A Poincaré Lemma for Whitney-de Rham complex

0.08 MB·English

by  Hou-Yi Chen

A Poincar´e Lemma for Whitney-de Rham
complex
By
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Hou-Yi Chen
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n Abstract
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LetM bearealanalyticmanifold,Z aclosedsubanalyticsubsetofM. Weshow

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that the Whitney-de Rham complex overZ is quasi-isomorphicto the constant
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sheaf C .
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Contents
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.
h 1 Introduction 1
t
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m 2 Review on Whitney and formal cohomology functors 2
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3 Main result 3

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1 Introduction
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. In [KS96], Kashiwara-Schapiraintroduced the Whitney functor (real case) and
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formal cohomology functor (complex case), then they introduced the notion of
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3 ind-sheavesandtheyalsodefinedGrothendiecksixoperationsinthisframework
1 in[KS01]. Asapplications,theydefinedtheWhitneyC∞functionsandWhitney
: holomorphic functions on the subanalytic site as examples of ind-sheaves. The
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i moreelementarystudyforsheavesonthesubanalyticsiteisperformedin[Pr08]
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and [Pr12].
r LetM bearealanalyticmanifold,byPoincar´eLemma,itiswell-knownthat
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the de Rham complex overM is isomorphicto C . The aimofthis paper is to
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showthatatheoremof[BP08]followseasilyfromadeepresultofKashiwaraon
regular holonomic D-module [K84] and the Whitney functor of [KS96]. More
precisely, we show that
Main theorem(=Theorem 3.3.) Let M be a real analytic manifold of
dimension n and Z a closed subanalytic subset of M. Then we have:
C −∼→(0−→W∞ −d→W(∞,1) −d→···−d→W(∞,n) −→0)
Z M,Z M,Z M,Z
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where W∞ denotes the sheaf of Whitney functions on Z and W(∞,i) denotes
M,Z M,Z
the sheaf of differential forms of degree i with coefficients in W∞ for each i,
M,Z
i.e., the Whitney-de Rham complex is isomorhpic to C .
Z
Acknowledgments. I would like to thank Pierre Schapira for suggesting this
problem to me and for many useful conversations.
2 Review on Whitney and formal cohomology
functors
In this section, we review some results on Whitney and formal cohomology
functors. References are made to [KS96], [KS01], [Pr08], [Pr12] and [S12].
Let M be a real analytic manifold, we denote by A ,C∞ the sheaf of
M M
complex-valuedreal analytic functions, C∞-functions on M. We denote by D
M
the sheaf of rings on M of finite-order differential operators with coefficients in
A .
M
We denote by ModR−c(CM) the abelian categoryof R-constructible sheaves
on M and Mod(D ) the abelian category of left D -modules. We also denote
M M
by Db (C ) the bounded derived category consisting of objects whose coho-
R−c M
mology groups belong to ModR−c(CM) and Db(DM) the derived category of
Mod(D ) with bounded cohomologies.
M
Definition 2.1. Let Z be a closed subset of M. We denote by I∞ the sheaf
M,Z
of C∞ functions on M vanishing up to infinite order on Z.
Definition 2.2. A Whitney function on a closed subset Z of M is an indexed
family F =(Fk)k∈N consisting of continuousfunctions on Z such that ∀m∈N,
∀k ∈ Nn, |k| ≤ m,∀x ∈ Z, ∀ε > 0 there exists a neighborhood such that
∀y,z ∈U ∩Z
(z−y)j
|Fk(z)− X Fj+k(y)|≤εd(y,z)m−|k|.
j!
|j+k|≤m
We denote by W∞ the space of Whitney C∞ functions on Z. We denote by
M,Z
W∞ the sheaf U 7→W∞ .
M,Z U,U∩Z
In[KS96],theauthorsdefinedtheWhitneytensorproductfunctorasfollows:
w
·⊗CM∞: ModR−c(CM)→Mod(DM)
in the following way: let U be an open subanalytic subset of M and Z =
w w
M \U. Then C ⊗ C∞ = I∞ and C ⊗ C∞ = W∞ . This functor is exact
U M M,Z Z M M,Z
and extends as a functor in the derived category, from Db (C ) to Db(D ).
R−c M M
w
Moreover the sheaf F ⊗C∞ is soft for any R-constructible sheaf F.
M
NowletX beacomplexmanifoldandwedenotebyD thesheafofringson
X
X offinite-orderdifferentialoperators. WestilldenotebyX therealunderlying
manifoldandwedenotebyX thecomplexmanifoldconjugatetoX. Onedefines
the functor of formal cohomology as follows:
Let F ∈Db (C ), we set
R−c X
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w w
F ⊗OX =RHomDX(OX,F ⊗CX∞),
where D denotes the sheaf of rings on X of finite-order differential operators.
X
Let M be a real analytic manifold, X a complexification of M, ı : M ֒→ X
the embedding. We recall the following result.
Theorem 2.3. ([KS96] Theorem 5.10.) Let F ∈Db (C ). Then we have
R−c M
w w
ı F ⊗O ≃ı (F ⊗C∞).
∗ X ∗ M
In particular,
w
C ⊗O ≃C∞.
M X M
(cid:3)
The following proposition is the key point of this note.
Proposition 2.4. ([KS96] Corollary 6.2) Let M be a regular holonomic D -
X
module, and let F be an object of Db (C ). Then, the natural morphism:
R−c X
w
(2.1) RHomDX(M,F ⊗OX)→RHomDX(M,F ⊗OX).
is an isomorphism. (cid:3)
3 Main result
Let X be a complex manifold of dimension n. We denote by D the sheaf of
X
ringsoffinite-orderdifferentialoperatorsandΘ thesheafofvectorfieldsonX.
X
First we recall the following basic result in D -module theory.
X
Proposition 3.1. ([K03] Proposition 1.6) The complex
n 2
0→DX⊗OX VΘX →···→DX⊗OX VΘX →DX⊗OXΘX →DX →OX →0
is exact. (cid:3)
Lemma 3.2. Let M be a left D -module. Then we have
X
n
RHomDX(OX,M)≃[M →Ω1X ⊗OX M →···→VΩ1X ⊗OX M].
Proof. By Proposition 3.1, we have
RHomDX(OX,M)
n
≃ [M →HomDX(DX ⊗OX ΘX,M)→···→HomDX(DX⊗OX VΘX,M)]
n
≃ [M →HomOX(ΘX,M)→···→HomOX(VΘX,M)]
n
≃ [M →Ω1X ⊗OX M →···→VΩ1X ⊗OX M]
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where Ω1X :=HomOX(ΘX,OX). (cid:3)
LetM bearealanalyticmanifold,X acomplexificationofM andZ aclosed
subanalytic subset of M. We denote by Ω1 the sheaf of differential one-form
X
on X and
i
(3.1) AZ(i) :=VΩ1X ⊗OX (CZ ⊗OX),
(3.2) WM(∞,Z,i) :=Vi Ω1X ⊗OX (CZ ⊗w OX)≃Vi Ω1X ⊗OX (CZ ⊗w CM∞).
Now we are ready to prove the main theorem of this note below.
Theorem 3.3. Let M be a real analytic manifold of dimension n and Z a
closed subanalytic subset of M. Then we have:
C −∼→(0−→W∞ −d→W(∞,1) −d→···−d→W(∞,n) −→0)
Z M,Z M,Z M,Z
where W∞ denotes the sheaf of Whitney functions on Z and W(∞,i) denotes
M,Z M,Z
the sheaf of differential forms of degree i with coefficients in W∞ for each i
M,Z
which are defined in (3.2), i.e., the Whitney-de Rham complex is isomorhpic to
C .
Z
Proof. Take M=O and F =C in Proposition 2.4.
X Z
On the one hand, we show that the left hand side of (2.1) is C . By Theorem
Z
2.3 and Lemma 3.2, we get the following complex
0−→C −→A(0) −d→···−d→A(n) −→0
M M M
which is exact by Poincar´e lemma where A(i)’s are defined in (3.1) by taking
M
Z =M. Tensoring C , we obtain the following exact sequence
Z
0−→C −→A(0) −d→···−d→A(n) −→0.
Z Z Z
Therefore,
C −∼→(0−→A(0) −d→···−d→A(n) −→0).
Z Z Z
Ontheotherhand,therighthandsideof(2.1)istheWhitney-deRhamcomplex
0−→W∞ −d→W(∞,1) −d→···−d→W(∞,n) −→0.
M,Z M,Z M,Z
Now the result follows from the isomoprhism of (2.1). (cid:3)
References
[BP08] J-PBrasseletandM.J.Pflaum,OnthehomologyofalgebrasofWhitney
functions over subanalytic sets, Annals of Mathematics, 167 (2008), 1-
52.
[K84] M. Kashiwara, The Riemann-Hilbert problem for holonomic systems,
Publ. Res. Inst. Math. Sci. 20 p. 319-365,(1984).
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[K03] , D-modules and microlocal calculus, Translations of Mathemati-
cal Monographs,Vol. 217, Amer. Math. Soc., 2003.
[KS96] M.KashiwaraandP.Schapira,Moderate andformal cohomology associ-
ated with constructible sheaves, M´emoiresSoc.Math.France64,(1996).
[KS01] , Ind-sheaves, Ast´erisque 271 (2001).
[Pr08] L.Prelli,Sheaves on subanalytic sites,RendicontidelSeminarioMatem-
atico dell’Universit di Padova, 120 p. 167-216(2008).
[Pr12] , Microlocalization of subanalytic sheaves, M´em. Soc. Math.
France (2012).
[S12] P.Schapira,Three lectures on Algebraic Microlocal Analysis,Northwest-
ern University May 2012, arXiv:1206.1435.
Institute of Mathematics, Academia Sinica, Taipei 106, Taiwan
E-mail address: [email protected]
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