256
12 Geometry of Moduli Spaces and Riemann Surfaces
for fixed τ α can be interpreted as a change of the coordinate σ α :
σ
α = F α (τ α ) + δF α (τ α ) = σ α + δF α (τ α ) = σ α + δF α
F
−1
α (σ α )
.
(12.20)
This transformation is generated by a vector field v (α) on the Riemann surface g,n :
σ
α = σ α + v
(α) (σ α ),
v
(α)
= δF α ◦ F
−1
α .
(12.21)
The situation is symmetrical, and one can obviously fix σ α and vary τ α . The vector
field is regular around the circle C α (to have a well-defined change of coordinates),
but it can have singularities away from the circle C α . Hence, the vector field v (α)
together with the circle C α defines a vector of P g,n :
V
(α)
∼
v
(α) , C (α)
.
(12.22)
This provides a basis of T P g,n . This is sufficient when using the coordinate
system (12.13), but, in more general situations, one needs to consider linear
combinations. For example, if a modulus appears in several transition functions,
then the associated vector field will be defined on the corresponding circles. A
general vector V is described by a vector field v with support on a subset C of
the circles C α :
V ∼
v, C
,
C ⊆
α
C α ,
(12.23)
and the restriction of v on the various circles is written as
v| C α = v
(α) .
(12.24)
Note that the vector field v (α) and its complex conjugate ¯
v (α) are independent and
are associated to different tangent vectors. This construction is called the Schiffer
variation.
The simplest tangent vectors ∂ s are given by varying one coordinate of P g,n while
keeping the other fixed:
x s −→ x s + δx s .
(12.25)
On each circle C α , this gives a deformation of the transition functions
C α : F α −→ F α + δF α ,
δF α =
∂F α
∂x s
δx s
(12.26)
(no sum over s), such that the change of coordinates reads
σ
α = σ α + v
(α)
s (σ α ) δx s ,
v
(α)
s (σ α ) =
∂F α
∂x s
F
−1
α (σ α )
.
(12.27)
12 Geometry of Moduli Spaces and Riemann Surfaces
for fixed τ α can be interpreted as a change of the coordinate σ α :
σ
α = F α (τ α ) + δF α (τ α ) = σ α + δF α (τ α ) = σ α + δF α
F
−1
α (σ α )
.
(12.20)
This transformation is generated by a vector field v (α) on the Riemann surface g,n :
σ
α = σ α + v
(α) (σ α ),
v
(α)
= δF α ◦ F
−1
α .
(12.21)
The situation is symmetrical, and one can obviously fix σ α and vary τ α . The vector
field is regular around the circle C α (to have a well-defined change of coordinates),
but it can have singularities away from the circle C α . Hence, the vector field v (α)
together with the circle C α defines a vector of P g,n :
V
(α)
∼
v
(α) , C (α)
.
(12.22)
This provides a basis of T P g,n . This is sufficient when using the coordinate
system (12.13), but, in more general situations, one needs to consider linear
combinations. For example, if a modulus appears in several transition functions,
then the associated vector field will be defined on the corresponding circles. A
general vector V is described by a vector field v with support on a subset C of
the circles C α :
V ∼
v, C
,
C ⊆
α
C α ,
(12.23)
and the restriction of v on the various circles is written as
v| C α = v
(α) .
(12.24)
Note that the vector field v (α) and its complex conjugate ¯
v (α) are independent and
are associated to different tangent vectors. This construction is called the Schiffer
variation.
The simplest tangent vectors ∂ s are given by varying one coordinate of P g,n while
keeping the other fixed:
x s −→ x s + δx s .
(12.25)
On each circle C α , this gives a deformation of the transition functions
C α : F α −→ F α + δF α ,
δF α =
∂F α
∂x s
δx s
(12.26)
(no sum over s), such that the change of coordinates reads
σ
α = σ α + v
(α)
s (σ α ) δx s ,
v
(α)
s (σ α ) =
∂F α
∂x s
F
−1
α (σ α )
.
(12.27)