Isomonodromic deformations of logarithmic connections and stability

Abstract

Let $X_0$ be a compact connected Riemann surface of genus g with $D_0 \subset X_0$ an ordered subset of cardinality n, and let $E_G$ be a holomorphic principal G-bundle on $X_0$, where G is a reductive affine algebraic group defined over $\mathbb C$, that is equipped with a logarithmic connection $\nabla _0$ with polar divisor $D_0$. Let $(\mathcal {E}_G , \nabla )$ be the universal isomonodromic deformation of $(E_G ,\nabla _0)$ over the universal Teichmüller curve $(\mathcal {X}, \mathcal {D})\,{\longrightarrow }\, \text {Teich}_{g,n}$, where $\text {Teich}_{g,n}$ is the Teichmüller space for genus g Riemann surfaces with n–marked points. We prove the following (see Sect. 5):

• (1) Assume that $g \ge 2$ and $n= 0$. Then there is a closed complex analytic subset $\mathcal {Y} \subset \text {Teich}_{g,n}$, of codimension at least g, such that for any $t\in \text {Teich}_{g,n} {\setminus } \mathcal {Y}$, the principal G-bundle $\mathcal {E}_G\vert _{{\mathcal X}_t}$ is semistable, where ${\mathcal X}_t$ is the compact Riemann surface over t.

• (2) Assume that $g\ge 1$, and if $g= 1$, then $n > 0$. Also, assume that the monodromy representation for $\nabla _0$ does not factor through some proper parabolic subgroup of G. Then there is a closed complex analytic subset $\mathcal {Y}' \subset \text {Teich}_{g,n}$, of codimension at least g, such that for any $t\in \text {Teich}_{g,n} {\setminus } \mathcal {Y}'$, the principal G-bundle $\mathcal {E}_G\vert _{{\mathcal X}_t}$ is semistable.

• (3) Assume that $g\ge 2$. Assume that the monodromy representation for $\nabla _0$ does not factor through some proper parabolic subgroup of G. Then there is a closed complex analytic subset $\mathcal {Y}'' \subset \text {Teich}_{g,n}$, of codimension at least $g-1$, such that for any $t\in \text {Teich}_{g,n} {\setminus } \mathcal {Y}'$, the principal G-bundle $\mathcal {E}_G\vert _{{\mathcal X}_t}$ is stable.

In [12], the second-named author proved the above results for $G= \text {GL}(2,{\mathbb C})$.