Table Of ContentLOCAL INDEX THEOREM FOR ORBIFOLD RIEMANN
SURFACES
7
1
0 LEON A. TAKHTAJANANDPETER ZOGRAF
2
r
a Abstract. WederivealocalindextheoreminQuillen’sformforfami-
M liesofCauchy-RiemannoperatorsonorbifoldRiemannsurfaces(orRie-
mann orbisurfaces) that are quotients of the hyperbolic plane by the
8 actionofcofinitefinitelygeneratedFuchsiangroups. Eachconicalpoint
2 (oraconjugacyclassofprimitiveellipticelementsintheFuchsiangroup)
gives rise to an extra term in the local index theorem that is propor-
]
tionaltothesymplecticformofanewK¨ahlermetriconthemodulispace
G
of Riemann orbisurfaces. We find a simple formula for a local K¨ahler
A
potential of theelliptic metric and show that when the order of elliptic
. element becomeslarge, theelliptic metricconvergestothecuspidalone
h
corresponding to a puncture on the orbisurface (or a conjugacy class
t
a of primitive parabolic elements). We also give a simple example of a
m
relation between the elliptic metric and special values of Selberg’s zeta
[ function.
2
v
1
7
7 1. Introduction
0
0 Quillen’s local index theorem for families of Cauchy-Riemann operators
.
1 [8]explicitlycomputesthefirstChernformofthecorrespondingdeterminant
0 linebundlesequippedwithQuillen’smetric. Theadvantageoflocalformulas
7
becomes apparent when the families parameter spaces are non-compact.
1
: In the language of algebraic geometry, Quillen’s local index theorem is a
v
manifestation of the “strong” Grothendieck-Riemann-Roch theorem that
i
X claims an isomorphism between metrized holomorphic line bundles.
r The literature on Quillen’s local index theorem is abundant, but mostly
a
deals with families of smooth compact varieties. In this paper we derive a
general local index theorem for families of Cauchy-Riemann operators on
Riemann orbisurfaces, both compact and with punctures, that appear as
quotients X = Γ H of the hyperbolic plane H by the action of finitely
\
generated cofinite Fuchsian groups Γ. The main result (cf. Theorem 2) is
the following formula on the moduli space associated with the group Γ:
Date: March 30, 2017.
1991 Mathematics Subject Classification. 14H10, 58J20, 58J52.
Key words and phrases. Fuchsian groups, determinant line bindles, Quillen’s metric,
local index theorems.
Supported byRScFgrant 16-11-10039.
1
2 LEONA.TAKHTAJANANDPETERZOGRAF
6k2 6k+1 1
Q
c (λ , ) = − ω ω
1 k ||·||k 12π2 WP − 9 cusp
l
1 k 1 1
m B − ωell , k 1.
− 4π j 2 m − 6m2 j ≥
j=1 (cid:18)(cid:26) j (cid:27)(cid:19) j!
X
Q
Here c (λ , ) is the first Chern form of the determinant line bundle
1 k || · ||k
λ of the vector bundle of square integrable meromorphic k-differentials on
k
X = Γ H equipped with the Quillen’s metric, ω is a symplectic form
WP
\
of the Weil-Petersson metric on the moduli space, ω is a symplectic
cusp
form of the cuspidal metric (also known as Takhtajan-Zograf metric), ωell
j
is the symplectic form of a Ka¨hler metric associated with elliptic fixpoints,
B (x) = x2 x+1 isthe2ndBernoullipolynomial, and x is thefractional
2 − 6 { }
part of x Q. We refer the reader to Sections 2.1–2.3 and 3.2 for the
∈
definitionsandprecisestatements. Notethattheaboveformulaisequivalent
to formula (3.13) for k 0 because the Hermitian line bundles λ and λ
k 1 k
≤ −
on the moduli space are isometrically isomorphic, see Remark 3.
Note that the case of smooth puncturedRiemann surfaces was treated by
us much earlier in [10], and now we add conical points into consideration.
ThemotivationtostudyfamiliesofRiemannorbisurfacescomesfromvarious
areas of mathematics and theoretical physics – from Arakelov geometry [6]
to the theory of quantum Hall effect [3]. In particular, the paper [6] that
establishes the Riemann-Roch type isometry for non-compact orbisurfaces
as Deligne isomorphism of metrized Q-line bundles stimulated us to extend
the results of [10] to the orbisurface setting.
The paper is organized as follows. Section 2 contains the necessary back-
groundmaterial. InSection3weprovethelocalindextheoremforfamiliesof
∂¯-operatorsonRiemannorbisurfacesthatarefactorsofthehyperbolicplane
by the action of finitely generated cofinite Fuchsian groups. Specifically, we
show that the contribution to the local index formula from elliptic elements
of Fuchsian groups is given by the symplectic form of a Ka¨hler metric on
the moduli space of orbisurfaces. Since the cases of smooth (both compact
and punctured) Riemann surfaces have been well understood by us quite
a while ago [14, 10], in Section 3.2 we mainly emphasize the computation
of the contribution from conical points corresponding to elliptic elements.
In Section 4.1 we find a simple formula for a local Ka¨hler potential of the
elliptic metric, and in Section 4.2 we show that in the limit when the or-
der of the elliptic element tends to the elliptic metric coincides with the
∞
corresponding cusp metric. Finally, in Section 4.3 we give a simple example
of a relation between the elliptic metric and special values of Selberg zeta
function for Fuchsian groups of signature (0;1;2,2,2).
Acknowledgments. We thank G. Freixas i Montplet for showing to us a
preliminaryversionof[6]andforstimulatingdiscussions. Ourspecialthanks
are to Lee-Peng Teo for carefully reading the manuscript and pointing out
to us a number of misprints.
LOCAL INDEX THEOREM FOR ORBIFOLD RIEMANN SURFACES 3
2. Preliminaries
2.1. Hyperbolic plane and Fuchsian groups. We will use two models
of the Lobachevsky (hyperbolic) plane: the upper half-plane H = z
{ ∈
dz 2
C Imz > 0 with the metric | | , and the Poincar´e unit disk D =
} (Imz)2
(cid:12) 4du 2
u(cid:12) C u < 1 with the metric | | . The biholomorphic isometry
{ ∈ | | } (1 u2)2
−| |
between(cid:12) the two models is given by the linear fractional transformation
z (cid:12)z
u= − 0 for any z H.
0
z z¯ ∈
0
AFu−chsiangroupΓofthefirstkindisafinitelygenerated cofinitediscrete
subgroupofPSL(2,R)actingonH(itcanalsobeconsideredasasubgroupof
PSU(1,1) acting on D). Such Γ has a standard presentation with 2g hyper-
bolic generators A ,B ,...,A ,B , n parabolic generators S ,...,S and l
1 1 g g 1 n
elliptic generators T ,...,T of orders m ,...,m satisfying the relations
1 l 1 l
A B A 1B 1...A B A 1B 1S ...S T ...T = I,
1 1 −1 1− g g −g g− 1 n 1 l
Tmi = I, i= 1,...,l,
i
where I is the identity element. The set (g;n;m ,...,m ), where 2 m
1 l 1
≤ ≤
m , is called the signature of Γ, and we will always assume that
l
··· ≤
l
1
2g 2+n+ 1 > 0.
− − m
i=1(cid:18) i(cid:19)
X
We will be interested in orbifolds X = Γ H (or X = Γ D, if we treat Γ
\ \
as acting on D) for Fuchsian groups Γ of the first kind. Such an orbifold
is a Riemann surface of genus g with n punctures and l conical points of
2π 2π
angles ,..., . By a (p,q)-differential on the orbifold Riemann sur-
m m
1 l
face X = Γ H we understand a smooth function φ on H that transforms
\ q
according to the rule φ(γz)γ (z)pγ (z) = φ(z). The space of harmonic
′ ′
(p,q)-differentials, square integrable with respect to the hyperbolic met-
ric on X = Γ H, we denote by Ωp,q(X). The dimension of the space of
\
square integrable meromorphic (with poles at punctures and conical points)
k-differentials on X, or cusp forms of weight 2k for Γ, is given by Riemann-
Roch formula for orbifolds:
l 1
(2k 1)(g 1)+(k 1)n+ k 1 , k > 1,
− − − − m
i=1(cid:20) (cid:18) i(cid:19)(cid:21)
dimΩk,0(X) = g, k = 1, P
1, k = 0,
0, k < 0,
where [r] denotes the integer part of r Q (see [9, Theorem 2.24]). In
∈
particular,
dimΩ2,0(X) = 3g 3+n+l.
−
4 LEONA.TAKHTAJANANDPETERZOGRAF
The elements of the space Ω 1,1(X) are called harmonic Beltrami differ-
−
entials and play an important role in the deformation theory of Fuchsian
groups, see Sect. 2.3. To study the behavior of harmonic Beltrami differen-
tials at the elliptic fixpoints we use the unit disk model. Take µ Ω 1,1(X)
−
∈
and let T Γ be an elliptic element of order m with fixpoint z H. The
0
∈ z z ∈
pushforwardofT toDbymeansofthemapu = − 0 isjustthemultiplica-
z z¯
0
−
tion by ω = e2π√ 1/m, the m-th primitive root of unity. Thepushforwardof
−
µ to D (that, slightly abusing notation, we will denote by the same symbol)
develops into a power series of the form
(1 u2)2 ∞
µ(u) = −| | a¯nu¯n−2.
4
n=2
X
Moreover, since µ(ωu) = µ(u)ω 2 we have a = 0 unless n 0 mod m, so
− n
≡
that
(1 u2)2 ∞
µ(u) = −| | a¯jmu¯jm−2. (2.1)
4
j=1
X
∂µ
In particular, µ(0) = 0 for m > 2 and (0) = 0 for m = 2.
∂u
As in [10], for µ,ν Ω 1,1(X) we put f = (∆ + 1) 1(µν¯), where
∈ − µν¯ 0 2 −
∂2
∆ = y2 , y = Imz ,
0
− ∂z∂z¯
is the Laplace operator (or rather 1/4 of the Laplacian) in the hyperbolic
metric acting on Ω0,0(X). The function f (u) is regular on D and satisfies
µν¯
f (ωu) = f (u).
µν¯ µν¯
The following result is analogues to Lemma 2 in [10] and describes the
behavior of f (u) at u = 0. We will use polar coordinates on D such that
µν¯
u= re√ 1θ.
−
Lemma 1. Let
f (u) = ∞ f (r)e√ 1jmθ (2.2)
µν¯ jm −
j=
X−∞
be the Fourier series of the function f (u) on D. Then
µν¯
(i) f (r) = c +c r2+O(r4) as r 0, where
0 0 2
→
2c , m > 2,
0
c = (2.3)
2
(2c0 4µ(0)ν¯(0), m = 2.
−
(ii) f (r)= O(rn) as r 0;
n | |
→
LOCAL INDEX THEOREM FOR ORBIFOLD RIEMANN SURFACES 5
(iii) For the constant term c = f (0) we have
0 0
c = G(0,u)µ(u)ν(u)dρ(u),
0
Z
X
where G(u,v) is the integral kernel of (∆ + 1) 1 on X = Γ D, and
0 2 − \
2√ 1
dρ(u) = − du du¯.
(1 u2)2 ∧
−| |
Proof. Since f (u) is a regular solution of the equation (∆ + 1)f = µν¯ at
µν¯ 0 2
u= 0, we have in polar coordinates
(1 r2)2 ∂2f 1∂f 1 ∂2f 1
− + + (r,θ)+ f(r,θ)=
− 16 ∂r2 r ∂r r2 ∂θ2 2
(cid:18) (cid:19)
= (1−r2)4 ∞ ∞ a¯imbjmr(i+j)m−4e√−1(j−i)mθ,
16
i=1 j=1
XX
where we used (2.1) for µ(u) and the analogous expansion
ν(u) = (1−|u|2)2 ∞ ¯bjmu¯jm−2. (2.4)
4
j=1
X
for ν(u). Then for the term f (r) of the Fourier series (2.2) we have the
0
differential equation
−(1−16r2)2 d2dfr02(r) + 1rdfd0(rr) + 12f0(r) = (1−16r2)4 ∞ a¯jmbjmr2jm−4.
(cid:18) (cid:19) j=1
X
From here we get that f (r)= c +c r2+O(r4) as r 0, where
0 0 2
→
2c , m > 2,
0
c =
2
(2c0 4µ(0)ν¯(0), m = 2.
−
For the coefficients f (r) with n= 0 we have
n
6
(1 r2)2 d2f 1df n2f (r) 1
n n n
− + + f (r) =
− 16 dr2 r dr − r2 2 n
(cid:18) (cid:19)
(1 r2)4 ∞
= − a¯jmbjm+nr2jm+n−4,
16
j=1
X
sothatf (r) = O(rn)asr 0. Thisprovesparts(i)and(ii)ofthelemma,
n | |
→
from where it follows that c = f (0). To prove part (iii) it is sufficient to
0 µν¯
observe that
f (0) = G(0,u)µ(u)ν(u)dρ(u).
µν
Z
X
(cid:3)
6 LEONA.TAKHTAJANANDPETERZOGRAF
2.2. Laplacians on Riemann orbisurfaces. Let us now switch to the
properties of the Laplace operators on the hyperbolic orbifold X = Γ H,
\
where Γ is a Fuchsian group of the first kind. Here we give only a brief
sketch, and thedetails can befoundin [14], [10]. Denote by p,q theHilbert
space of (p,q)-differentials on X, and let ∂¯ : k,0 k,1Hbe the Cauchy-
k
H → H
Riemann operator acting on (k,0)-differentials (in terms of the coordinate
z on H we have ∂¯ = ∂/∂z¯). Denote by ∂¯ : k,1 k,0 the formal adjoint
to ∂¯ and define tkhe Laplace operator actk∗ingHon (→k,0H)-differentials on X by
k
the formula ∆ = ∂¯ ∂¯ .
k k∗ k
1
We denote by Qk(z,z′;s) the integral kernel of ∆k + (s−1)(4s−2k)I − on
theentireupperhalf-planeH(whereI is theidenti(cid:16)ty operator intheH(cid:17)ilbert
spaceofk-differentialsonH). ThekernelQ (z,z ;s)issmoothforz = z and
k ′ ′
6
has an important property that Q (z,z ;s) = Q (σz,σz ;s)σ (z)kσ (z )k for
k ′ k ′ ′ ′ ′
any σ PSL(2,R). For k 0 and s = 1 we have the explicit formula
∈ ≥
2k
∂ ∂ 1 1 z z¯
∂zy−2k∂z′Q−k(z,z′;1) = −π · (z−z′)2 (cid:18)z¯′−−z′′(cid:19) , (2.5)
where y = Imz.
Furthermore, denote by G (z,z ;s) the integral kernel of the resolvent
k ′
1
∆k + (s−1)(4s−2k)I − of ∆k on X = Γ\H (where I is the identity operator
i(cid:16)n the Hilbert spa(cid:17)ce k,0). For k < 0 and s = 1 the Green’s function
H
G (z,z ;s) is a smooth function on X X away from the diagonal (i. e. for
k ′
×
z = z ). For k = 0 we have the following Laurent expansion near s = 1:
′
6
4 1
G (z,z ;s) = +G (z,z )+O(s 1) (2.6)
0 ′ 0 ′
X · s(s 1) −
| | −
as s 1, where X = 2π 2g 2+n+ l (1 1/m ) is the hyperbolic
→ | | − i=1 − i
area of X = Γ H. Then fo(cid:16)r any integer k 0 we have (cid:17)
\ P≥
2k
∂ ∂ 1 1 γz¯ γz
∂zy−2k∂z′ G−k(z,z′;1) = −π γ Γ (z−γz′)2 (cid:18) z¯′−−γz′′(cid:19) γ′(z′)γ′(z¯′)−k .
X∈
(2.7)
This series converges absolutely and uniformly on compact sets for z =
6
γz , γ Γ.
′
∈
We now recall the definition of the Selberg zeta function. Let Γ be a
Fuchsiangroupofthefirstkind,andletχ :Γ U(1)beaunitarycharacter.
→
Put
∞
Z(s,Γ,χ) = 1 χ(γ)N(γ) s i , (2.8)
− −
−
γ i=0
Y{ } Y(cid:0) (cid:1)
where γ runs over the set of classes of conjugate hyperbolic elements of
{ }
Γ, and N(γ) is the norm of γ defined by the conditions N(γ)+1/N(γ) =
LOCAL INDEX THEOREM FOR ORBIFOLD RIEMANN SURFACES 7
trγ , N(γ) > 1 (in other words, logN(γ) is the length of the closed ge-
| |
odesic in the free homotopy class associated with γ). The product (2.8)
converges absolutely for Res > 1 and admits a meromorphic continuation
to the complex s-plane.
Except for the last section, in what follows we will always assume that
χ 1 and will denote Z(s,Γ,1) simply by Z(s). The Selberg trace formula
≡
relates Z(s) to the spectrum of the Laplacians on Γ H, and it is natural
\
(cf. [4]) to define the regularized determinants of the operators ∆ by the
k
−
formula
Z (1), k = 0,
′
det∆ = (2.9)
k
− (Z(k+1), k 1,
≥
(note that Z(s) has a simple zero at s= 1).
2.3. Deformation theory. We proceed with the basics of the deformation
theory of Fuchsian groups. Let Γ be a Fuchsian group of the first kind of
signature(g;n;m ,...,m ). Considerthespaceofquasiconformalmappings
1 l
oftheupperhalf-planeHthatfix0,1and . Twoquasiconformalmappings
∞
are equivalent if they coincide on the real axis. A mapping f is compatible
with Γ if f 1 γ f PSL(2,R) for all γ Γ. The space of equivalence
−
◦ ◦ ∈ ∈
classes of Γ-compatible mappings is called the Teichmu¨ller space of Γ and
is denoted by T(Γ). The space T(Γ) is isomorphic to a bounded complex
domaininC3g 3+n+l. TheTeichmu¨ller modulargroupMod(Γ)acts onT(Γ)
−
by complex isomorphisms. Denote by Mod (Γ) the subgroup of Mod(Γ)
0
consisting of pure mapping classes (i. e. those fixing the punctures and
elliptic points on X pointwise). The factor T(Γ)/Mod (Γ) is isomorphic to
0
themodulispace ofsmoothcomplexalgebraiccurvesofgenusg with
g,n+l
M
n+l labeled points.
Remark 1. Note that T(Γ), as well as the quotient space T(Γ)/Mod(Γ),
actually depends not on the signature of Γ, but rather on its signature type,
the unordered set r = r ,r ,... , where r = n and r is the number of
1 2 1 i
{ }
elliptic points of order i, i = 2,3,... (see [2]).
The holomorphic tangent and cotangent spaces to T(Γ) at the origin are
isomorphic to Ω 1,1(X) and Ω2,0(X) respectively (where, as before, X =
−
Γ H). Let B 1,1(X) be the unit ball in Ω 1,1(X) with respect to the L
− − ∞
\
norm and let β : B 1,1(X) T(Γ) be the Bers map. It defines complex
−
→
coordinates in the neighborhood of the origin in T(Γ) by the assignment
(ε ,...,ε ) Γµ = fµ Γ (fµ) 1,
1 d −
7→ ◦ ◦
where µ = ε µ + +ε µ , µ ,...,µ is a basis for Ω 1,1(X), and fµ is a
1 1 d d 1 d −
···
quasiconformal mapping of H that fixes 0, 1, and satisfies the Beltrami
∞
equation
fµ = µfµ.
z¯ z
8 LEONA.TAKHTAJANANDPETERZOGRAF
∂ ∂
For µ Ω 1,1(X) denote by and the partial derivatives along the
−
∈ ∂ε ∂ε¯
µ µ
holomorphic curve β(εµ) in T(Γ), where ε C is a small parameter.
TheCauchy-Riemann operators ∂¯ form∈a holomorphicMod(Γ)-invariant
k
family of operators on T(Γ). The determinant bundle λ associated with
k
∂¯ is a holomorphic Mod(Γ)-invariant line bundle on T(Γ) whose fibers are
k
given by the determinant lines maxker∂¯ ( maxcoker∂¯ ) 1. Since the
k k −
kernel and cokernel of ∂¯ are th∧e spaces of⊗har∧monic differentials Ωk,0(X)
k
and Ωk,1(X) respectively, the line bundle λ is Hermitian with the metric
k
induced by the Hodge scalar products in the spaces Ωp,q(X) (note that each
orbifold Riemann surface X = Γ H inherits a natural metric of constant
\
negative curvature 1). The corresponding norm in λ we will denote by
k
−
. Notethatbyduality betweenΩk,0(X)andΩ1 k,1(X)thedeterminant
k −
||·||
line bundles λ and λ are isometrically isomorphic.
k 1 k
−
The Quillen norm in λ is defined by the formula
k
Q k
= ||·|| (2.10)
||·||k √det∆
k
for k 0 and is extended for all k by the isometry λ = λ . The de-
≤ k ∼ 1−k
terminant det∆ defined via the Selberg zeta fuction is a smooth Mod(Γ)-
k
invariant function on T(Γ).
3. Main results
Our objective is to compute the canonical connection and the curvature
(or the first Chern form) of the Hermitian holomorphic line bundle λ on
k
T(Γ). By Remark 1 λ can be thought of as holomorphic Q-line bundle on
k
the moduli space T(Γ)/Mod(Γ).
3.1. Connection formon the determinantbundle. Westartwithcom-
puting the connection form on the determinant line bundle λ for k > 0
k
−
relative to the Quillen metric. The following result generalizes Lemma 3 in
[10]:
Theorem 1. For any integer k 0 and µ Ω 1,1(X) we have
−
≥ ∈
∂
logdet∆ =
k
∂εµ εµ=0 −
(cid:12)
(cid:12)
= (cid:12) ∂y−2k∂′ G k(z,z′;1) Q k(z,z′;1) µ(z)d2z , (3.1)
−ZX − − − (cid:12)z′=z
∂ ∂(cid:0) dz dz¯ (cid:1)(cid:12)
where ∂ = , ∂ = , and d2z = ∧ is the (cid:12)Euclidean area form on
′
∂z ∂z 2√ 1
′
H. − −
Remark 2. Theintegral in(3.1)is absolutely convergent if m > 2for alli =
i
1,...,l. If m = 2 for some i, then this integral should be understood in the
i
principalvaluesenseasfollows. Letz bethefixpointoftheellipticgenerator
i
LOCAL INDEX THEOREM FOR ORBIFOLD RIEMANN SURFACES 9
z z
T Γ of order 2, and consider the mapping h : H D, h (z) = − i.
i i i
∈ → z z¯
i
−
Denote by B = u D u < δ the disk of radius δ in D with center at
δ
{ ∈ | | }
0. Since Γ is discrete, for δ small enough we have h 1(B ) γh 1(B ) =
(cid:12) −i δ ∩ −j δ ∅
unless i = j and γ is eith(cid:12)er I or T . The subset
i
H = H γh 1(B )
δ \ i mi=2 γ Γi Γ −i δ
{ | } ∈ \
is Γ-invariant, where Γi den(cid:0)oStes the cycSlic group of order(cid:1)2 generated by Ti.
The factor X = Γ H is an orbifold Riemann surface with holes centered
δ δ
\
at the conical points of angle π. The integral in the right hand side of (3.1)
we then define as
lim ∂y 2k∂ G (z,z ;1) Q (z,z ;1) µ(z)d2z .
− ′ k ′ k ′
δ→0 ZXδ − − − (cid:12)z′=z
(cid:0) (cid:1)(cid:12)
Proof. We will use the results of [10] profoundly. Re(cid:12)peating verbatim the
proof of Lemma 3 in [10] we get for k 0
≥
∂
logdet∆ = ∂∂ G (z,z ;k+1) Q (z,z ;k+1)
k ′ 0 ′ 0 ′
∂εµ(cid:12)εµ=0 − −XZ (cid:16) − −
(cid:12)
(cid:12) Q (z,γz ;k+1) Q (z,γz ;k+1) µ(z)d2z .
0 ′ 0 ′
− − z′=z
γpγaXr∈aΓb,olic γγeXl∈liΓp,tic (cid:17)(cid:12)(cid:12)
(cid:12)
(3.2)
Note that by Lemma 3 in [10] the contribution from parabolic elements to
the right hand side of (3.2) vanishes, i. e.
∂∂ Q (z,γz ;k+1) µ(z)d2z = 0 .
′ 0 ′
z′=z
XZ γpγaXr∈aΓb,olic (cid:12)(cid:12)
(cid:12)
By Lemma 4 in [10] we can further rewrite (3.2) as follows:
∂
logdet∆ = ∂y 2k∂ G (z,z ;1) Q (z,z ;1)
k − ′ k ′ k ′
∂εµ(cid:12)εµ=0 − −XZ (cid:16) − − −
(cid:12)
(cid:12) Q (z,γz ;1)γ (z¯) k µ(z)d2z .
k ′ ′ ′ −
− − z′=z
γγeXl∈liΓp,tic (cid:17)(cid:12)(cid:12)
(cid:12)
Theintegrand in theright handsideis smooth and theintegral is absolutely
convergent, cf. (2.7). We need to show that
∂y 2k∂ Q (z,γz ;1)γ (z¯) k µ(z)d2z = 0 (3.3)
− ′ k ′ ′ ′ −
− z′=z
ΓZ\H (cid:16)γγeXl∈liΓpt,ic (cid:17)(cid:12)(cid:12)
(cid:12)
(if there is m = 2 we understand this integral as the principle value, see
i
Remark 2).
10 LEONA.TAKHTAJANANDPETERZOGRAF
Withoutlossofgenerality wemayassumethatl = 1andΓhasoneelliptic
generator T of order m with fixpoint z H. Then by (2.5) we have
0
∈
∂y 2k∂ Q (z,γz ;1)γ (z¯) k
− ′ k ′ ′ ′ −
− − z′=z
γγeXl∈liΓp,tic (cid:17)(cid:12)(cid:12)
(cid:12)
2k
1 1 γz γz¯
= − γ′(z)γ′(z¯)−k
π (z γz)2 z¯ γz
γ Γ, − (cid:18) − (cid:19)
γeXl∈liptic
(z z¯)2k γ (z)k+1
′
= −
π (z γz)2(z¯ γz)2k
γ Γ, − −
γeXl∈liptic
(z z¯)2k m−1 (σ−1Tiσ)′(z)k+1
= −
π (z σ 1Tiσz)2(z¯ σ 1Tiσz)2k
− −
σ∈XΓ0\Γ Xi=1 − −
(z z¯)2k m−1 σ′(z)σ′(z¯)k(Tiσ)′(z)k+1
= −
π (σz Tiσz)2(σz¯ Tiσz)2k
σ∈XΓ0\Γ Xi=1 − −
1 m−1(σz σz¯)2k(Ti)′(σz)k+1σ′(z)2
= −
π (σz Tiσz)2(σz¯ Tiσz)2k
σ∈XΓ0\Γ Xi=1 − −
1
= φ(σz)σ (z)2 ,
′
π
σ∈XΓ0\Γ
where Γ = Z/mZ is the cyclic group generated by T (the stabilizer of z in
0 ∼ 0
Γ), and
m−1 (z z¯)2k(Ti)′(z)k+1
φ(z) = −
(z Tiz)2(z¯ Tiz)2k
i=1 − −
X
Since φ(Tz)T (z)2 = φ(z), it is easy to check that the last expression in
′
the above formula is a (meromorphic) quadratic differential on X. Using
z z
0
the standard substitution u= − we get
z z¯
0
−
(z z¯)2k(Ti)(z)k+1 ωi(k+1) (1 u2)2k du 2
′
− = −| | . (3.4)
(z Tiz)2(z¯ Tiz)2k (1 ωi)2u2 · (1 ωi u2)2k dz
− − − − | | (cid:18) (cid:19)
Sinceµ(0) =0 for m > 2, see (2.1), the integral in the left hand side of (3.3)
is absolutely convergent, and we have
∂y 2k∂ Q (z,γz ;1)γ (z¯) k µ(z)d2z
− ′ k ′ ′ ′ −
− z′=z
XZ (cid:16)γγeXl∈liΓpt,ic (cid:17)(cid:12)(cid:12)
(cid:12)
