Projectively Invariant Cocycles of Holomorphic Vector Fields on an Open Riemann Surface

8 Pages·0.12 MB·English

by  S. Bouarroudj

Projectively Invariant Cocycles of Holomorphic Vector Fields
on an Open Riemann Surface
1
0
0
2

n
a S. BOUARROUDJ
J
Department of Mathematics, Keio University, Faculty of Science & Technology. 3-14-1, Hiyoshi,

8
Kohoku-ku, Yokohama, 223-8522,Japan. E-mail:[email protected]

]
V
H. GARGOUBI
C
I.P.E.M., route de Kairouan, 5019 Monastir, Tunisia. E-mail:[email protected]
.
h
t Abstract
a
m
Let Σ be an open Riemann surface and Hol(Σ) be the Lie algebra of holomorphic
[ vector fields on Σ. We fix a projective structure (i.e. a localSL2(C)−structure)on Σ.

We calculate the first groupof cohomology of Hol(Σ) with coefficients in the space of
1
linear holomorphic operators acting on tensor densities, vanishing on the Lie algebra
v
7 sl2(C). The result is independent on the choice of the projective structure. We give
5 explicit formulæ of 1-cocycles generating this cohomology group.
0
1
0 1 Introduction
1
0
/ The first group of cohomology of the Lie algebra of (formal) vector fields on the circle S1
h
with coefficients in the space Hom(F ,F ), whereF is the space of tensor densities of de-
t λ µ λ
a
gree λ on S1, was firstcalculated in [6]. This group of cohomology measures all extensions
m
of exact sequences 0 → F → · → F → 0 of modules. The first group of cohomology of
: µ λ
v the Lie algebra Vect(S1) of (smooth) vectors fields on the circle with coefficients in the
i
X space of linear differential operators acting from λ−densities to µ−densities, vanishing
r on the subalgebra sl2(R) ⊂ Vect(S1), was calculated in [2]. This group of cohomology
a
appears as an obstruction to the equivariant quantization (see [2], [8]). The computation
is based on the following observation: any 1-cocycle vanishing on the subalgebra sl (R) is
2
ansl (R)−invariant operator. Thesl (R)−invariant differential operatorsacting ontensor
2 2
densities, whicharecalled “Transvectants”, wereclassified byGordan(see[7,11]). Tofind
the 1-cocycles generating the group of cohomology means, therefore, to determine which
from the Transvectants are 1-cocycles.
In this paper, we study the complex analog of the above group of cohomology on an
open Riemann surface Σ endowed with a flat projective structure.
The aim of this paper is to describe the first group of cohomology
H1(Hol(Σ),sl (C);D ), (1.1)
2 λ,µ
1
of holomorphic vector fields on Σ with coefficients in the space of linear differential op-
erators acting on tensor densities, vanishing on the Lie algebra sl (C). We give explicit
2
formulæof 1-cocycles generating thegroup(1.1). These1-cocycles are thecomplex analog
of the 1-cocycles given in [2].
The main tool of this paper is the existence of affine and projective connection on
any open Riemann surface (see [10, 13]). These notions has been recently used in [16] to
compute the second group of cohomology of the Lie algebra Hol(Σ) with coefficients in
the space of λ−densities.
2 Affine and Projective Structure
Let Σ be a Riemann surface, and let {U ,z } be an atlas of Σ.
α α
Aholomorphic affineconnection is afamily of holomorphic functionsΓ(z ) on U such
α α
that for non-empty U ∩U , we have
α β
dz d2z dz
β β α
Γ(z ) = Γ(z )+ ·
β dz α dz2 dz
α α β
Affine connection exists in any open Riemann surface in contrast with the compact case
where affine connection exists only if the genus of Σ is one (see [10]).
A holomorphic projective connection is a family of holomorphic functions R(z ) on U
α α
such that for non-empty U ∩U , we have
α β
2
dz
b
R(z ) = R(z )+S(z ,z ),
β α β α
(cid:18)dz (cid:19)
α
where S(z ,z )= d2zβ ′− 1 d2zβ 2 is the Schwarzian derivative.
β α dz2 2 dz2
(cid:16) α (cid:17) (cid:16) α (cid:17)
Recall that any holomorphic affine connection Γ defines naturally a holomorphic pro-
jective connection given in a local coordinates z by
dΓ(z) 1
R(z)= − Γ2(z). (2.1)
dz 2
For any projective connection R there exits locally an affine connection Γ satisfying (2.1).
We say that two affine connections are projectively equivalent if they define the same
projective connection.
Let us define the notion of projective structure.
A Riemann surface admits a projective structure if there exists an atlas of charts
{U ,z } such that the coordinate change z ◦z−1 are projective transformations.
α α α β
Inthiscase,theLiealgebrasl (C),ineachchartoftheprojectivestructure,isgenerated
2
by the following vector fields
d d d
, z , z2 . (2.2)
dz dz dz
Thereexistsa1-1correspondencebetweenprojectivestructuresandprojectiveconnections
on an open Riemann surface (cf. [10]). We use implicitly this correspondence along this
paper.
2
3 Hol(Σ)-module Structures on the Space of Differential
Operators
The modules of linear differential operators on the space of tensor densities on a (real)
smooth manifold has been studied in series of recent papers (see [1, 2, 3, 4, 8, 9, 14, 15]).
Note that this space viewed as a module over the Lie algebra of vector fields has already
been studied in the classical monograph [17].
Let us give the definition of the natural two-parameter family of modules over the Lie
algebra of holomorphic vector fields on the space of linear differential operators.
3.1 Tensor Densities
Let Σ be an open Riemann surface. Fix an affine connection Γ on it.
Space of tensor densities on Σ, noted F , is the space of sections of the line bundle
λ
(T∗Σ)⊗λ, where λ ∈ C. This bundle is of course trivial, since any (holomorphic) bundle
on an open Riemann surface is holomorphically trivial.
Fix a global section dzλ on F . Any λ−density can be written in the form φ dzλ. Let
λ
us recall the definition of a covariant derivative of tensor densities. Let ∇ bethe covariant
derivative associated to the affine connection Γ. If φ ∈ F , then ∇φ ∈ Ω1(Σ)⊗F given
λ λ
by the formula
dφ
∇φ= −λΓφ.
dz
The standard action of Hol(Σ) on F reads as follows (cf. [16]):
λ
Lλ (φ) = X∇φ+λφ∇X, (3.1)
X
d
where X = X(z) ∈ Hol(Σ).
dz
3.2 Hol(Σ)−Module of Differential Operators
Consider differential operators acting on tensor densities:
A: F → F . (3.2)
λ µ
In local coordinates z, any operator A can be written in the form
dk
A= a (z) +···+a (z),
k dzk 0
where a , for i= 1,k, are holomorphic functions on z.
i
Atwo-parameterfamilyofactionsofHol(Σ)onthespaceofdifferentialoperatorsisdefined
by
Lλ,µ(A) = Lµ ◦A−A◦Lλ . (3.3)
X X X
Denote by D (Σ) the space of operators (3.2) endowed with the defined Hol(Σ)-module
λ,µ
structure (3.3).
3
4 Main Result
Assume Σ endowed with a projective structure, this defines locally an action of the Lie
group SL (C) on Σ.
2
Consider cochains on Hol(Σ) with values in D , vanishing on the Lie algebra sl (C).
λ,µ 2
One,therefore,obtainstheso-calledrelativecohomologyoftheLiealgebraHol(Σ),namely
H1(Hol(Σ),sl (C);D (Σ)),
2 λ,µ
(see [5]).
The purpose of this paper is the following:
Theorem 4.1 Thefirstgroupofcohomology H1(Hol(Σ),sl (C);D (Σ))isone-dimensional
2 λ,µ
in the following cases:
(a) µ−λ =2,λ 6= −1/2,
(b) µ−λ = 3,λ 6= −1,
(c) µ−λ = 4,λ 6= −1/2,
(d) (λ,µ) =(−4,1),(0,5).
Otherwise, this cohomology group is trivial.
This Theorem generalizes the result of [2] in the case of the circle S1.
Note that this result does not depend on the choice of the projective structure.
5 Construction of the 1-cocycles
In this section, we give explicit formula for the 1-cocycles generating the nontrivial coho-
mology classes from Theorem 4.1. Given a projective structure on Σ, we prove that there
is a canonical choice of the 1-cocycles vanishing on sl (C).
2
Fix (locally) an affine connection Γ related to the projective structure. Denote by R
the projective connection associated to Γ (see section 2.)
Lemma 5.1 The following linear differential operators
I = R,
3
λ
I = R∇ − ∇R,
4
2
2λ+1 λ(2λ+1) λ(λ+3)
I = R∇2 − ∇R∇ + ∇2R+ R2,
5
2 10 5
3 3 4
I = R∇3 − ∇R∇2 + ∇2R+ R2 ∇,
6
2 (cid:18)10 5 (cid:19)
9 63 4 14 8
I′ = R∇3 + ∇R∇2 + ∇2R+ R2 ∇ + ∇3R+ R∇R,
6 2 (cid:18)10 5 (cid:19) 5 5
are globally defined in Dλ,λ+2(Σ), Dλ,λ+3(Σ), Dλ,λ+4(Σ), D0,5(Σ), D−4,1(Σ), respectively,
and depend only on the projective class of the connection Γ.
Proof. Since the surface Σ is projectively flat, the connection R defines a 2-density on
Σ. Then the operators of the above Lemma are globally defined. Let us prove that the
operators depend only on the projective class of the connection Γ.
4
LetI = R∇+α∇R.DenotebyI˜ theoperatorI writtenwithrespecttoaconnection
4 4 4
Γ˜ which is projectively equivalent to Γ. After an easy calculation one has
I = I˜ +(2α+λ)(Γ˜ −Γ)R.
4 4
Then I = I˜ if and only if α= −λ/2. The proof is analogous for the operators I ,I and
4 4 5 6
I′.
6
Theorem 5.2 (i) For every λ, there exist unique (up to constant) 1-cocycles
J : Hol(Σ) → D
3 λ,λ+2
J : Hol(Σ) → D
4 λ,λ+3
J : Hol(Σ) → D
5 λ,λ+4
vanishing on sl (C). They are given by the formulæ:
2
J (X) = ∇3X −Lλ,λ+2(I ),
3 X 3
λ
J (X) = ∇3X∇ − ∇4X −Lλ,λ+3(I ),
4 2 X 4
2λ+1 λ(2λ+1)
J (X) = ∇3X∇2 − ∇4X∇ + ∇5X −Lλ,λ+4(I ).
5 2 10 X 5
For (λ,µ) = (0,5),(−4,1), respectively, the 1-cocycles vanishing on sl (C) are given by
2
3 3
J0(X) = ∇3X∇3 − ∇4X∇2 + ∇5X∇ −L0,5(I ),
6 2 10 X 6
9 63 14
J−4(X) = ∇3X∇3 + ∇4X∇2 + ∇5X∇ + ∇6X −L−4,1(I′).
6 2 10 5 X 6
(ii) The 1-cocycles J , J and J are nontrivial for every λ except λ = −1/2,λ = −1 and
3 4 5
λ = −3/2, respectively. The 1-cocycles J0 and J−4 are nontrivial.
6 6
(iii) These 1-cocycles are independent on the choice of the projective structure.
6 sl (R)−invariant operators on S1
2
For almost all λ and µ, there exits unique (up to constant) sl (R)−invariant bilinear
2
λ,µ
differential operators J : F ⊗F → F given by
m λ µ λ+µ+m
2λ+m−1 2µ+m−1
Jλ,µ(φ,ψ) = (−1)im! φ(i)ψ(j), (6.1)
m (cid:18) i (cid:19)(cid:18) j (cid:19)
i+Xj=m
called “Transvectants” (see [9, 11]).
Let us recall the results of [2, 9].
The first group of cohomology H1(Vect(S1),sl (R);D (S1)) is one-dimensional in the
2 λ,µ
following cases:
(a) µ−λ = 2,λ 6= −1/2,
(b) µ−λ = 3,λ 6= −1,
(c) µ−λ = 4,λ 6= −1/2,
(d) (λ,µ) = (−4,1),(0,5).
Otherwise, this cohomology group is trivial (see [2]).
5
Thisgroup of cohomology is generated by thefollowing 1-cocycles whichare particular
cases of the Transvectants (6.1):
′′′
J (X,φ) = X φ,
3
λ
J (X,φ) = X′′′φ′− XIVφ,
4
2
2λ+1 λ(2λ+1)
J (X,φ) = X′′′φ′′− XIVφ′+ XVφ,
5 (6.2)
2 10
3 3
J (X,φ) = X′′′φ′′′− XIVφ′′+ XVφ′,
6
2 10
9 63 14
J′(X,φ) = X′′′φ′′′+ XIVφ′′+ XVφ′+ XVIφ.
6 2 10 5
(see [2, 9]).
The 1-cocycles given in Theorem 5.2 are the complex analogue of the 1-cocycles (6.2).
7 Proof of the main Theorems
In this section, we will prove Theorem 4.1 and Theorem 5.2.
7.1 Proof of the Theorem 5.2
To prove that the operators J , for k = 3,4,5,6, are 1-cocycles one has to check the
k
1-cocycle relation. It reads as follows
λ,λ+k−1 λ,λ+k−1
J [X,Y]−L (J (Y))+L (J (X)) = 0, (7.1)
k X k Y k
where X,Y ∈ Hol(Σ).
λ,λ+2
Let us verify (7.1) for the operator J . It is obvious that L (R) is a 1-cocycle. It
3 X
suffices then to verify the relation (7.1) for the 0−order operator ∇3X :
∇3([X,Y])φ−Lλ,λ+2(∇3Yφ)+Lλ,λ+2(∇3Xφ)= (2∇X∇3Y +X∇4Y −2∇Y∇3X −Y∇4X)φ
X Y
−X∇(∇3φ)−(λ+2)∇X∇3Yφ
+∇3Y(X∇φ+λ∇Xφ)+Y∇(∇3Xφ)
+(λ+2)∇Y∇3Xφ−∇3X(Y∇φ+λ∇Yφ)
= 0.
Let us prove that the 1-cocycle J vanishes on sl (C). Let X be one of the vector fields
3 2
(2.2). After calculation one has∇3X = 2R∇X+X∇R.Itis easy to see that Lλ,λ+2(I ) =
X 3
2R∇X +X∇R. Hence, one obtains J (X) = 0.
3
In the same manner we prove that the operators J , for k = 4,5,6, are 1-cocycles
k
vanishing on sl (C). Theorem 5.2 (i) is proven.
2
Let us prove the non-triviality of the 1-cocycle J for λ 6= −1/2. Suppose that the
3
1-cocycle J (Σ) is trivial, then there exists an operator A ∈ D such that
3 λ,λ+2
λ,λ+2
J (X) = L (A). (7.2)
3 X
Inaneighborhoodof apointz, onecan choice alocal coordinates suchthattheconnection
R = 0.Inthesecoordinates, the1-cocycle J coincides withthe1-cocyle J of(6.2). Hence
3 3
6
the relation (7.2) implies that the 1-cocycle J is trivial which is absurd (see section 6).
3
For λ = −1/2, take A = −2∇2 ∈ D−1,3(Σ). One can easily check that J3(X) =
2 2
−1/2,3/2
L (A).
X
With the same arguments we prove the non-triviality of the 1-cocyles J , for k = 4,5,6.
k
Theorem 5.2 (ii) is proven.
7.2 Proof of Theorem 4.1
Letusprovethatthedimensionofthegroupofcohomology H1(Hol(Σ),sl (C);D (Σ))is
2 λ,µ
boundedbythe dimensionof thegroupof cohomology H1(Vect(S1),sl (R);D (S1)). Let
2 λ,µ
C andC′betwo1-cocyclesinH1(Hol(Σ),sl (C);D (Σ)).WewillprovethatC andC′are
2 λ,µ
cohomologous. Denote C˜ and C˜′ the restriction of C and C′ on a neighborhood of a point
of Σ. The operators C˜ and C˜′ define 1-cocycle in H1(Vect(S1),sl (R);D (S1)). These
2 λ,µ
1-cocycles are equal (up to constant); since the unique sl (R)−invariant linear differential
2
operators are given as in (6.2). It follows that the 1-cocyle C and C′ are cohomologous.
Now from the construction of the 1-cocyles given in Theorem (5.2) follows Theorem (4.1).
8 Final Remark
The group of cohomology H1(Diff(S1),PSL (R);D ) is one-dimension, for generic λ,
2 λ,µ
generated by the following 1-cocyles
S (f)= S(f),
λ
d λ
′
T (f)= S(f) − S(f),
λ
dx 2
d2 2λ+1 d λ(2λ+1) λ(λ+3)
U (f)= S(f) − S(f)′ + S(f)′′− S(f)2,
λ dx2 2 dx 10 5 (8.1)
d3 3 d2 3 4 d
V0(f)= S(f)dx3 − 2S(f)′dx2 +(cid:18)10S(f)′′+ 5S(f)2(cid:19)dx
d3 9 d2 63 4 d 14 8
V−4 = S(f)dx3 + 2S(f)′ dx2 +(cid:18)10S(f)′′+ 5S(f)2(cid:19)dx + 5 S(f)′′′+ 5S(f)S(f)′,
where S(f) is the Schwarzian derivative (see [2]). It is a remarkable fact to see that the
1-cocycles (8.1) on the group Diff(S1) have the same expression (up to change of sign)
than the operators of the Lemma 5.1 if one replaces the connection R by −S(f).
Sincethegroupof biholomorphicmapson aRiemann surfaceisfinitedimension(see [12]),
this group does not integrate the Lie algebra of holomorphic vector fields. In some sense,
the cohomology group (1.1) contains informations coming from the cohomology of the
diffeomorphisms of S1 and the cohomology of the Lie algebra of vector fields on S1.
Acknowledgments. It is a pleasure to acknowledge numerous fruitful discussions with
Ch. Duval, V. Ovsienko and F. Wagemann. The first author is grateful to the JSPS for
the research support, and Prof. Y. Maeda and Keio University for their hospitality.
7
References
[1] S. Bouarroudj, Projectively equivariant quantization map, Lett. Math. Phy. 51: (4),
2000, 265-274.
[2] S.Bouarroudj &V.Ovsienko, Three cocycles on Diff(S1) generalizing the Schwarzian
derivative, IMRN 1998, No.1, 25–39.
[3] C. Duval & V. Ovsienko, Space of second order linear differential operators as a
module over the Lie algebra of vector fields, Adv. in Math. 132:2 (1997) 316–333.
[4] C. Duval & V. Ovsienko, Conformally equivariant quantization, Preprint CPT, 1998.
[5] D.B.Fuchs,Cohomology ofinfinite-dimensionalLiealgebras,Contemp.SovietMath.,
Consultants Bureau, New york, 1986.
[6] B. L. Feigin & D. B. Fuchs, Homology of Lie algebras on vector fields on the line,
Funkts. Anal. Prilozhen., 16, No. 2, 47-63 (1982).
[7] P. Gordan, Invariantentheorie, teubner, Leipzig, 1887.
[8] H. Gargoubi, Sur la g´eom´etrie de l’espace des op´erateurs diff´erentiels lin´eaires sur R.
Bull. Soc. Roy. Sci. Liege. Vol. 69, 1, 2000, pp. 21-47.
[9] H. Gargoubi & V. Ovsienko, Space of linear differentiel operators on the real line as
a module over the Lie algebra of vector fields, IMRN, (1996) N.5, pp. 235-251.
[10] R. C. Gunning, Lectures on Riemann surfaces, Princeton Mathematical Notes 1966.
[11] S. Janson & J. Peetre, A new generalization of Hankel operators (the case of higher
weights), Math. Nachr. 132 (1987) 313-328.
[12] S. Kobayashi, Transformation groups in differential geometry, Springer-Verlag,
Berlin, 1995.
[13] S. Kobayashi, C. Horst, Topics in complex differential geometry, in: Complex Differ-
ential Geometry, Birkh¨auser Verlag, 1983, 4–66.
[14] P.B.A. Lecomte, P. Mathonet & E. Tousset, Comparison of some modules of the Lie
algebra of vector fields, Indag. Math., N.S., 7:4 (1996), 461–471.
[15] P.B.A. Lecomte & V. Ovsienko, Projectively invariant symbol calculus , Lett. Math.
Phy. 49 (3) (1999), 173-196.
[16] F. Wagemann, Explicit formulæfor cocycles of holomorphic vector fields with values
in λ−densities. To appear in Journal of Lie Theory.
[17] E.J.Wilczynski, Projective differential geometry of curves and ruled surfaces, Leipzig
- Teubner - 1906.
8