Abstract
Let
X
be a compact Riemann surface of genus
g
≥
3
{g\geq 3}
and let
S
be a finite subset of
X
. Let ξ be a fixed line bundle over
X
of degree
d
. We consider the moduli space
ℳ
Hod
(
X
,
S
,
ξ
)
{\mathcal{M}_{\rm Hod}(X,S,\xi)}
of logarithmic λ-connections singular over
S
⊂
X
{S\subset X}
of rank
n
and degree
d
with fixed residues in the center of
𝔤
𝔩
(
n
,
ℂ
)
{\mathfrak{gl}(n,\mathbb{C})}
, where
n
and
d
are mutually coprime. We determine the Picard group and investigate the automorphism
group of
ℳ
Hod
(
X
,
S
,
ξ
)
{\mathcal{M}_{\rm Hod}(X,S,\xi)}
. Let
ℳ
Hod
′
(
X
,
S
,
ξ
)
⊂
ℳ
Hod
(
X
,
S
,
ξ
)
{\mathcal{M}^{\prime}_{\rm Hod}(X,S,\xi)\subset\mathcal{M}_{\rm Hod}(X,S,\xi)}
be the open subvariety consisting of those logarithmic λ-connections whose underlying vector bundle is stable. We show that there is a natural compactification of the moduli space
ℳ
Hod
′
(
X
,
S
,
ξ
)
{\mathcal{M}^{\prime}_{\rm Hod}(X,S,\xi)}
. We also recall
the logarithmic version of Riemann–Hilbert correspondence which arises from Deligne’s extension theorem, and using this we prove the Torelli-type theorem for
ℳ
Hod
(
X
,
S
,
ξ
)
{\mathcal{M}_{\rm Hod}(X,S,\xi)}
.