always true). Since g G Bn, as a ft-invariant analytic subset V we can take any Schubert variety in X.
Example 3. Let us consider a special case of Example 2 above. Let X = G3/B3 be the flag manifold consisting of full flags in C3 and ft = : X —► X the left action by the element
ia. 0 0\
g = I 0 a 0 1 G B3 C G3,
\0 0 p)
where a = p are non-zero complex numbers. In this case, the fixed point set M C X of ft is the disjoint union of 3-copies of CP1 's. Let X = |JB3CTB3 = |JXCT be the Bruhat decomposition of X = G3/B3. Here an element a of the symmetric group 63 is identified with the matrix №,a-(j))i<i,j<3 G G3 (see e.g. [8] for the detail of this subject), where ¿¿j is Kronecker’s delta. In this case, X(i,3) is the unique open dense Schubert cell in X. Set V = X \ X(i,3) = |Ja=(i 3) XCT. Then V is a ft-invariant analytic subset of X and we can check that the assumptions of Corollary 1 are satisfied.
Example 4. Let N ~ Zn be a Z-lattice, A a complete fan in Nr = R N and X(A) the toric variety associated with A. Denote the open dense algebraic torus of X(A) by T ~ (C*)n. Since T acts on X(A) itself, for each g G T we can consider the left action ft := : X(A) —> X(A) by
g. Assume that X(A) is smooth. Then for any g G T the fixed point set M of ft = is smooth and we can easily verify that 1 G Evx for any x G M. We assume also that there exists a Laurent polynomial f: T ~ (C*)n —> C which satisfies the condition
3C G C s.t. f (aixi, • • • , anxn) = C • f (xi, • • • , xn),
where we set g = (ai, • • • , an) G T ~ (C*)n (for example, if g = (ami, am2, • • • , amn) we can take f to be any quasi-homogeneous polynomial of weight (mi, m2, • • • , mn) ). Then the toric hypersurface V = {x G T | f (x) = 0} C X(A) associated with f is invariant by ft = : X(A) —► X(A). Hence
we can use Corollary 1 to conclude that the Lefschetz number tr(ft|y) of ft|v : V —> V is x(VnM).
3. Microlocal Study of Lefschetz Fixed Point Formulas
3.1. Definition of Lefschetz Cycles
In this section, we construct certain Lagrangian cycles which encode the local contributions discussed in previous sections into topological objects. We inherit the notations in Sections 1 and 2. For the sake of simplicity, we assume that the fixed point set M = {x G X | ft(x) = x} of ft: X —► X is a submanifold of X. However, here we do not assume that M is connected. We also assume that the diagonal set Ax ~ X C X x X intersects with the graph = {(ft(x), x) gX x X | xGX} of ft cleanly along M in X x X. Note that the last condition is equivalent to the one: 1 G Evx for any x G M. Identifying r^ with X by the second projection X x X —> X, we obtain a natural identification M = r^ n Ax. We also identify Tax (X x X) (resp. (X x X)) with TX (resp. T*X) by the first projection T(X x X) ~ TX x TJX —► TX (resp. T*(X x X) ~ T*X x T*X —► T*X) as usual. Then, under the above assumptions, we see that the natural morphism
Tmr0 -^ TAx (X x X) ~ TX
induced by the inclusion map c—> X x X is injective and the image of this morphism is a
subbundle of M xx TX. We denote this vector bundle over M by E. The following lemma will be obvious.
Lemma 1. The subset T* (X x X) n T^x (X x X) of (r^ n Ax) x a x Tax (X x X) ~ M xx T*X is naturally identified with the subset of M xx T*X consisting of covectors which are orthogonal to the vectors in E C M xx TX with respect to the natural pairing (M xx TX) x (M xx T*X) —> R.
By this lemma, we see that Tj*^ (X x X) n T*x (X x X) is a subbundle of M xx T*X. We denote it by F and call it the Lefschetz bundle associated with ft: X —► X. The Lefschetz bundles satisfy the following nice property.
Proposition 2. The natural surjective morphism p: M xxT*X —» T*M induces an isomorphism T ^ T* M.
From now on, by Proposition 2 we shall always identify the Lefschetz bundle F with T*M. Now let F be an object of DR_c(X) and $: ft-1F —> F a morphism in DR_c(X). To the pair (F, $), we can associate a conic Lagrangian cycle in the Lefschetz bundle F — T*M as follows. Let
Max : Db(X x X) -^ Db(Tlx(X x X))
be the microlocalization functor along Ax. Recall that the micro-support SS(F) of F is a closed conic subanalytic Lagrangian subset of T*X and the support of max (F H DF) is contained in SS(F) C T*X — (X x X). Let ¿x : X —> X x X be the diagonal embedding and h: X —>
X x X a morphism defined by x i—► (ft(x), x). We denote by nM : F — T*M —► M the projection. Then we have a chain of natural morphisms:
RHomCx (F, F) — Rr(X; ¿x (F H DF))
— RFss(f)(T*X; max (F H DF))
-^ rfss(f)(T*X; max (h*h-i(F H DF)))
— RrSS(F)(T*X; MAX (h*(ft 1f ® DF)))
---► RrSS(F)(T*X; MAX (h*(F ® DF)))
---► RFSS(F)(T*X; MAX (h*wx))
— RFSS(F)nF(F; nM^M),
where we used the isomorphism max(h*^x)|f — n—1^m in the last step. Taking the 0-th hypercohomology groups of both sides, we obtain a morphism
HomDb(x) (F,F) ► HSS(F)nF(F; f—1wM). (3)
Definition 6. We denote by LC(F, $) the image of idF G HomDb(x)(F, F) in Hss(f)nF(F; n— WM) by (3) (since SS(F)nF is contained in a closed conic subanalytic Lagrangian subset n/F — T*M, LC(F, $) is a Lagrangian cycle in F — T*M). We call LC(F, $) the Lefschetz cycle associated with the pair (F, $).
Note that a similar construction of microlocal Lefschetz classes was also given by Guillermou
[7]. The difference from his construction is that we explicitly realize such microlocal characteristic classes as geometric objects in the cotangent bundle T* M. Note also that if ft = idx, M = X and $ = idF, our Lefschetz cycle LC(F, $)M coincides with the characteristic cycle CC(F) of F introduced by Kashiwara [9] (for the applications of characteristic cycles to projective duality, see [4], [13], [14] etc.) By our Lefschetz cycles, we can generalize almost all nice properties of characteristic cycles into more general situations (see the subsequent subsections).
As a basic property of Lefschetz cycles, we have the following homotopy invariance. Let I = [0,1] and let ft: X x I —► X be the restriction of a morphism of real analytic manifolds X x R —► X. For t G I, let it: X c—► X x I be the injection defined by x i—► (x, t) and set ftt := fto it: X —► X. Assume that the fixed point set of ftt in X is smooth and does not depend on t G I. We denote this fixed point set by M. Let F G DR_c(X) and consider a morphism $: ft-1F —► p-1F in DR-c(X x I), where p: X x I —► X is the projection. We set
$t := $|xx{i} : ft-iF -► F
for t G I. We denote the Lefschetz bundle associated with ftt by Ft — T*M.
Proposition 3. Assume that supp(F) n M is compact and Ft does not depend on t G I. Then the Lefschetz cycle LC(F, $t) G H^s(f)nF*m(T*M; i—'^m) does not depend on t G I.
3.2. Microlocal Index Formula for Local Contributions
In this subsection, we introduce a microlocal index theorem for local contributions. Our theorem is a natural generalization of Kashiwara’s microlocal index theorem for characteristic cycles (see
[11, Theorem 9.5.3]). Let M = |Ji£/ Mi be the decomposition of M into connected components. Set Fi := Mi xM F. Then we get a decomposition F = |Jie/ Fi — LIie/ T*Mi of F. By the direct sum decomposition
HSS(F)nF(F; ^a/^M) — HSS(F)nFi (F; r!) — HS!3(F)nFi (F; nM*°rMi )>
ie/ ie/
we obtain a decomposition
LC(F, $) = ^ LC(F, $)ms
ie/
of LC (F, $), where LC (F, $)Mi € H^S^p^ri f (Fi; n-1 orMi ). Now let us fix a fixed point component Mi. We shall show how the local contribution c(F, $)Mi € C of (F, $) from Mi can be expressed by LC(F, $)Mi. In order to state our results, for the sake of simplicity, we denote Mi, Fi, LC (F, $)Mi , c(F, $)Mi simply by M, F, LC (F, $), c(F, $) respectively. Recall that to any continuous section a : M —► F — T*M of the vector bundle F, we can associate a cycle
[a] € H°(M)(T*M; nM(Cm)) by the isomorphism H°(M)(T*M; nMCm) — H0(M;(iM oa)!CM) — H0(M; CM) and 1 € H0(M; CM) (see [11, Definition 9.3.5]). If a(M) n supp(LC(F, $)) is compact, we can define the intersection number ft([a] n LC(F, $)) of [a] with LC(F, $) as the image of [a] <g> LC(F, $) by the chain of natural morphisms
H°(M )(F ; nM CM ) <8> Hsupp(LC(F,^)) (F ; nM^M ) ► H°(M )nsupp(LC(F,$))(F ; )
C.
Theorem 3. Assume that supp(F) n M is compact. Then for any continuous section a : M —► F — T*M of F, we have
c(F, $) = H([a] n LC(F, $)).
As an application of Theorem 3, we shall give a useful formula which enables us to describe the Lefschetz cycle LC(F, $) explicitly in the special case where ft : X —► X is the identity map of X and M = X. For this purpose, until the end of this subsection, we shall consider the situation where ft = idX, M = X and $: F —► F is an endomorphism of F € DRR_c(X). In this case, LC(F, $) is a Lagrangian cycle in T*X. For a real-valued real analytic function p : X —► R on X, we define a section av : X —► T*X of T*X by av(x) := (x; dp(x)) (x € X) and set
Av := av(X) = {(x; dp(x)) | x € X}.
Note that Av is a Lagrangian submanifold of T*X. Then we have the following analogue of [11, Theorem 9.5.3].
Let X = LI aeA Xa be a ^-stratification (for the definition see [11, Definition 8.3.19]) of X such that
supp(LC(F, $)) C SS(F) C |_| TXaX.
aeA
Then A = \JaeA TJ X is a closed conic subanalytic Lagrangian subset of T*X. Moreover there exists an open dense smooth subanalytic subset A0 of A whose decomposition A0 = i / Ai into connected components satisfies the condition
”For any i € I, there exists ai € A such that Ai C T^ X. ”
Definition 7. For i € I and ai € A etc. as above, we define a complex number mi € C by
mj := ^(_1)jtr{H|V>V(x)}(F)x *■ H{V>V(x)|(F)x},
jez
where x € nx (Aj) C Xai and the real-valued real analytic function p : X —► R (defined in an open neighborhood of x in X) are defined as follows. Take a point p € Ai and set x = nx (p) € Xai. Then p : X —> R is a real analytic function which satisfies the following conditions:
(i) p = (x; dp(x)) € Aj.
(ii) The Hessian Hess(p|xa ) of p|xa is positive definite.
Theorem 4. In the situation as above, for any i € I there exists an open neighborhood Uj of Aj in T * X such that
LC(F, $)= mj • [T*X]
in Uj.
3.3. Explicit Description of Lefschetz Cycles
In this subsection, we explicitly describe the Lefschetz cycle LC(F, $)m introduced in Section in many cases. Let M be a possibly singular fixed point component of ft : X —> X. Throughout this subsection, we assume the condition
”1 € Evx for any x € supp(F) n Mreg.”
By making use of some localization theorems for Lefschetz cycles similar to the one in Section 2, we obtain the following explicit description of LC(F, $)M.
Theorem 5. Let xo G Mreg be a point of Mreg such that
Evæo n R>1 = 0.
Then we have
LC(F, $)m = LC(F|m, $|m)m in an open neighborhood of n-'1(xo) in T*Mreg.
In the complex case, we have the following stronger result.
Theorem 6. In the above situation, assume that X and ft : X F G DC(X) i.e. F is C-constructible. Then we have
(4)
X are complex analytic and
LC (F, Ф)- = LC (F I-, Ф-)-
globally on T»Mri
Combining Theorem 5, 6 with Theorem 4, we thus obtain the explicit description of Lefschetz cycles. By this description, we also obtain the following result.
Corollary 2. Let X, ф and M be as above and Fi —F2 —L F3 —U +1 a distinguished triangle in DL c(x ). Assume that we are given a commutative diagram
ф 1 a
ф-1в
ф Y
ф-1í11-----5- ф F2----5- ф F3-----5- ф-1F1[1]
Ф1[1]
F1
F2
F3
Fl[1]
in Di
,(X). Then for any x0 Є Mreg such that EvXo П R>1 = 0, we have
LC(F2, Ф2)- = LC(Fl, Фі)- + LC(F3, Ф3)
)-
in an open neighborhood of nM1(x0) in T»Mreg.
Ф
2
з
в
Y
a
3.4. Direct Image Theorem
From now on, we study functorial properties of our Lefschetz cycles.
Let f: Y —> X be a morphism of real analytic manifolds.. Assume that we are given two morphisms ftx : X —> X and fty: Y —> Y such that the diagram
^ X
^ X
commutes. Let us take an object G of DRR_C(Y) such that f is proper on supp(G) and a morphism
: ft-1G —► G
in DR_c(Y). Then ñf*G € Dr_c(X) and we obtain a natural morphism
: ftx1ßf*G ñf*G
induced by . Our aim in this subsection is to compare the Lefschetz cycle of (G, ) with that of (ñf*G, ). Let M be a smooth fixed point component of such that f (supp(G))nM is compact. Also let be the set of all fixed point components N of such that N C f-1(M) and
supp(G) nN = 0. Note that I is a finite set by our assumptions. Set N := [Jie1 N and assume that N is smooth. We also assume that r^X C X x X (resp. r^Y C Y x Y) intersects with AX in X x X (resp. Ay in Y x Y) cleanly along M (resp. N) as in previous sections. For the sake of simplicity, we denote M xX {T* (X x X) nT*x (X x X)} ~ T*M, N xY {T* (Y x Y) n (Y x Y)} ~ T*N
simply by F, G respectively. Then we obtain two Lefschetz bundles
FC T*^ (X x X) n T£x (X x X),
and the Lefschetz cycles
G С Tr»^ (Y x Y) n (Y x Y)
LC(G, Фу )N є Я°з(С)П0(G; ), LC(Rf»G, ФХ)- Є ^^(R/^nF(F; пд^1wм)..
Note that by setting G* := N xN G we have the direct sum decompositions G = Uie/ G* — uiei T*Nj and
LC(G, )n = ^ LC(G, )n,
ЇЄ/
where LC(G, Фу)Ni Є Я^(С)пЄ, (G*; п—1wN¿). Now, set g = f IN : N —> M and consider the natural morphisms
t і
T»N ^— N x- T»M —^ T»M
induced by g. Take a closed conic subanalytic Lagrangian subset Л = УЛ* of T»N = УT»N*
such that SS(G) n G С Л and set Л' = 4g' 1(Л) and Л" = gn(Л'). Then there exists a morphism
g» : Я« (T»N; п-1.„) —^ Я°„(T»M; п-W)
of Lagrangian cycles induced by g (see [11, Proposition 9.3.2 (i)])..
Theorem 7. In the above situation, we have
LC (Rf»G, Фх )- = g»(LC (G, Фу )n )
in T* M. More precisely, for the morphism
(gi)* : (T*N*; „(T*M; )
of Lagrangian cycles induced by gi := f |n : N* —> M we have
LC(Rf*G, )m = ^(g*)*(LC(G, )n). (5)
ie/
Applying Theorem 3 to both sides of (5), we obtain Corollary 3. For the local contributions c(G, $y )n and c(Rf*G, )m, we have
c(Rf*G, )m = ^ c(G, )n .
ie/
In fact, Corollary 3 can be proved more directly without using Lefschetz cycles. We can even generalize it as follows (see [16]).
Theorem 8 ([16]). We inherit the notations and the situation as above. However, we do not
assume that M (resp. N = |Jj£/ Nj) is smooth nor 1 G Evx for x G M (resp. 1 G Evy for y G N)
here. Then we have
c(Rf*G, )m = ^ c(G, )n .
ie/
3.5. Inverse Image Theorem
In this subsection, we introduce the inverse image theorem for Lefschetz cycles. We inherit the situation treated in Subsection 3.4. However, here M and N are smooth fixed point components of and respectively satisfying the condition f (N) C M. We take an object F of DRR_c(X) and a morphism
: ftxF -^ F
in DR_c(X). Then f-1F G DR_c(Y) and we obtain a natural morphism
: ft-1f-1F -^ f-1F
induced by . Assuming the same conditions on ftx, etc.. and keeping the same notations
for F, G etc. as in Subsection 3.4, we obtain the Lefschetz cycles
LC(F, )mG Hs0s(f)rf(F;
nM WM ),
LC(f )N G hSs(/-1F)nG(G; ).
Set g = f |n : N —► M as before and consider the natural morphisms
t /
T*N N xM T*M -^ T*M
induced by g. Take a closed conic subanalytic Lagrangian subset A of T*M such that SS(F)nF C A and set A' = g—1 (A) and A" = ig/(A/). If ig/ is proper on A' (e.g. if f is non-characteristic for F on an open neighborhood of N), then there exists a morphism
g* : (T*M; nMW) „(T*N; n-W)
of Lagrangian cycles induced by g (see [11, Proposition 9.3.2 (ii)]).
Theorem 9. In the above situation, assume moreover that f is non-characteristic for F on an open neighborhood of N. Then we have
LC(f-1F, )n = sgn(id - ftX) ■ sgn(id - ftY) ■ g*(LC(F, )m)
in T*N, where sgn(id - ) = ±1 (resp. sgn(id - ftY) = ±1) is the sign of the determinant of
id - : TmX -^ TmX (resp. id - ftY: TnY -^ TWY).
As a special case of this theorem, we obtain the following result which drops the condition (4) of Theorem 5.
Corollary 4. Under the assumptions in Theorem 5, instead of assuming the condition (4), assume that the inclusion map i : M c—> X of the fixed point manifold M is non-characteristic for F. Then we have
LC(F, $)m = sgn(id - ft') • LC(F|M, $|M)M in T* M. In particular, if moreover supp(F) 0 M is compact, we have
c(F, $)m = sgn(id - ft') • tr(F|m, $|m).
Note that the last term tr(F|M, $|m) in Corollary 4 can be easily computed by Proposition 1. And note that Corollary 4 is not true if we do not assume that i : M c—► X is non-characteristic for F. See e.g. [11, Example 9.6.18].
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