Some mixed Hodge structure on l^2-cohomology of covering of Kähler manifolds

0.39 MB·English

by  Pascal Dingoyan

SOME MIXED HODGE STRUCTURES ON l2−COHOMOLOGY OF
COVERING OF KA¨HLER MANIFOLDS I.
P.DINGOYAN
Abstract. Wegivemethodstocomputel2−cohomologygroupsofacoveringmanifoldobtained
byremovingpullbackofa(normalcrossing)divisortoacoveringofacompactK¨ahlermanifold.
0
We prove that in suitable quotient categories, these groups admits natural mixed Hodge
1
structurewhosegradedpiecesaregivenbyexpected Gysinmaps.
0
2

p
e
1. introduction.
S
0 1.1. Let p:X˜ →(X,ω) be a Galois coverof a compactK¨ahler manifolds with coveringgroupG.
2 LetN(G)bethe(left)groupVonNeumannalgebraofG. Let(Sk−q,q(X˜),d)and(Sk−q,(p,q)(X˜),∂)
be the complexes built on differential q−forms (resp. (p,q)-forms) of Sobolev class k−p (k ∈ N
]
V big enough). We have the following morphism of (left) N(G)−modules:
C Imd Im∂
Kodaira decomposition: Hr (X˜)=Hr (X˜)⊕ and H(p,q)(X˜)=H(p,q)⊕
h. d(2) d(2) Imd ∂(2) ∂(2) Im∂
t Hodge to De Rham spectral sequence: Hp,q (X˜) ⇒ Hp+q(X˜)
a (2)∂ (2)d
m
K¨ahler condition: Hr (X˜) = ⊕ H(p,q)(X˜)
[ d(2) p+q=r ∂(2)

2 with Hdr(2)(X˜) and H∂(p(,2q))(X˜) respective harmonic spaces associated to Laplace operators ∆d
v and ∆ .
4 ∂
A torsion theory τ = (T,F) on Mod(N(G)) (category of N(G)−modules, T is class of torsion
8
modules, F class of free modules; see 2.4 and 3.1) is given by a Serre class T of N(G)−modules:
7
2 T is a stable by extension, quotient and submodule. Then in quotient category Mod(N(G))/τ, a
. module A ∈ T is isomorphic to {0}, a morphism α ∈ Hom (.,.) defines an inversible [α] ∈
6 N(G)
0 HomN(G)/T(.,.) iff its kernel and cokernel are in T. Moreover reduction functor Mod(N(G)) →
0 Mod(N(G)) is exact.
/τ
1 Then Theorem 3.4 shows that interpretation of K¨ahler condition gives:
:
v
Im∂
Xi Theorem 1. Let τ =(T,F)be areal torsion theorysuch that ∈T. Then Hodge tode Rham
Im∂
r spectral sequence Hp,q (X˜) ⇒ Hp+q(X˜) degenerates at E in quotient category Mod(N(G))
a (2)∂ (2)d 1 /τ
and isomorphism ⊕ H(p,q)(X˜)≃Hr (X˜) in Mod(N(G)) defines a τ−Hodge structure on
p+q=r ∂(2) d(2) /τ
Hr (X˜).
d(2)
1.2. RecallthatamixedHodgestructureisbuiltassuccessiveextensionofpureHodgestructures.
ThepurposeofthisworkistoputamixedHodgestructuremoduloatorsiontheory(localisation
at a full thick subcategory) on l2−cohomology groups H. (X^\D) when D ⊂ X is a normal
d(2)
crossing divisor.
Let p R → X, p C → X be the sheaves of L2−direct image of locally constant function
∗(2) ∗(2)
(locally isomorphic to the constant sheaf l2(G,R) and l2(G,C)). Let (p Ω. (logD),W,d) → X
∗(2) X
be the sheaves of L2−direct image ([4]) of the complex of logarithmic forms with pole on D with
its weight filtration W. Theorem 4.5 implies
Theorem 2. Let p : X˜ → X be a Galois covering of a compact K¨ahler manifold. Let G be its
group of deck transforms and let N(G) be its von Neumann algebra.
1
2 P.DINGOYAN
Let D =D ∪...∪D →X be a normal crossing divisor in X. Let τ =(T,F) be a real torsion
1 k
theory on Mod(N(G)) and let Mod(N(G)) be the quotient category.
/τ
A) Isomorphism
(1) H.(X \D,p∗(2)C)≃N(G,C) H.(X,(p∗(2)Ω.X(logD),d))
(valid for any complex manifold X) induces a weight filtration W and a Hodge filtration F .
ind
B) Assume the Hodge to de Rham spectral sequences of each D˜ :=p−1(∩ D ) (D˜ :=X˜) degen-
I i∈I i ∅
erates in Mod(N(G)) :
/τ
(2) ∀I ∈P({1,...,k}),∀r∈N, Hr (D˜ )≃⊕Hp,q (D˜ ) in Mod(N(G)) .
d(2) I ∂(2) I /τ
Then
i) weight spectral sequence
(3) E−p,p+n(W)=Hn(X,(p GrWΩ. (logD))≃Hn−p(⊔ D )⇒Hn(X \D,p C)
1 ∗(2) p X d(2) |I|=p I ∗(2)
degenerates at E in Mod(N(G)) . f
2 /τ
ii) Hodge spectral sequence Ep,q = Hq(X,p Ωp (logD)) ⇒ Hp+q(X^\D) degenerates at E in
1 ∗(2) X d(2) 1
Mod(N(G)) : induced filtration F and recurrente filtration F are equal in Mod(N(G)) .
/τ ind rec /τ
1.3. Above spectral sequences are valid on general complex manifold X and may be realised
through various Sobolev spaces.
From Deligne [6], one deduce quickly that category of mixed Hodge structure mod τ (see def.
4.2.1 ) is abelian.
We will give some details of the following basic principle in theory of mixed Hodge structures.
Degenerescence (mod some torsion theory τ) of Hodge to de Rham spectral sequence of each
coveringD˜ →D impliesdegenerescenceofthe weightspectralsequencesatE (insimple words:
I I 2
It is enough to control first constituant within a torsion theory to obtain control of the whole
spectral sequence).
1.4. It is a priori not clear wich object becomes isomorphic to {0} in a quotient category . If X
is a compact K¨ahler manifold, there exists a scale of torsion theories wich gives non trivial mixed
Hodge structure:
Let τ be the torsiontheorysuchthat module ofzeroG−dimension aretorsion. FromAtiyah
dim
Imd
[1], reduced cohomology spaces are of finite G−dimension and dim =0,
N(G) Imd
Im∂
dim = 0. Reduction of above spectral sequence to Mod(N(G)) always defines a
N(G) Im∂ τdim
mixed Hodge structure and isomorphism E ≃E in Mod(N(G)) gives expected realisation
2 ∞ /τdim
in term of harmonic spaces and Gysin maps:
Ep,q(W)≃Hq−2p(D˜ )
1 (2) p
Ep,q(F)≃Hq (Ωp (logD˜))
1 ∂(2) X
(after a choice of hermitian metric on vector bundles Ω. (logD)). Count of dimensions prove
X
for example that GrWHn (X^\D) is isomorphic (in Mod(N(G)) ) to H (T,l2(G)) the first
n (2) /τdim 1
(reduced) homology group of the dual CW-complex associated to D˜ →D.
1.5. Finer torsion theories are available: We prove in lemma 3.1, that for any α ∈ Im∂, there
exists r ∈N(G) an almost isomorphism (injective with dense range) such that rα∈Im∂.
This is a particular case of the T−lemma 2.2.8 wich is well known to operator algebraists.
Let U(G) be the ring of affiliated operator of N(G): it is the quotient ring of N(G) by the
multiplicative set of almost isomorphism (see Lu¨ck [23]Chap. 8). Above lemma reads
Im∂
U(G)⊗ =0.
N(G)
Im∂
SOME MIXED HODGE STRUCTURES ON l2−COHOMOLOGY OF COVERING OF KA¨HLER MANIFOLDS I. 3
Therefore workingin Mod(N(G)) allows easy computation with Hilbert Modules (one could
/τU(G)
also work directly in category of U(G)−modules and associated sheaves).
1.6. Let N(G) be the local constant sheaf associated to N(G). There is a natural exact functor
N from sheaves on X to sheaves of N(G)−modules givenby F →N(G)⊗C[G]p!p∗F. (p! is direct
image with compact support). Let K be a cohomological Hodge complex of sheaves on X. Then
N(K)viewasacomplexesofN(G)−sheavesmoduloτ shouldbeaminimal(τ ,τ )−CHMC
dim dim dim
(see 4.7). In this article we obtain a (0,γ∗τ )−CHMC, wich might be view as a maximal object
∂
(4.7).
1.7. In last section, we treat simple examples through various choice of torsion theories.
1.8. Inthis article,wefocus onthe technicalpartofmixedHodgestructuremodulo sometorsion
theory. In a subsequent paper, functoriality, geometricaland analyticalapplications will be given.
1.9. We give a short list of references:
- We tried to follow articles of Deligne [6], [7]. The reference books of El Zein [12],Voisin [36],
and Peters Steenbrick [28] contains steps to mixed Hodge structures and we extracted from [28]
presentation of this theory.
- ArticleofFarber[14]definesanabeliancategory(objectsaremorphisms(f :A →A )offinitely
1 2
generated N(G)−module, morphisms are the natural one) in order to deal with non closed range.
According to Freyd [15] , this is the smallest abelian category wich contains category of finitely
generated hilbertian modules. Farber already develops homological algebra for theses modules.
However calculus in this categoryare not so easy (Objects are 2−termcomplexes) and localisa-
tion seems more appropriate in order to use sheaf theory.
- Article of Atiyah [1], Cheeger Gromov[5], Shubin [32] give foundational results. In [5], [32] exact
sequences of Fredholm complexes are strudied.
- ArticleofShubin[32]containsalreadyresultsonexistenceofdistributionalsectionswithprescribed
singularity. It covers (in a short format) needed analytical results on Von Neumann algebra,
G−Fredholm operators and Sobolev spaces (see also Ma Marinescu [24]).
- Article ofEyssidieux[13]propose,followingmethodofFarber,a constructionofL2−directimage
for coherent sheaves and compute its derived category (It is also suggested in the introduction to
developpe L2−mixed Hodge structure and may be L2−intersection cohomology).
- Article of Campana Demailly [4] defines L2−direct image of coherent sheaf and gives functorial
properties that will be used in this article.
- In Chapter 6 of Lu¨ck [23], an algebraic treatment of l2−cohomology is given. A dimension for
any (abstract) N(G)−module is defined, and it is proved that the ring N(G) is semi hereditary
(finitely generated submodule of a projective module is projective). Then subsequent results are
essentially stated using dimension torsion theory.
Inchapter8andarticleofSauer,Thom[29], torsiontheoryandhomologicalalgebraassociated
with dimension function or affiliated operators is used.
1.10. My hearty thanks go to S. Vassout and G. Skandalis for illuminating discussions on Von
Neumann algebra.
2. preliminaries.
2.1. real structures.
2.1.1. Godement resolution. (See Godement [17]or Bredon[2]): Let X be a topologicalspace. Let
R be a ring and R be the sheaf of rings it defines. Let A be a sheaf of (left) R−modules, let
C∗(X,A):=(C .(X,A),d) be Godement resolution ([17] p. 168) by sheaves of R−module with
God
differential R−linear. Then
A→C∗(X,A), A→C∗(X,A)=(Γ(C ∗(X,A)),d)
God
are covariant additive exact functors with value in the category of differential R−sheaf, resp.
cochain R−complexes. If X is clear from the context, we write C∗(A),...
4 P.DINGOYAN
If(F.,d)is adifferentialsheaf,letC.(F.)be the totalcomplexassociatedto the doublecomplex
C q(X,Fp). Letj :Y →X beacontinuousmapand(F.,d)beadiferentialsheafofR−modules
God
on Y. One sets R.j F. :=j C.(F.)
∗ ∗
2.1.2. real structure. Assume that R is a R−algebra,then any sheafof R−modules is also a sheaf
of R−vector spaces so that A⊗RC is a sheaf of R⊗RC−left modules. Let i:A→A⊗RC.
Then
(4) j∗(A)⊗RC → j∗(A⊗RC)
(5) C∗(X,A)⊗RCC.−(i)→⊗1C C∗(X,A⊗RC) C∗(X,A)⊗RCC.−(i→)⊗1C C∗(X,A⊗RC)
are R⊗C−isomorphisms (resp. R⊗C−isomorphisms).
2.1.3. A real structure on a R⊗RC sheaf B is a R−subsheaf A→i B such that A⊗RCi⊗→1C B is
an isomorphism. It induces a real structure on Godement resolution.
Then H.(X, A)⊗RC→H.(X, A⊗RC) is a R⊗RC isomorphism.
2.1.4. Let G be a discret group with left action of Z[G]. Let N(G) = N (G) be the left von
l
Neumann algebra of G.
ThenR[G]−isomorphisml2(G,C)∋a→Re(a)+iIm(a)∈l2(G,R)⊗C inducesdecompositions
(6) EndC(l2(G,R)⊗C) ≃ EndR(l2(G,R))⊗C
(7) EndC[G](l2(G,R)⊗C) ≃ EndR[G](l2(G,R))⊗C
(8) N (G,C) ≃ N (G,R)⊗C
r r
Left action of G on l2(G,R) defines a left action of G on EndR(l2(G,R)) and EndCl2(G,C)
so that g(m(a)) = (gm)(ga). Continuous invariant morphism for this action are N (G,R) and
r
N (G,C). An element of ρ(f) ∈ N(G,C) is represented by right convolution with a function
r
f ∈l2(G,C) wich is moderate (Dixmier [9]13.8.3): ∃C ≥ 0: ∀g ∈C[G], ||g⋆f|| ≤ C||g|| . Then
2 2
ρ(f)∈N (G,R) is equivalent to f real valued.
r
Above isomorphism induces isomorphism N(G,C)≃N(G,R)⊗C. Note that C[G] is identified
to a subring of N(G,C).
2.2. A T−lemmaon von Neumann algebra. SurveyonvonNeumannalgebraisgiveninLu¨ck
[23]chap.9. See also Pedersen [27], Dixmier [9].
Definition 2.2.1. Let A be a Von Neumann algebra on a separable Hilbert space. Let A be the
+
cone of positive operator in A.
1) A trace is a function t : A → [0,+∞] such that if λ > 0 and x,y ∈ A then t(λx) = λt(x),
+ +
t(x+y)=t(x)+t(y) and for all unitary u∈A, t(u∗xu)=t(x).
2) A state is a positive functional of norm one: ϕ∈A′ such that ϕ(A )⊂R+ and ϕ(1)=1.
+
3) A trace or a state is normal if it is continuous on increasing limit of net.
3) Defines the left kernel of a trace or a state ϕ by L = {x ∈ A : ϕ(x∗x) = 0}. One says that ϕ
ϕ
is faithful if L =0.
ϕ
4) A faithfull trace is called finite if t(1)<+∞ (and A is then called finite von Neumann algebra).
Itis called semifiniteifAt ={y ∈A : t(y)<+∞}is weakly denseinA . Then for all x∈A ,
+ + + +
t(x)= sup t(y).
y≤x, y∈At
+
Example 2.2.2.
1) Let e ∈ l2(G) be the function g → δ (g) and e the unit element. Then trace of n ∈ N (G) or
e l
N (G) is tr n:=<n(e),e>.
r N(G)
2) LetTrH be the HilbertSchmidt traceonB(H): TrHα= ||α(ei)||2 with(ei)i∈N anorthonor-
X
i
mal basis. Then Tr ⊗tr defines a semi-finite trace on B(H)⊗N(G). If t∈B(H)⊗N(G) is
H N(G)
a positive element represented(in a orthonormalbasis) by aninfinite matrix (n ) of element in
ij i,j
N(G) then TrT = tr n .
N(G) ii
X
i
SOME MIXED HODGE STRUCTURES ON l2−COHOMOLOGY OF COVERING OF KA¨HLER MANIFOLDS I. 5
3) Ifπ :A→B(H)isarepresentation,letζ ∈H ofunitnorm. Thenx7→<π(x)ζ,ζ >isapositive
functional of norm 1, called the vector state associated to (π,H,ζ).
Traces and states satisfies Cauchy-Schwartzinequality:
(9) |ϕ(y∗x)|2 ≤ϕ(x∗x)ϕ(y∗y) on Aϕ ={x∈A: ϕ(x∗x)<+∞}.
2
Therefore, left kernel L of ϕ is a linear space. Define a scalar product on Aϕ/L (wich is equal
ϕ 2 ϕ
to A/Kerϕ if ϕ is a state) by (ζ ,ζ ):=ϕ(y∗x) with x7→ζ be the quotient map.
x y x
TheGNS(Gelfand-Naimark-Segal)constructionisobtainbycompletionofthispre-Hilbertspace
(see Pedersen [27]3.3):
Lemme 2.2.3.
1) Let ϕ be a positive functional on A, then there exists a cyclic representation (π ,H ,ζ ) such
ϕ ϕ ϕ
that <π (x)ζ ,ζ >=ϕ(x).
ϕ ϕ ϕ
2) Letϕ′ beapositivefunctionalsuchthatϕ′ ≤ϕ, thenthereexistsauniquea∈π (A)′, 0≤a≤1,
ϕ
such that ϕ′(x)=(π aζ ,ζ ).
ϕ(x) ϕ ϕ
3) Hence (πϕ′,Hϕ′,ζϕ′) is a subrepresentation of (πϕ,Hϕ,ζϕ).
4) Let ϕ be the vector stateassociated to (π,H,ζ), then A/Kerϕ∋x7→xζ ∈H defines a A−linear
H
isometry between (π ,H ,ζ ) and (π,π(A)ζ ,ζ).
ϕ ϕ ϕ
5) Assume that ϕ is a faithfull state, then for any state ϕ′ on A, (πϕ′, Hϕ′, ζϕ′) is a sub- represen-
tation of (π ,H ,ζ ).
ϕ ϕ ϕ
Recall that a representation (π,H,ζ) of A is said to be cyclic if π(A)ζ is dense in H.
Proof.
1) Let H be the Hilbert space completion of (A/(Kerϕ),(.,.)). Then A/(Kerϕ) → H is dense.
ϕ ϕ
Defines π (x)ζ = ζ . One checks that this defines a continuous operator π (x) on A/Kerϕ. It
ϕ y xy ϕ
extends as a continuous operatoron H . Identity (π (x)ζ ,ζ )=ϕ(z∗xy)=(ζ ,π (x∗)ζ ) proves
ϕ ϕ y z y ϕ z
that π is a ⋆−representation. Let e be unit of A. One set ζ = ζ . Then orbit of ζ is dense in
ϕ ϕ e ϕ
H .
ϕ
2) Densely defined positive bilinear form B(ζ ,ζ ) := ϕ′(y∗x) is bounded by (ζ ,ζ ). Friedrich
x y x y
extension gives a unique operator a on H such that B(ζ ,ζ ) = (aζ ,ζ ). One checks that
ϕ x y x y
a∈π (A)′ and 0≤a≤1.
ϕ
3) Let a12 be some positive square roots in πϕ(A)′. Then
(πϕ′(z)ζx′,ζy′)ϕ′ =ϕ′(y∗zx)=(πϕ(z)a12ζx,a12ζy)ϕ. Hence densely defined morphism
a21 :A/(Kerϕ′)→A/(Kerϕ) extends as an isometric morphism U :Hϕ′ →Hϕ wich interwines
1
πϕ′ and πϕ and such that U(ζϕ′)=a2ζϕ.
4) is clear.
5) From (2), (3), one gets a subrepresentation (πϕ,Hϕ,ζϕ) → (πϕ+ϕ′,Hϕ+ϕ′,ζϕ+ϕ′) with dense
image (for ϕ is faithfull). Hence representation πϕ and πϕ+ϕ′ are isomorphic. But πϕ′ is a sub
representation of πϕ+ϕ′. (cid:3)
Note thatHilbert spacecompletionis solutionofa universalproblem. This implies thatnatural
bounded morphism (A/(Kerϕ),(.,.)ϕ)→(A/(Kerϕ′),(.,.)ϕ′) induces a morphism Hϕ →Hϕ′.
2.2.4. Standard form. A similar statements holds for the GNS construction associated to a trace.
Note however that unitary invariance of a trace t implies that t(xy) = t(yx) on At. This gives
2
further property to associated GNS space: Let (A,t) be a von Neumann algebra with a normal
faithfull tracial state (a continuous positive linear formϕ such that ϕ(1) = 1 and ϕ is a normal
trace). Then (x,y)7→t(y∗x)=(x,y) is a scalar product.
t
Let (π ,l2(A,t),ζ ) be the Hilbert space obtains through GNS construction and called it the
t t
standard form associated to (A,t).
A vector ζ ∈l2(A,t) defines two closed, densely defined, unbounded operators:
(10) λ(ζ):D(λ(ζ))=ζ A′ ∋ ζ x′ 7→ζx′ ∈l2(A,t)
t t
(11) ρ(ζ):D(ρ(ζ))=Aζ ∋ xζ 7→xζ ∈l2(A,t)
t t
6 P.DINGOYAN
The isometric densely defined operator A ∋ x → x∗ ∈ A extends to J : L2(A,t) → L2(A,t) wich
is conjugate linear, isometric and involutive.
Lemme 2.2.5.
1) U(A) = λ(A) and V(A) = ρ(A) are von Neumann subalgebra on l2(A,t) such that JU(A)J =
V(A) and U(A)′ =V(A).
2) Let ζ ∈l2(A,t). Then ρ(ζ) is bounded iff λ(ζ) is bounded.
Proof.
1) see Dixmier [9]I.6.2 theorem 2 and I.5.2.
2) see Dixmier [9]I.5.3. (cid:3)
Therefore we assume that A is a von Neumann sub-algebra of B(L2(A,t)) and let A′ acts on
L2(A,t) on the right.
Example 2.2.6. Let G be a discret group, let N (G), N (G) left (resp. right) von Neumann
l r
algebra generated by left (resp. right) translation.
Then N (G)′ =N (G) and N (G)′ =N (G) (Dixmier [9]I.5.2).
l r r l
If L∈N (G), there exists a unique l ∈l2(G) such that L(f)=l⋆f (convolution product). Let
l
h∈l2(G) then
λ(h)(f):=h⋆f ∈l2(G) and ρ(h)(f):=f ⋆h∈l2(G)
on respective domains. Then ρ(h) is bounded iff λ(h) is bounded and in this case, λ(h) ∈N (G),
l
ρ(h)∈N(G) (Dixmier[9]III7.6). Itisapparentonx→h⋆x⋆h′ thatrightandleftrepresentation
d
commute. Note that standard form of (N (G),tr) is (λ,l2(G),e).
l
Closeddenselydefinedunboundedoperatorρ(x)withx∈l2(G)areparticularcasesofoperators
affiliated to N (G):
r
Definition 2.2.7. (Murray-Von Neumann [26]chap.XVI) Let A be a finite von Neumann algebra
on H. A closed densely defined operator h:D(h)→H is said to be affiliated to A if it commutes
with A′: for all unitary u∈A′, uD(h)=D(h) and uh=hu.
Then (see Pedersen [27]5.3.10) from the bicommutant theorem, it follows that h is affiliated to
A iff f(h) ∈ A for every bounded Borel function on Spec(h). In particular, if h ≥ 0 then h is
affiliated to A iff (1+ǫh)−1h∈A for some ǫ>0.
An essential property of affiliated operator to (A,t), a finite von Neumann algebra,is that they
form an algebra (a property valid for more general semi-finite von Neumann algebra, see Lu¨ck
[23]Chap.8, Murray-Von Neumann [26]Chap. XVI, Shubin [32]):
The followinglemma is avariationonthe descriptionofN (G)f intermofaffiliatedoperatoras
l
in Murray-Von Neumann [26]9.2, developped as Radon-Nykodim theorems in Dye [11], Pedersen
[27]5.3 :
Lemme 2.2.8 (The T−lemma). Let (A,t) be a finite von Neumann algebra. Let π : A → B(H)
be a faithfull represention of A as a von Neumann subalgebra of B(H). Let T be an operator in the
commutant π(A)′ of π(A). For any ζ ∈ImT, there exists r ∈A such that π (r)(2.2.4) is injective
t
with dense range and π(r)ζ ∈Im(T).
Proof.
1) Assume that H =l2(A,t) and that ρ(ζ), T ∈A′ have dense ranges. Then T =ρ(r) with r ∈A
(2.2.5). LetT−1◦ρ(ζ)=upbethepolardecompositionofoperatorT−1ρ(ζ)affiliatedtoA′. Then
u is an isometry, p is positive, u, p are affiliated to A′. But up = [u(1+p)−1p](1+p). Note
that 1+p ≥ 1 hence (1+p)−1, (1+p)−1p ∈ A′. From (2.2.5), there exists a,y ∈ A, such that
(1+p)−1 =ρ(a)andρ(y)=u(1+p)−1p. Thenρ(ζ)◦ρ(a)=T◦ρ(y). Butρ(a)◦ρ(ζ)=ρ(a.ζ)and
T ◦ρ(y)=ρ(T(y)) (identity is valid on the dense subset A). Then Im(ρ(a))=D(1+p) is dense.
2) Let ζ ∈ImT. Then π(A)ζ ⊂ImT. Let T :KerT⊥ →ImT. Then
⊥
T′ =T ⊕id :T−1[π(A)ζ]⊕l2(A,t)→[π(A)ζ]⊕l2(A,t)
⊥ l2(A,t) ⊥
is a A−linear injective morphism with dense range. Moreover vector state associated to (ζ,1)
(x →< π(x)ζ,ζ > +t(x)) dominates t. Using GNS construction 2.2.3 (5), one deduces that there
SOME MIXED HODGE STRUCTURES ON l2−COHOMOLOGY OF COVERING OF KA¨HLER MANIFOLDS I. 7
exists A−isomorphism (not isometric) U : T−1[π(A)ζ]⊕l2(A,t) → l2(A,t) and U : [π(A)ζ]⊕
1 ⊥ 2
l2(A,t)→l2(A,t). Setting ζ˜=U (ζ,1) and T˜ =U ◦T′◦U−1, we apply (1) to conclude.
1 2 1
(cid:3)
In particular, if x∈l2(G), conductor of ρ(x)∈U(G) to N (G) is non trivial.
r
2.3. N(G)−Hilbert Modules and von Neumann dimension. A(left)N(G)−Hilbertmodule
is a Hilbert spaceV with anunitary actionU(.) ofG suchthat V is G−isometricto a G invariant
subspaceofthefreeHilbertN(G)−moduleH⊗ˆl2(G). ThenthevonNeumanngeneratedby{U(.)}
is isomorphic to Nl(G). If dimCH <+∞, then V is said to be finitely generated.
Example 2.3.1. (see Shubin [32] section 3) If E˜ → X˜ is a pullback of smooth vector bundle
E → X under a G−covering map X˜ → X, then space Ws(X˜,E˜) of section of E˜ with coefficient
in Sobolev space are N(G)−Hilbert modules.
If V is embedded as a closed G−invariant subset of H ⊗l2(G), let P ∈ N (G)⊗B(H) be its
r
projector. Defines dim V :=TrP.
N(G)
AleftN(G)−HilbertsubmoduleV ofl2(G)nisdefinedbyaG−equivariantorthogonalprojection
p onto V. Then p is identified with a matrix (p ) ∈ M (N (G)) and dim V = tr p :=
ij n r N(G) N(G)
n
tr p .
N(G) ii
X
i=1
A finitely generated projective N(G)−module is represented by an idempotent matrix A ∈
M (N(G)). Thendim P :=tr A. LetM ∈Mod(N(G)). InLu¨ck[23]Chap. 6adimension
n N(G) N(G)
function on Mod(N(G)) is defined:
dim M :=sup{dim P : P ⊂M a finitely generated projective submodule}.
N(G) N(G)
Then if V is a Hilbert N(G)−module the two dimension functions agree: both of them satisfies
thatdimensionissupremumoverfinitedimensionalsubspaces(Lu¨ck[23]p.21th.1.12andth. 6.24).
2.4. Localisation at a Torsion theory.
Definition 2.4.1. A Serre subcategory T of an abelian category A is a full subcategory T of A
such that for any exact sequence 0→A→B →C →0 in A, then
B ∈T ⇐⇒ A∈T and C ∈T .
2) In category Mod(R) of R−modules over a ring R, a Serre class T defines a hereditary torsion
theory τ =(T,F) on R (Vas [35], Golan [18]): Module in T are torsion modules. Defines class of
free modules F ={F ∈Mod(R) :∀T ∈T, Hom (T,F)=0}.
R
Lemme 2.4.2. Let τ =(T,F) be a hereditary torsion theory, then
1) F is closed under submodules, direct products and extension.
2) Any M ∈Mod(R) has a unique maximal τ−torsion submodule, denoted T (M).
τ
3) Ahereditary torsion theoryis cogeneratedbyaninjective moduleE: τ =(T,F)withT ∋S ⇐⇒
Hom (C,E)=0.
R
4) Functor T (.):Mod(R)→T is a left exact functor (N ⊂M implies that T (M)∩N =T (N)).
τ τ τ
Proof.
2) Let {N , i ∈ Λ} be the set of all torsion submodule of M. Trivial module {0} belongs to this
i
set. Then T (M):= N satisfies requiredproperties for it is a homomorphic images of ⊕ N
τ i i∈Λ i
X
i∈Λ
wich is torsion.
3), 4) See Golan [18] p.5 and p. 24. (cid:3)
If τ and τ are torsion theories, then τ is smaller than τ (τ ≤τ ) if T ⊂T iff F ⊃F .
1 2 1 2 1 2 1 2 1 2
Then if C is a class in Mod(R), the hereditary torsion theory generated by C is the smallest
hereditary torsion theory τ such that C ⊂T.
8 P.DINGOYAN
2.4.3. Quotient category. According to Grothendieck [21](1.11) and Gabriel[16]Chap.3, if T is a
SerresubcategoryofanAbeliancategoryA,thereexistsaquotientcategoryA/T withsameobject
than A and obtained by extending class of isomorphism. Moreover functor A→A/T is exact.
Definition 2.4.4. Let τ = (T,F) be a hereditary torsion theory on Mod(R). Defines Mod(R)
/τ
to be the quotient category of Mod(R) by the Serre class T:
i) Object of Mod(R) are identical with object of Mod(R);
/τ
ii) Morphisms are elements of the following inductive limite
lim{Hom (M′,N/N′) :M′ ⊂M, N′ ⊂N and M/M′, N′ ∈T}.
−→ R
Let α∈Hom (M,N) and let [α] be its image in quotient category. Then [α] is a monomor-
N(G)
phism (resp. epimorphism, resp. isomorphism)iff Kerα (resp. Cokerα, resp. Kerα and Cokerα)
is a torsion object.
2.4.5. Torsion theory and complexification. Let R be a ring wich is also an R−algebra then R⊗R
C is a C−algebra. If A ∈ Mod(R) then C ⊗R A has a structure of R ⊗R C−module through
(R,C)−isomorphism A⊗R(R⊗RC)→A⊗RC.
Forgetful functor φ : Mod(R⊗C) ∋ B → B ∈ Mod(R) is faithfull and exact and has a left
R
adjoint
(12) (C⊗RR)⊗ . :HomR(A,ER) ≃ HomC⊗RR((C⊗RR)⊗RA,E)
Let J = φ(iIdB) be automorphism in BR defined by multiplication by i in B and let JC =
J ⊗1C : BR ⊗R C → BR ⊗C be its complex linear extension. Let B¯ be the complex conjugate
R⊗RC−moduleassociatedtoB: ItsunderliyingabeliangroupisB,andmodulestructureisgiven
by the following representation ρ of R⊗RC in End(B): ρ(r⊗c)(b) = (r⊗c¯).b. Then there is a
naturalantilinearR−isomorphismB →B¯. ThisdefinesafunctoronMod(S). Moreoverantilinear
R−linear map from B to B′ are in bijection with R⊗C−linear map from B to B′ (or from B to
B′). A R⊗C−isomorphismfromB to B¯ is equivalent witha realstructureB ≃R⊗C A⊗RC onB
(A∈Mod(R)). ThisdefinesantilinearR−isomorphismHomR⊗C(A⊗C,E¯)≃HomR⊗C(A⊗C,E).
Lemme 2.4.6. Let B ∈Mod(R⊗C), then BR⊗RC≃R⊗C B⊕B¯.
Id∓iJC
Proof. Let P± = : BR ⊗C → BR ⊗C be projectors onto eigenspaces {JC = ±iId}.
2
One checks that B ∋ b → P+(b⊗1) ∈ {JC = iId} and B¯ ∋ b → P−(b⊗1) ∈ {JC = iId} are
R⊗C−isomorphisms. (cid:3)
Note that B ∋b→P+(b⊗1)∈BR⊗RC is a splitting of natural surjection BR⊗RC→B.
Definition 2.4.7. (see [18].) Let γ :R→R⊗RC be above ring extension.
i) Let τ =(T,F) be a torsion theory on Mod(R). Then γ∗τ is the torsion theory on Mod(R⊗RC)
such that B ∈Mod(S) is γ τ torsion iff B is τ−torsion.
∗ R
ii) Let τC = (TC,FC) be a torsion theory on Mod(R⊗R C). Then γ∗τC is the torsion theory on
Mod(R) such that A∈Mod(R) is γ∗τC torsion iff A⊗RC is τC−torsion.
One checks that if τ is cogenerated by injective I (2.4.2), then γ τ is cogenerated by injective
∗
I ⊗RC. Hence γ∗γ∗τ = τ for I ⊗RC ≃R I2. However if τC is a torsion theory on Mod(R⊗RC),
γ∗γ∗τC is in general strictly smaller than τC: B torsion does not implies B⊕B¯ torsion.
Definition 2.4.8. Let τC = (TτC,FτC) be a torsion theory on Mod(R⊗RC) with torsion functor
T (2.4.2). The following properties are equivalent:
τC
i) γ∗γ∗τC =τC.
ii) There exists a torsion theory τ on Mod(R) such that τC =γ∗τ.
iii) T is stable by conjugation.
τC
iv) ∀B ∈Mod(R⊗RC), TτC(B¯)=TτC(B).
A (hereditary) torsion theory on Mod(R⊗RC) is real if it satisfies one of the above properties.
Example 2.4.9. Let τC be any torsion theory then γ∗γ∗τC is a real torsion theory.
SOME MIXED HODGE STRUCTURES ON l2−COHOMOLOGY OF COVERING OF KA¨HLER MANIFOLDS I. 9
Corollary 2.5.
1) Let (τ,τC,τ′) be a torsion theories on (R,R⊗C, R) such that τ ≤ γ∗τC and τC ≤ γ∗τ′ (iff
Tτ ⊗C⊂TτC and [TτC]R ⊂Tτ′). There exists exact functors
.⊗C:Mod(R)/τ →Mod(R⊗C)/τC (.)R :Mod(R⊗C)/τC →Mod(R)τ′
such that the following diagrams
Mod(R) .⊗C- Mod(R⊗C) Mod(R⊗C) (.)R- Mod(R)
? ? ? ?
Mod(R)/τ .⊗C- Mod(R⊗C)/τC Mod(R⊗C)/τC (.)R- Mod(R)/τ′
commute up to an equivalence of functors.
2) In particular if τ =τ′ and τC =γ∗τ, then (.⊗C, (.)R) are adjoint functors.
3) Let (τ′,γ τ′) be a torsion theories on (R,R⊗C) such that τ′ ≤ τ. Then there exists an exact
∗
functorfrom(Mod(R)/τ′,Mod(C⊗R)/γ∗τ′)to(Mod(R)/τ,Mod(C⊗R)/γ∗τ)wichcommutetotensor
product up to an equivalence.
Proof. Kernel of Mod(R)C→⊗.Mod(C⊗RR)→Mod(C⊗R)/τC is spanned by module M such that
C⊗M is τC torsion wich follows from M being τ torsion. One conclude from Gabriel[16] p. 368
Cor. 2 and Cor. 3 or Faith15.9. Other assertion are proved in the same way. (cid:3)
Notations 2.5.1. Let τ be a (hereditary) torsion theory on Mod(R). Then τ ⊗C := γ τ is the
∗
real torsion theory on Mod(R⊗RC) associated to τ.
Definition 2.5.2. Defines a category R of modules with real structure modτ:
Objectsarepair(B,α),B ∈Mod(C⊗R)τ⊗C withα:B →C⊗AanisomorphisminMod(C⊗R)τ⊗C.
A morphism f : (B,α) → (B′,α′) is a map f : B → B′ such that there exists g : A → A′ with
1C⊗g◦α=α′◦f.
Lemme 2.5.3. Then R is abelian.
2.6. Examples of torsion theories.
2.6.1. From Dickson [8]section 3, if C is a class of modules defines
(13) L(C):={B ∈Mod(R): ∀C ∈C, Hom (B,C)=0},
R
(14) R(C):={B ∈Mod(R): ∀C ∈C, Hom (C,B)=0}.
R
Then torsion theory generated by τ is given by T =LR(C).
C C
2.6.2. A multiplicative system S is a subset in R stable by multiplication. Then T = {M ∈
S
Mod(R) : ∀m ∈ M, ∃s ∈ S s.t.sm = 0} defines a Serre class. We are interested in torsion theory
Imf
i
generated by A = ⊕ with f a bounded morphism between N(G)−Hilbert module. If
1≤i≤j i
Imf
i
A∈Mod(R), let
S ={r∈N(G)with dense range: ∃a∈A with ra=0}⊂∪ (0:a).
a∈A
Then properties of almost G−dense subset (see Shubin [32]or [26] Chap. XVI) imply that S is
a multiplicative system. A N(G)−module M is τ = (T ,F ) torsion iff ∀m ∈ M, ∃s ∈ S s.t.
S S S
sm=0. Note that intersection of multiplicative systems is a multiplicative system.
2.6.3. According to Lu¨ck [23] Chap. 6, there is a dimension function dim : Mod(N(G)) →
N(G)
[0,+∞] defined on N(G)−modules wich is additive on short exact sequences and wich coincides
with Von Neumann dimension for N(G)−Hilbert modules.
Therefore T = {M ∈ Mod(N(G)) : dim M = 0} is a Serre class and defines a torsion
dim N(G)
theory τ on N(G). This torsion theory is real for a N(G)−module P is projective finitely
dim
generated iff P¯ is (see 2.3).
10 P.DINGOYAN
A¯
Standard exemples of τ torsion module are given in 3.1. Modules of the shape with A a
dim
A
G−invariant subspace of a finite G−dimensional G−Hilbert modules are zero dimensional. This
follows from the normality of the dimension function (see a proof in 3.1).
2.6.4. AlgebraU(G) ofaffiliatedoperatortoN(G) isstudiedinMurray-VonNeumann[26],Lu¨ck
[23] Chap. 8. and Reich [30].
Elements f ∈ U(G) is a G−equivariant unbounded operator f : dom(f) ⊂ l2(G) → l2(G). Let
M ∈ Mod(N(G)). Defines T (M):= Ker(M →U(G)⊗ M). This defines a torsion class T
U N(G) U
and torsion theory τ .
U
An element m ∈ M belongs to T (M) iff there exists an r ∈ N(G) wich is a non right zero
U
divisor (iff r is a weak isomorphism) such that rm=0. Indeed r becomes inversible in U(G).
According to Vas [35]p.9, a module F ∈Mod(N(G)) is τ −torsion free iff F is flat.
U
This torsion theory is real for F is flat iff F¯ is.
Exemples of τ torsion modules are given in 3.1.
U
2.7. Direct L2 image. Let p:X˜ →X be a covering map between complex manifolds. Let G be
the group of deck transformations.
Let N(G) := N(G,C) be the its (left) von Neumann algebra, and let N(G) be the sheaf of
rings it defines. According to Campana Demailly [4], if F is a coherent analytic sheaf on X, there
existsasheafp F calleddirectL2 imagesuchthatp (.)isanexactfunctoronthecategoryof
∗(2) ∗(2)
coherent analytic sheaves to the category of sheaves (Campana Demailly [4]prop2.6). We change
from notation of Campana Demailly [4]: our p F is written there p F˜.
∗(2) ∗(2)
Campana Demailly [4]Cor 2.7 proves
Lemme 2.7.1. For any analytic coherent sheaf F, morphism p O⊗F →p F is an isomor-
∗(2) ∗(2)
phism.
This isomorphism defines on p F a structure of N(G)− sheaf compatible with natural struc-
∗(2)
ture of Z[G]−sheaf.
Definition 2.7.2. Let K be a subring of C. Let p K be the sheaf defined by the presheaf
∗(2)
U →{f ∈L2(π−1(U,K)and locally constant}
Then p (R) is a N(G,R)−module and p (C) is a N(G,C)−module.
∗(2) ∗(2)
Lemme 2.7.3. Let D : E → F be a differential operator with holomorphic coefficiants between
holomorphic vector bundles. Then there exists an operator p D :p E →p F
∗(2) ∗(2) ∗(2)
Proof. In local trivialisation On ≃ E, Om ≃ F, D is given has a ∂ with a holomorphic
I I I
X
|I|≤r
function. p E, and p F are O modules, hence It suffice to study D = ∂ . But the claim is
∗(2) ∗(2) I
then a consequence of Cauchy inequalities. (cid:3)
Lemme 2.7.4.
0→p C→p O →d p Ω1...→d p Ωn →0
∗(2) ∗(2) ∗(2) ∗(2)
is well defined and exact.
Question: It seems natural to extends functor direct L2−images to an exact functor from cat-
egory of D coherent modules to category of N(G)−sheaves. So that above lemma and proof of
X
prop. 4.1 will be simple consequences.
3. Hodge to de rham spectral sequence.
Letp:X˜ →X beaG−cover. Fixahermitian(laterK¨ahler)metriconX,takespullbackmetric
on X˜. Let ∆ be the maximalextensionof the Hodge laplaciandd∗+d∗d on complex forms. Then
A=(1+∆)12 isapositiveinjectiveoperatorwithdensedomain. LetA′p,q bethe sheafofcurrents
of bidegree (p,q), and let A′k be the sheaf of currentof degree k. Let s∈R. Define Sobolev space
Ss(X˜)tobespaceofcurrentαsuchthatAsα∈L2(X˜)LetU anopensubsetinX. Defineuniform