Logarithmic connections on principal bundles over a Riemann surface

TL;DR

This paper establishes criteria for the existence of specific logarithmic connections on principal bundles over Riemann surfaces, considering invariance and residue conditions, advancing the understanding of connections with prescribed singularities.

Contribution

It provides necessary and sufficient conditions for the existence of $T$-invariant and general logarithmic connections with prescribed residues on principal bundles over Riemann surfaces.

Findings

Criteria for $T$-invariant logarithmic connections established.

Conditions for general logarithmic connections with residues given.

Residue conditions depend on $T$-rigidity of elements.

Abstract

Let $E_G$ be a holomorphic principal $G$-bundle on a compact connected Riemann surface $X$, where $G$ is a connected reductive complex affine algebraic group. Fix a finite subset $D \subset X$, and for each $x\in D$ fix $w_x \in \text{ad}(E_G)_x$. Let $T$ be a maximal torus in the group of all holomorphic automorphisms of $E_G$. We give a necessary and sufficient condition for the existence of a $T$-invariant logarithmic connection on $E_G$ singular over $D$ such that the residue over each $x \in D$ is $w_x$. We also give a necessary and sufficient condition for the existence of a logarithmic connection on $E_G$ singular over $D$ such that the residue over each $x \in D$ is $w_x$, under the assumption that each $w_x$ is $T$-rigid.

Equations

$$E_H(\chi )=E_H{\times }^{\chi }\mathbb{C}⟶X$$

$$w_x\in \text{ad}(E_G{)}_x$$

$$T\subset \text{Aut}(E_G).$$

$$\mathrm{degree}(E_H(\chi ))+\sum _{x\in D}d\chi (w_x)=0$$

$$\mathrm{degree}(E_H(\chi ))+\sum _{x\in D}d\chi (w_x)=0,$$

$$D:=\{x_1,\cdots ,x_n\}\subset X.$$

$$p:E_H⟶X$$

$$\mathrm{d}p:\mathrm{T}{E}_H⟶p^{\ast }\mathrm{T}X$$

$$\text{ad}(E_H):=\text{kernel}(\mathrm{d}p)/H⟶X,$$

$$\text{At}(E_H):=(\mathrm{T}{E}_H)/H⟶X.$$

$$\text{At}(E_H)=(p_{\ast }\mathrm{T}{E}_H{)}^H\subset p_{\ast }\mathrm{T}{E}_H$$

$$0⟶\text{ad}(E_H)⟶\text{At}(E_H)\stackrel{{\mathrm{d}}^{\prime }p}{⟶}\mathrm{T}X⟶0,$$

$$\text{At}(E_H,D):=({\mathrm{d}}^{\prime }p{)}^{-1}(\mathrm{T}X(-D))\subset \text{At}(E_H),$$

$$0⟶\text{ad}(E_H)\stackrel{i_0}{⟶}\text{At}(E_H,D)\stackrel{\sigma }{⟶}\mathrm{T}X(-D)⟶0,$$

$$\theta :\mathrm{T}X(-D)⟶\text{At}(E_H,D)$$

$$\begin{matrix}0&\longrightarrow&\text{ad}(E_{H})&\stackrel{{\scriptstyle i_{0}}}{{\longrightarrow}}&\text{At}(E_{H},\,D)&\stackrel{{\scriptstyle\sigma}}{{\longrightarrow}}&\mathrm{T}X(-D)&\longrightarrow&0\\ &&\|&&~{}\Big{\downarrow}j&&~{}\Big{\downarrow}\iota\\ 0&\longrightarrow&\text{ad}(E_{H})&\stackrel{{\scriptstyle i}}{{\longrightarrow}}&\text{At}(E_{H})&\stackrel{{\scriptstyle\mathrm{d}^{\prime}p}}{{\longrightarrow}}&\mathrm{T}X&\longrightarrow&0\\ \end{matrix}$$

$${\mathrm{d}}^{\prime }p(x)\circ j(x)=\iota (x)\circ \sigma (x):\text{At}(E_H,D{)}_x⟶(\mathrm{T}X{)}_x={\mathrm{T}}_{x}X.$$

$$R_x:\text{At}(E_H,D{)}_x⟶\text{ad}(E_H{)}_x$$

$$R_x\circ i_0(x)={\text{Id}}_{\text{ad}(E_H{)}_x},$$

$$\text{At}(E_H,D{)}_x=\text{ad}(E_H{)}_x\oplus \text{kernel}(R_x)=\text{ad}(E_H{)}_x\oplus \mathrm{T}X(-D{)}_x;$$

$$\text{At}(E_H,D{)}_x=\text{ad}(E_H{)}_x\oplus \mathbb{C}$$

$$\text{Res}(\theta ,x):=R_x(\theta (1))\in \text{ad}(E_H{)}_x,$$

$$\rho :H⟶M$$

$$E_M:=E_H{\times }^{\rho }M⟶X$$

$$\hat{\rho }:E_H⟶E_M,y⟼\widetilde{(y,e_M)},$$

$$\alpha :\text{ad}(E_H)⟶\text{ad}(E_M).$$

$$\tilde{A}:\text{At}(E_H)⟶\text{At}(E_M).$$

$$A:\text{At}(E_H,D)⟶\text{At}(E_M,D),$$

$$\begin{matrix}0&\longrightarrow&\text{ad}(E_{H})&\stackrel{{\scriptstyle i_{0}}}{{\longrightarrow}}&\text{At}(E_{H},\,D)&\stackrel{{\scriptstyle\sigma}}{{\longrightarrow}}&\mathrm{T}X(-D)&\longrightarrow&0\\ &&~{}\Big{\downarrow}\alpha&&~{}\Big{\downarrow}A&&\|\\ 0&\longrightarrow&\text{ad}(E_{M})&\longrightarrow&\text{At}(E_{M},\,D)&\longrightarrow&\mathrm{T}X(-D)&\longrightarrow&0\\ \end{matrix}$$

$$A\circ \theta :\mathrm{T}X(-D)⟶\text{At}(E_M,D)$$

Full text

1. Introduction

Let $X$ be a compact connected Riemann surface. Given a holomorphic vector bundle $E$ on $X$, a theorem of Weil and Atiyah says that $E$ admits a holomorphic connection if and only if the degree of every indecomposable component of $E$ is zero (see [We], [At]). Now let $G$ be a connected complex affine algebraic group and $E_{G}$ a holomorphic principal $G$–bundle on $X$. Then $E_{G}$ admits a holomorphic connection if and only if for every holomorphic reduction of structure group $E_{H}\,\subset\,E_{G}$, where $H$ is a Levi factor of some parabolic subgroup of $G$, and for every holomorphic character $\chi$ of $H$, the degree of the associated line bundle

[TABLE]

is zero [AB]. Our aim here is to investigate the logarithmic connections on $E_{G}$ with fixed residues, where $(G,\,E_{G})$ is as above. More precisely, fix a finite subset $D\,\subset\,X$ and also fix

[TABLE]

for each $x\,\in\,D$, where $\text{ad}(E_{G})$ is the adjoint vector bundle for $E_{G}$. We investigate the existence of logarithmic connections on $E_{G}$ singular over $D$ such that residue is $w_{x}$ for every $x\,\in\,D$.

Let $\text{Aut}(E_{G})$ denote the group of all holomorphic automorphisms of $E_{G}$; it is a complex affine algebraic group. Fix a maximal torus

[TABLE]

This choice produces a Levi factor $H$ of a parabolic subgroup of $G$ as well as a holomorphic reduction of structure group $E_{H}\,\subset\,E_{G}$ to $H$ [BBN]. This pair $(H,\,E_{H})$ is determined by $T$ uniquely up to a holomorphic automorphism of $E_{G}$.

The group $\text{Aut}(E_{G})$ acts on the vector bundle $\text{ad}(E_{G})$. An element of $\text{ad}(E_{G})$ will be called $T$–rigid if it is fixed by the action of $T$; some examples are given in Section 4.2.

We prove the following (see Theorem 5.3):

Theorem 1.1**.**

The following two are equivalent:

1. (1)

There is a $T$–invariant logarithmic connection on $E_{G}$ singular over $D$ with residue $w_{x}$ at every $x\,\in\,D$. 2. (2)

The element $w_{x}$ is $T$–rigid for each $x\,\in\,D$, and

[TABLE]

for every holomorphic character $\chi$ of $H$, where $E_{H}(\chi)$ is the line bundle in (1.1), and $d\chi\,:\,{\rm Lie}(H)\,\longrightarrow\,{\mathbb{C}}$ is the homomorphism of Lie algebras corresponding to $\chi$.

The Lie algebra $\mathbb{C}$ being abelian the homomorphism $d\chi$ factors through the conjugacy classes in $\text{Lie}(H)$, so it can be evaluated on the elements of $\text{ad}(E_{H})$.

We also the prove the following (see Theorem 5.1):

Theorem 1.2**.**

Assume that each $w_{x}$ is $T$–rigid. Then there is a logarithmic connection on $E_{G}$ singular over $D$, with residue $w_{x}$ at every $x\,\in\,D$, if and only if

[TABLE]

where $E_{H}(\chi)$ and $d\chi(w_{x})$ are as in Theorem 1.1.

2. Logarithmic connections and residue

2.1. Preliminaries

Let $G$ be a connected reductive affine algebraic group defined over $\mathbb{C}$. A Zariski closed connected subgroup $P\,\subset\,G$ is called a parabolic subgroup if $G/P$ is a projective variety [Bo, 11.2], [Hu]. The unipotent radical of a parabolic subgroup $P\,\subset\,G$ will be denoted by $R_{u}(P)$. The quotient group $P/R_{u}(P)$ is called the Levi quotient of $P$. A Levi factor of $P$ is a Zariski closed connected subgroup $L\,\subset\,P$ such that the composition $L\,\hookrightarrow\,P\,\longrightarrow\,P/R_{u}(P)$ is an isomorphism [Hu, p. 184]. We note that $P$ admits Levi factors, and any two Levi factors of $P$ are conjugate by an element of $R_{u}(P)$ [Hu, § 30.2, p. 185, Theorem].

The multiplicative group ${\mathbb{C}}\setminus\{0\}$ will be denoted by ${\mathbb{G}}_{m}$. A torus is a product of copies of ${\mathbb{G}}_{m}$. Any two maximal tori in a complex algebraic group are conjugate [Bo, p. 158, Proposition 11.23(ii)].

By a homomorphism between algebraic groups or by a character we will always mean a holomorphic homomorphism or a holomorphic character.

2.2. Logarithmic connections

Let $X$ be a compact connected Riemann surface. Fix a finite subset

[TABLE]

The reduced effective divisor $x_{1}+\ldots+x_{n}$ will also be denoted by $D$.

Let

[TABLE]

be a holomorphic principal $H$–bundle on $X$, where $H$ is a connected affine algebraic group defined over $\mathbb{C}$. The Lie algebra of $H$ will be denoted by $\mathfrak{h}$. Let

[TABLE]

be the differential of the map $p$ in (2.1), where $\mathrm{T}E_{H}$ and $\mathrm{T}X$ are the holomorphic tangent bundles of $E_{H}$ and $X$ respectively; note that $\mathrm{d}p$ is surjective. The action of $H$ on $E_{H}$ produces an action of $H$ on $\mathrm{T}E_{H}$. This action on $\mathrm{T}E_{H}$ clearly preserves the subbundle $\text{kernel}(\mathrm{d}p)$. Define

[TABLE]

which is a holomorphic vector bundle on $X$; it is called the adjoint vector bundle for $E_{H}$. We note that $\text{ad}(E_{H})$ is identified with the vector bundle $E_{H}\times^{H}{\mathfrak{h}}\,\longrightarrow\,X$ associated to $E_{H}$ for the adjoint action of $H$ on its Lie algebra $\mathfrak{h}$. So the fibers of $\text{ad}(E_{H})$ are Lie algebras isomorphic to $\mathfrak{h}$. Define the Atiyah bundle for $E_{H}$

[TABLE]

The action of $H$ on $\mathrm{T}E_{H}$ produces an action of $H$ on the direct image $p_{*}\mathrm{T}E_{H}$. We note that

[TABLE]

(see [At]). Taking quotient by $H$, the homomorphism $\mathrm{d}p$ in (2.2) produces a short exact sequence

[TABLE]

where $\mathrm{d}^{\prime}p$ is constructed from $\mathrm{d}p$; this is known as the Atiyah exact sequence for $E_{H}$.

The subsheaf $\mathrm{T}X\otimes{\mathcal{O}}_{X}(-D)$ of $\mathrm{T}X$ will be denoted by $\mathrm{T}X(-D)$. Now define

[TABLE]

where $\mathrm{d}^{\prime}p$ is the projection in (2.3). So from (2.3) we have the exact sequence of vector bundles on $X$

[TABLE]

where $\sigma$ is the restriction of $\mathrm{d}^{\prime}p$; this will be called the logarithmic Atiyah exact sequence for $E_{H}$.

A logarithmic connection on $E_{H}$ singular over $D$ is a holomorphic homomorphism

[TABLE]

such that $\sigma\circ\theta\,=\,\text{Id}_{\mathrm{T}X(-D)}$, where $\sigma$ is the homomorphism in (2.4). Note that giving such a homomorphism $\theta$ is equivalent to giving a homomorphism $\varpi\,:\,\text{At}(E_{H},\,D)\,\longrightarrow\,\text{ad}(E_{H})$ such that $\varpi\circ i_{0}\,=\,\text{Id}_{\text{ad}(E_{H})}$, where $i_{0}$ is the homomorphism in (2.4).

2.3. Residue of a logarithmic connection

Given a vector bundle $W$ on $X$, the fiber of $W$ over any point $x\,\in\,X$ will be denoted by $W_{x}$. For any ${\mathcal{O}}_{X}$–linear homomorphism $f\,:\,W\,\longrightarrow\,V$ of holomorphic vector bundles, its restriction $W_{x}\,\longrightarrow\,V_{x}$ will be denoted by $f(x)$.

From (2.3) and (2.4) we have the commutative diagram of homomorphisms

[TABLE]

on $X$. So for any point $x\,\in\,X$, we have

[TABLE]

Note that $\iota(x)\,=\,0$ if $x\,\in\,D$, therefore in that case $\mathrm{d}^{\prime}p(x)\circ j(x)\,=\,0$. Consequently, for every $x\,\in\,D$ there is a homomorphism

[TABLE]

uniquely defined by the identity $i(x)\circ R_{x}(v)\,=\,j(x)(v)$ for all $v\,\in\,\text{At}(E_{H},\,D)_{x}$. Note that

[TABLE]

where $i_{0}$ is the homomorphism in (2.6). Therefore, from (2.4) we have

[TABLE]

note that the composition $\text{kernel}(R_{x})\,\hookrightarrow\,\text{At}(E_{H},\,D)_{x}\,\stackrel{{\scriptstyle\sigma(x)}}{{\longrightarrow}}\,\mathrm{T}X(-D)_{x}$ is an isomorphism.

For any $x\,\in\,D$, the fiber $\mathrm{T}X(-D)_{x}$ is identified with $\mathbb{C}$ using the Poincaré adjunction formula [GH, p. 146]. Indeed, for any holomorphic coordinate $z$ around $x$ with $z(x)\,=\,0$, the image of $z\frac{\partial}{\partial z}$ in $\mathrm{T}X(-D)_{x}$ is independent of the choice of the coordinate function $z$; the above mentioned identification between $\mathrm{T}X(-D)_{x}$ and $\mathbb{C}$ sends this independent image to $1\,\in\,\mathbb{C}$. Therefore, from (2.8) we have

[TABLE]

for all $x\,\in\,D$.

For a logarithmic connection $\theta\,:\,\mathrm{T}X(-D)\,\longrightarrow\,\text{At}(E_{H},\,D)$ as in (2.5), and any $x\,\in\,D$, define

[TABLE]

where $R_{x}$ is the homomorphism in (2.7); in the above definition $1$ is considered as an element of $\mathrm{T}X(-D)_{x}$ using the identification of $\mathbb{C}$ with $\mathrm{T}X(-D)_{x}$ mentioned earlier.

The element $\text{Res}(\theta,\,x)$ in (2.10) is called the residue, at $x$, of the logarithmic connection $\theta$.

2.4. Extension of structure group

Let $M$ be a complex affine algebraic group and

[TABLE]

a homomorphism. As before, $E_{H}$ is a holomorphic principal $H$–bundle on $X$. Let

[TABLE]

be the holomorphic principal $M$–bundle obtained by extending the structure group of $E_{H}$ using $\rho$. So $E_{M}$ is the quotient of $E_{H}\times M$ obtained by identifying $(y,\,m)$ and $(yh^{-1},\,\rho(h)m)$, where $y$, $m$ and $h$ run over $E_{H}$, $M$ and $H$ respectively. Therefore, we have a morphism

[TABLE]

where $\widetilde{(y,\,e_{M})}$ is the equivalence class of $(y,\,e_{M})$ with $e_{M}$ being the identity element of $M$. The homomorphism of Lie algebras $d\rho\,:\,{\mathfrak{h}}\,\longrightarrow\,{\mathfrak{m}}\,:=\,\text{Lie}(M)$ associated to $\rho$ produces a homomorphism of vector bundles

[TABLE]

The maps $\widehat{\rho}$ and $d\rho$ together produce a homomorphism of vector bundles

[TABLE]

This map $\widetilde{A}$ produces a homomorphism

[TABLE]

which fits in the following commutative diagram of homomorphisms

[TABLE]

where the top exact sequence is the one in (2.4) and the bottom one is the corresponding sequence for $E_{M}$.

If $\theta\,:\,\mathrm{T}X(-D)\,\longrightarrow\,\text{At}(E_{H},\,D)$ is a logarithmic connection on $E_{H}$ as in (2.5), then

[TABLE]

is a logarithmic connection on $E_{M}$ singular over $D$. From the definition of residue in (2.10) it follows immediately that

[TABLE]

for all $x\,\in\,D$. This proves the following:

Lemma 2.1**.**

With the above notation, if $E_{H}$ admits a logarithmic connection $\theta$ singular over $D$ with residue $w_{x}\in{\rm ad}(E_{H})_{x}$ at each $x\,\in\,D$, then $E_{M}$ admits a logarithmic connection $\theta^{\prime}\,=\,A\circ\theta$ singular over $D$ with residue $\alpha(w_{x})$ at each $x\,\in\,D$.

3. Connections with given residues

3.1. Formulation of residue condition

Fix a holomorphic principal $H$–bundle $E_{H}$ on $X$, and fix elements

[TABLE]

for all $x\,\in\,D$. Consider the decomposition of $\text{At}(E_{H},\,D)_{x}$ in (2.9). For any $x\,\in\,D$, let

[TABLE]

be the line in the fiber $\text{At}(E_{H},\,D)_{x}$. Let

[TABLE]

be the subsheaf that fits in the short exact sequence

[TABLE]

Note that the composition

[TABLE]

is injective, hence it is an isomorphism, where $i_{0}$ is the homomorphism in (2.13); this composition will be denoted by $\phi_{x}$. Therefore, from (2.4) and (3.1) we have a commutative diagram

[TABLE]

where all the rows and columns are exact; the restriction of $\sigma$ to the subsheaf $\mathcal{A}$ is denoted by $\sigma_{1}$.

Lemma 3.1**.**

Consider the space of all logarithmic connections on $E_{H}$ singular over $D$ such that the residue over every $x\,\in\,D$ is $w_{x}$. It is in bijection with the space of all holomorphic splittings of the short exact sequence of vector bundles

[TABLE]

on $X$ in (3.2).

Proof.

Let $\theta\,:\,\mathrm{T}X(-D)\,\longrightarrow\,\text{At}(E_{H},\,D)$ be a logarithmic connection on $E_{H}$ singular over $D$ such that the residue over every $x\,\in\,D$ is $w_{x}$. From the definition of residue and the construction of $\mathcal{A}$ it follows that

[TABLE]

Therefore, $\theta$ defines a holomorphic homomorphism

[TABLE]

Evidently, we have $\sigma_{1}\circ\theta^{\prime}\,=\,\text{Id}_{\mathrm{T}X(-D)}$. So $\theta^{\prime}$ is a holomorphic splitting of the exact sequence in the lemma.

To prove the converse, let

[TABLE]

be a holomorphic homomorphism such that $\sigma_{1}\circ\theta_{1}\,=\,\text{Id}_{\mathrm{T}X(-D)}$. Consider the composition

[TABLE]

where $\nu$ is the homomorphism in (3.2). This defines a logarithmic connection on $E_{H}$ singular over $D$, because $\sigma\circ\nu\circ\theta_{1}\,=\,\sigma_{1}\circ\theta_{1}\,=\,\text{Id}_{\mathrm{T}X(-D)}$ by the commutativity of (3.2). From (3.2) it follows immediately that $\nu\circ\theta_{1}(\mathrm{T}X(-D)_{x})\,=\,\ell_{x}\,\subset\,\text{At}(E_{H},\,D)_{x}$ for every $x\,\in\,D$. Now from the definition of residue it follows that the residue of the connection $\nu\circ\theta_{1}$ at any $x\,\in\,D$ is $w_{x}$. ∎

3.2. Extension class

The short exact sequence in Lemma 3.1 determines a cohomology class

[TABLE]

where $K_{X}\,=\,({\rm T}X)^{*}$ is the holomorphic cotangent bundle of $X$. Therefore, $E_{H}$ admits a logarithmic connection singular over $D$ with residue $w_{x}\in{\rm ad}(E_{H})_{x}$ at each $x\in D$ if and only if the cohomology class $\beta$ in (3.3) vanishes.

Let $\rho:H\longrightarrow M$ be a homomorphism of affine algebraic groups. Let $E_{M}:=E_{H}\times^{\rho}M$ be the principal $M$–bundle over $X$ obtained by extending the structure group of $E_{H}$ to $M$ by $\rho$. Consider the homomorphism $\alpha$ in (2.11). It produces a homomorphism

[TABLE]

From (2.13) we have a commutative diagram

[TABLE]

where the top exact sequence is the one in Lemma 3.1 and the bottom one is the same sequence for $E_{M}$. From this diagram it follows that the cohomology class in $H^{1}(X,\,{\rm ad}(E_{M})\otimes K_{X})$ for the short exact sequence in Lemma 3.1 for $E_{M}$ coincides with $\overline{\rho}(\beta)$, where $\beta$ is the cohomology class in (3.3).

Corollary 3.2**.**

Assume that $E_{H}$ admits a holomorphic reduction of structure group $E_{J}\,\subset\,E_{H}$ to a complex algebraic subgroup $J\,\subset\,H$. The cohomology class $\beta$ in (3.3) is contained in the image of the natural homomorphism $H^{1}(X,\,{\rm ad}(E_{J})\otimes K_{X})\,\hookrightarrow\,H^{1}(X,\,{\rm ad}(E_{H})\otimes K_{X})$

3.3. A necessary condition for logarithmic connections with given

residue

Let $\theta$ be a logarithmic connection on $E_{H}$ singular over $D$. Take any character

[TABLE]

The group $H$ acts on $\mathbb{C}$; the action of $h\,\in\,H$ sends any $c\,\in\,\mathbb{C}$ to $\chi(h)\cdot c$. Let

[TABLE]

be the holomorphic line bundle over $X$ associated to $E_{H}$ for this action of $H$ on $\mathbb{C}$. Since ${\mathbb{G}}_{m}$ is abelian, the adjoint vector bundle for $E_{H}(\chi)$ is the trivial holomorphic line bundle ${\mathcal{O}}_{X}$ over $X$. The above logarithmic connection $\theta$ induces a logarithmic connection on $E_{H}(\chi)$ (see (2.14)); this induced logarithmic connection on $E_{H}(\chi)$ will be denoted by $\theta^{\chi}$.

For any $x\,\in\,D$, let

[TABLE]

be the residue of $\theta^{\chi}$. The residue $\text{Res}(\theta,\,x)$ defines a conjugacy class in the Lie algebra $\mathfrak{h}$, because any fiber of $\text{ad}(E_{H})$ is identified with $\mathfrak{h}$ uniquely up to conjugation. From (2.15) it follows immediately that $\text{Res}(\theta^{\chi},\,x)$ coincides with with $d\chi(\text{Res}(\theta,\,x))$, where $d\chi\,:\,{\mathfrak{h}}\,\longrightarrow\,\mathbb{C}$ is the homomorphism of Lie algebras associated to $\chi$; note that since the Lie algebra $\mathbb{C}$ is abelian, the homomorphism $d\chi$ factors through the conjugacy classes in $\mathfrak{h}$.

As $\theta^{\chi}$ is a logarithmic connection on the line bundle $E_{H}(\chi)$ with residue $d\chi(\text{Res}(\theta,\,x))$ at each $x\,\in\,D$, using a computation in [Oh] it follows that

[TABLE]

(see [BDP, Lemma 2.3]).

Therefore, we have the following:

Lemma 3.3**.**

Let $E_{H}$ be a holomorphic principal $H$–bundle on $X$. Fix

[TABLE]

for every $x\,\in\,D$. If there is a logarithmic connection on $E_{H}$ singular over $D$ with residue $w_{x}$ at every $x\,\in\,D$, then

[TABLE]

for every character $\chi$ of $H$, where $E_{H}(\chi)$ is the associated holomorphic line bundle, and $d\chi$ is the homomorphism of Lie algebras corresponding to $\chi$.

Let $\rho\,:\,H\,\longrightarrow\,M$ be an injective homomorphism to a connected complex algebraic group $M$ and $E_{M}\,:=\,E_{H}\times^{\rho}M$ the holomorphic principal $M$–bundle on $X$ obtained by extending the structure group of $E_{H}$ using $\rho$. As before, the Lie algebras of $H$ and $M$ will be denoted by $\mathfrak{h}$ and $\mathfrak{m}$ respectively. Using the injective homomorphism of Lie algebras

[TABLE]

associated to $\rho$, we have an injective homomorphism $\alpha$ as in (2.11). For every $x\,\in\,D$, fix an element $w_{x}\,\in\,\text{ad}(E_{H})_{x}$.

Lemma 3.4**.**

Assume that $H$ is reductive. There is a logarithmic connection on $E_{H}$ singular over $D$ with residue $w_{x}$ at each $x\,\in\,D$ if there is a logarithmic connection on $E_{M}$ singular over $D$ with residue $\alpha(x)(w_{x})$ at each $x\,\in\,D$, where $\alpha$ is the homomorphism in (2.11).

Proof.

The adjoint action of $H$ on $\mathfrak{h}$ makes it an $H$–module. On the other hand, the homomorphism $\rho$ composed with the adjoint action of $M$ on $\mathfrak{m}$ produces an action of $H$ on $\mathfrak{m}$. The injective homomorphism $d\rho$ in (3.4) is a homomorphism of $H$–modules. Since $H$ is reductive, there is an $H$–submodule $V\,\subset\,\mathfrak{m}$ which is a complement of $d\rho({\mathfrak{h}})$, meaning

[TABLE]

Let $\eta\,:\,{\mathfrak{m}}\,\longrightarrow\,{\mathfrak{h}}$ be the projection constructed from this decomposition of $\mathfrak{m}$; in particular, we have $\eta\circ d\rho\,=\,\text{Id}_{\mathfrak{h}}$.

Since the above $\eta$ is a homomorphism of $H$–modules, it produces a projection

[TABLE]

such that $\widehat{\eta}\circ\alpha\,=\,\text{Id}_{\text{ad}(E_{H})}$, where $\alpha$ is the homomorphism in (2.11).

Now if $\theta\,:\,\text{At}(E_{M},\,D)\,\longrightarrow\,\text{ad}(E_{M})$ is a logarithmic connection on $E_{M}$ singular over $D$ with residue $\alpha(x)(w_{x})$ at each $x\,\in\,D$, consider the composition

[TABLE]

where $A$ is constructed in (2.12). Evidently, it is a logarithmic connection on $E_{H}$ singular over $D$ with residue $w_{x}$ at each $x\,\in\,D$. ∎

4. $T$–rigid elements of adjoint bundle

4.1. Definition

As before, $H$ is a complex affine algebraic group and $p\,:\,E_{H}\,\longrightarrow\,X$ a holomorphic principal $H$–bundle on $X$. An automorphism of $E_{H}$ is a holomorphic map $F\,:\,E_{H}\,\longrightarrow\,E_{H}$ such that

• •

$p\circ F\,=\,p$, and

• •

$F(zh)\,=\,F(z)h$ for all $z\,\in\,E_{H}$ and $h\,\in\,H$.

Let $\text{Aut}(E_{H})$ be the group of all automorphisms of $E_{H}$. We will show that $\text{Aut}(E_{H})$ is a complex affine algebraic group.

First consider the case of $H\,=\,\text{GL}(r,{\mathbb{C}})$. For a holomorphic principal $\text{GL}(r,{\mathbb{C}})$–bundle $E_{\rm GL}$ on $X$, let $E\,:=\,E_{\rm GL}\times^{\text{GL}(r,{\mathbb{C}})}{\mathbb{C}}^{r}$ be the holomorphic vector bundle of rank $r$ on $X$ associated to $E_{\rm GL}$ for the standard action of $\text{GL}(r,{\mathbb{C}})$ on ${\mathbb{C}}^{r}$. Then $\text{Aut}(E_{\rm GL})$ is identified with the group of all holomorphic automorphisms $\text{Aut}(E)$ of the vector bundle $E$ over the identity map of $X$. Note that $\text{Aut}(E)$ is the Zariski open subset of the complex affine space $H^{0}(X,\,\text{End}(E))$ consisting of all global endomorphisms $f$ of $E$ such that $\det(f(x_{0}))\,\not=\,0$ for a fixed point $x_{0}\,\in\,X$; since $x\,\longmapsto\,\det(f(x))$ is a holomorphic function on $X$, it is in fact a constant function. Therefore, $\text{Aut}(E_{\rm GL})$ is an affine algebraic variety over $\mathbb{C}$.

For a general $H$, fix an algebraic embedding $\rho\,:\,H\,\hookrightarrow\,\text{GL}(r,{\mathbb{C}})$ for some $r$. For a holomorphic principal $H$–bundle $E_{H}$ on $X$, let $E_{\rm GL}\,:=\,E_{H}\times^{\rho}\,\text{GL}(r,{\mathbb{C}})$ be the holomorphic principal $\text{GL}(r,{\mathbb{C}})$–bundle on $X$ obtained by extending the structure group of $E_{H}$ using $\rho$. The injective homomorphism $\rho$ produces an injective homomorphism

[TABLE]

The image of $\rho^{\prime}$ is Zariski closed in the algebraic group $\text{Aut}(E_{\rm GL})$. Hence $\rho^{\prime}$ produces the structure of a complex affine algebraic group on $\text{Aut}(E_{H})$. This structure of a complex algebraic group is independent of the choices of $r$, $\rho$. Therefore, $\text{Aut}(E_{H})$ is an affine algebraic group. Note that $\text{Aut}(E_{H})$ need not be connected, although the automorphism group of a holomorphic vector bundle is always connected (as it is a Zariski open subset of a complex affine space).

The Lie algebra of $\text{Aut}(E_{H})$ is $H^{0}(X,\,\text{ad}(E_{H}))$. The group $\text{Aut}(E_{H})$ acts on any fiber bundle associated to $E_{H}$. In particular, $\text{Aut}(E_{H})$ acts on the adjoint vector bundle $\text{ad}(E_{H})$. This action evidently preserves the Lie algebra structure on the fibers of $\text{ad}(E_{H})$.

Let $\text{Aut}(E_{H})^{0}\,\subset\,\text{Aut}(E_{H})$ be the connected component containing the identity element. Fix a maximal torus

[TABLE]

An element $w\,\in\,\text{ad}(E_{H})_{x}$, where $x\,\in\,X$, will be called $T$–rigid if the action of $T$ on $\text{ad}(E_{H})_{x}$ fixes $w$.

Consider the adjoint action of $H$ on itself. Let

[TABLE]

be the associated holomorphic fiber bundle. Since this adjoint action preserves the group structure of $H$, the fibers of $\text{Ad}(E_{H})$ are complex algebraic groups isomorphic to $H$. More precisely, each fiber of $\text{Ad}(E_{H})$ is identified with $H$ uniquely up to an inner automorphism of $H$. The corresponding Lie algebra bundle on $X$ is $\text{ad}(E_{H})$.

The group $\text{Aut}(E_{H})$ is the space of all holomorphic sections of $\text{Ad}(E_{H})$. For any $x\,\in\,X$, the action of $\text{Aut}(E_{H})$ on the fiber $\text{ad}(E_{H})_{x}$ coincides with the one obtained via the composition

[TABLE]

where ${\rm ev}_{x}$ is the evaluation map that sends a section of $\text{Ad}(E_{H})$ to its evaluation at $x$, and ${\rm ad}$ is the adjoint action of the group $\text{Ad}(E_{H})_{x}$ on its Lie algebra $\text{ad}(E_{H})_{x}$.

Therefore, an element $w\,\in\,\text{ad}(E_{H})_{x}$ is $T$–rigid if and only if the adjoint action of ${\rm ev}_{x}(T)\,\subset\,\text{Ad}(E_{H})_{x}$ on $\text{ad}(E_{H})_{x}$ fixes $w$.

4.2. Examples

The center of $H$ will be denoted by $Z_{H}$. Let $E_{H}$ be a holomorphic principal $H$–bundle on $X$. Since $Z_{H}$ commutes with $H$, for any $t\,\in\,Z_{H}$, the map $E_{H}\,\longrightarrow\,E_{H}$, $z\,\longmapsto\,zt$ is $H$–equivariant. Therefore, we have $Z_{H}\,\subset\,\text{Aut}(E_{H})$. The principal $H$–bundle $E_{H}$ is called simple if $Z_{H}\,=\,\text{Aut}(E_{H})$. Note that if $E_{H}$ is simple then every element of $\text{ad}(E_{H})$ is $T$–rigid, where $T$ is any maximal torus in $\text{Aut}(E_{H})^{0}$.

Let $H$ be connected reductive, and let $E_{H}$ be stable. Then $Z_{H}$ is a finite index subgroup of $\text{Aut}(E_{H})$. Therefore, any maximal torus of $\text{Aut}(E_{H})^{0}$ is contained in $Z_{H}$. This implies that every element of $\text{ad}(E_{H})$ is $T$–rigid, where $T$ is any maximal torus in $\text{Aut}(E_{H})^{0}$.

Take $E_{H}$ to be the trivial holomorphic principal $H$–bundle $X\times H$. Then the left–translation action of $H$ identifies $H$ with $\text{Aut}(E_{H})$. Also, $\text{ad}(E_{H})$ is the trivial vector $X\times\mathfrak{h}$, where $\mathfrak{h}$ is the Lie algebra of $H$. Let $T$ be a maximal torus of $H\,=\,\text{Aut}(E_{H})$. Then an element $v\,\in\,{\mathfrak{h}}\,=\,\text{ad}(E_{H})_{x}$ is $T$–rigid if and only if $v\,\in\,\text{Lie}(T)$.

5. A criterion for logarithmic connections with given residue

5.1. Logarithmic connections with $T$–rigid residue

As in Section 2.1, $G$ is a connected reductive affine algebraic group defined over $\mathbb{C}$. Let $E_{G}$ be a holomorphic principal $G$–bundle over $X$. Fix a maximal torus

[TABLE]

where $\text{Aut}(E_{G})^{0}$ as before is the connected component containing the identity element of the group of automorphisms of $E_{G}$.

We now recall some results from [BBN], [BP].

As in (4.1), define the adjoint bundle $\text{Ad}(E_{G})\,=\,E_{G}\times^{G}G$. For any point $y\,\in\,X$, consider the evaluation homomorphism

[TABLE]

Then $\varphi_{y}$ is injective and its image is a torus in $G$ [BBN, p. 230, Section 3]. Since $G$ is identified with $\text{Ad}(E_{G})_{y}$ uniquely up to an inner automorphism, the image $\varphi_{y}(T)$ determines a conjugacy class of tori in $G$; this conjugacy class is independent of the choice of $y$ [BBN, p. 230, Section 3], [BP, p. 63, Theorem 4.1]. Fix a torus

[TABLE]

in this conjugacy class of tori. The centralizer

[TABLE]

of $T_{G}$ in $G$ is a Levi factor of a parabolic subgroup of $G$ [BBN, p. 230, Section 3], [BP, p. 63, Theorem 4.1]. The principal $G$–bundle $E_{G}$ admits a holomorphic reduction of structure group

[TABLE]

to the above subgroup $H$ [BBN, p. 230, Theorem 3.2], [BP, p. 63, Theorem 4.1]. Since $T_{G}$ is in the center of $H$, the action of $T_{G}$ on $E_{H}$ commutes with the action of $H$, so $T_{G}\,\subset\,\text{Aut}^{0}(E_{H})$ (this was noted in Section 4.2). The image of $T_{G}$ in $\text{Aut}^{0}(E_{H})$ coincides with $T$. This reduction $E_{H}$ is minimal in the sense that there is no Levi factor $L$ of some parabolic subgroup of $G$ such that

• •

$L\,\subsetneq\,H$, and

• •

$E_{G}$ admits a holomorphic reduction of structure group to $L$.

(See [BBN, p. 230, Theorem 3.2].)

The above reduction $E_{H}$ is unique in the following sense. Let $L$ be a Levi factor of a parabolic subgroup of $G$ and $E_{L}\,\subset\,E_{G}$ a holomorphic reduction of structure group to $L$ satisfying the condition that $E_{G}$ does not admit any holomorphic reduction of structure group to a Levi factor $L^{\prime}$ of some parabolic subgroup of $G$ such that $L^{\prime}\,\subsetneq\,L$. Then there is an automorphism $\varphi\,\in\,\text{Aut}(E_{G})^{0}$ such that $E_{L}\,=\,\varphi(E_{H})$ [BP, p. 63, Theorem 4.1]. In particular, the subgroup $L\,\subset\,G$ is conjugate to $H$.

The Lie algebras of $G$ and $H$ will be denoted by $\mathfrak{g}$ and $\mathfrak{h}$ respectively. The inclusion of $\mathfrak{h}$ in $\mathfrak{g}$ and the reduction in (5.3) together produce an inclusion $\text{ad}(E_{H})\,\hookrightarrow\,\text{ad}(E_{G})$. This subbundle $\text{ad}(E_{H})$ of $\text{ad}(E_{G})$ coincides with the invariant subbundle $\text{ad}(E_{G})^{T}$ for the action of $T$ on $\text{ad}(E_{G})$ [BBN, p. 230, Theorem 3.2], [BP, p. 61, Proposition 3.3] (this action is explained in Section 4.1), in other words,

[TABLE]

For every $x\,\in\,D$ fix a $T$–rigid element

[TABLE]

(see Section 4.1). Since each $w_{x}$ is $T$–rigid, from (5.4) we conclude that

[TABLE]

So $w_{x}$ determines a conjugacy class in $\mathfrak{h}$. For any character $\chi$ of $H$, the corresponding homomorphism of Lie algebras $d\chi\,:\,{\mathfrak{h}}\,\longrightarrow\,\mathbb{C}$ factors through the conjugacy classes in $\mathfrak{h}$, because $\mathbb{C}$ is abelian. Therefore, we have $d\chi(w_{x})\,\in\,\mathbb{C}$.

Theorem 5.1**.**

There is a logarithmic connection on $E_{G}$ singular over $D$, and with $T$–rigid residue $w_{x}$ at every $x\,\in\,D$ (see (5.5)), if and only if

[TABLE]

for every character $\chi$ of $H$, where $E_{H}(\chi)$ is the holomorphic line bundle on $X$ associated to $E_{H}$ for $\chi$, and $d\chi$ is the homomorphism of Lie algebras corresponding to $\chi$.

Proof.

Assume that there is a logarithmic connection on $E_{G}$ singular over $D$ such that the residue at each $x\,\in\,D$ is $w_{x}$. Since the group $H$ is reductive, from Lemma 3.4 it follows that $E_{H}$ admits a logarithmic connection singular over $D$ such that the residue at each $x\,\in\,D$ is $w_{x}$ (see (5.6)). Now from Lemma 3.3 we know that (5.7) holds for every character $\chi$ of $G$.

To prove the converse, assume that (5.7) holds for every character $\chi$ of $H$. We will show that $E_{G}$ admits a logarithmic connection singular over $D$ such that the residue at each $x\,\in\,D$ is $w_{x}$.

Since $E_{G}$ is the extension of structure group of $E_{H}$ using the inclusion of $H$ in $G$ (see (5.3)), a logarithmic connection on $E_{H}$ induces a logarithmic connection on $E_{G}$ (this is explained in Section 2.4). Therefore, in view of (2.15) and (5.6), the following proposition completes the proof of the theorem.

Proposition 5.2**.**

There is a logarithmic connection on $E_{H}$ singular over $D$ such that the residue over any $x\,\in\,D$ is $w_{x}\,\in\,{\rm ad}(E_{H})_{x}$.

Proof.

The connected component of the center of $H$ containing the identity element coincides with $T_{G}$ in (5.1). Define the quotient groups

[TABLE]

So $S$ is semisimple, and $Z$ is a torus. The projections of $H$ to $S$ and $Z$ will be denoted by $p_{S}$ and $p_{Z}$ respectively. Let $E_{S}$ (respectively, $E_{Z}$) be the principal $S$–bundle (respectively, $Z$–bundle) on $X$ obtained by extending the structure group of $E_{H}$ using $p_{S}$ (respectively, $p_{Z}$). Consider the homomorphism

[TABLE]

It is surjective with finite kernel, hence it induces an isomorphism of Lie algebras. Let $E_{S\times Z}$ be the principal $S\times Z$–bundle on $X$ obtained by extending the structure group of $E_{H}$ using $\varphi$. Note that $E_{S\times Z}\,\cong\,E_{S}\times_{X}E_{Z}$. Since $\varphi$ induces an isomorphism of Lie algebras, we have

[TABLE]

and

[TABLE]

where $\mathcal{A}_{E_{S\times Z}}$ is constructed as in (3.1) for $(E_{S\times Z},\,\{w_{x}\}_{x\in D})$ (see (5.9)). Consequently, $E_{H}$ admits a logarithmic connection singular over $D$, with residue $w_{x}$ for all $x\,\in\,D$, if and only if $E_{S\times Z}$ admits a logarithmic connection singular over $D$ with residue $w_{x}$ for all $x\,\in\,D$.

For $x\,\in\,D$, let

[TABLE]

be the decomposition given by (5.9). Consider the short exact sequences

[TABLE]

and

[TABLE]

as in Lemma 3.1 for the data $(E_{S},\,\{w_{x}^{s}\}_{x\in D})$ and $(E_{Z},\,\{w_{x}^{z}\}_{x\in D})$ respectively. Let

[TABLE]

be the projections. Note that

[TABLE]

Therefore, giving a holomorphic splitting of the exact sequence

[TABLE]

(see Lemma 3.1 for it) is equivalent to giving holomorphic splittings of both (5.10) and (5.11). Consequently, $E_{S\times Z}$ admits a logarithmic connection singular over $D$ with residue $w_{x}$ over every $x\,\in\,D$ if and only if both $E_{S}$ and $E_{Z}$ admit logarithmic connections singular over $D$ such that their residues over any $x\,\in\,D$ are $w^{s}_{x}$ and $w^{z}_{x}$ respectively.

Consider the homomorphism of character groups $\text{Hom}(Z,\,{\mathbb{G}}_{m})\,\longrightarrow\,\text{Hom}(H,\,{\mathbb{G}}_{m})$ given by the projection $p_{Z}$. It is an isomorphism because being semisimple $[H,\,H]$ does not admit any nontrivial character. Since $E_{Z}\,=\,E_{H}\times^{p_{Z}}Z$, for any character $\chi\,\in\,\text{Hom}(Z,\,\mathbb{G}_{m})$, the holomorphic line bundle $E_{Z}(\chi)\,=\,E_{Z}\times^{\chi}\mathbb{C}$ is identified with the holomorphic line bundle $E_{H}(\chi\circ p_{Z})$.

A holomorphic line bundle $L$ on $X$ admits a logarithmic connection singular over $D$ with residue $\lambda_{x}\,\in\,\mathbb{C}$ for every $x\,\in\,D$ if and only if

[TABLE]

(see [BDP, Lemma 2.3]). Since $Z=H/[H,\,H]\,=\,(\mathbb{G}_{m})^{d}$ for some $d$, it follows that $E_{Z}$ admits a logarithmic connection singular over $D$ with residue $w_{x}^{z}$ at each $x\,\in\,D$ if and only if for each $1\,\leq\,i\,\leq\,d$, the line bundle $E_{Z}(\pi_{i})$ admits a logarithmic connection singular over $D$ with residue ${\rm d}\pi_{i}(w_{x}^{z})$ at each $x\in D$, where $\pi_{i}\,:\,Z\,=\,(\mathbb{G}_{m})^{d}\,\longrightarrow\,{\mathbb{G}}_{m}$ is the projection to the $i$-th factor. From this and the given condition in (5.7) we conclude that $E_{Z}$ admits logarithmic connection singular over $D$ with residue $w^{z}_{x}$ for all $x\,\in\,D$.

To complete the proof of the proposition we need to show that $E_{S}$ admits logarithmic connection singular over $D$ such that the residues over each $x\,\in\,D$ is $w^{s}_{x}$. We will show that (5.10) splits holomorphically.

Let

[TABLE]

be the extension class for (5.10) as in (3.3). The exact sequence in (5.10) splits holomorphically if and only if

[TABLE]

The Lie algebra of $S$ will be denoted by $\mathfrak{s}$. Consider $\mathfrak{s}$ as a $S$–module using the adjoint action of $S$ on $\mathfrak{s}$. Since $S$ is semisimple, the Killing form

[TABLE]

is nondegenerate, where ${\rm ad}_{u}(u^{\prime})\,:=\,[u,\,u^{\prime}]$. Therefore, the Killing form induces an isomorphism ${\mathfrak{s}}\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,{\mathfrak{s}}^{*}$ of $S$–modules. This isomorphism produces a holomorphic isomorphism of ${\rm ad}(E_{S})$ with the dual vector bundle ${\rm ad}(E_{S})^{*}$. Now Serre duality gives

[TABLE]

Let

[TABLE]

be the element corresponding to $\beta$ (defined in (5.12)) by the above isomorphism. Then

[TABLE]

where $\widehat{\beta}$ is an ${\rm ad}(E_{S})$–valued $(1,\,1)$–form on $X$ which represents the cohomology class $\beta$ using the Dolbeault isomorphism.

As before, $\text{Aut}(E_{H})^{0}\,\subset\,\text{Aut}(E_{H})$ (respectively, $\text{Aut}(E_{G})^{0}\,\subset\,\text{Aut}(E_{G})$) is the connected component containing the identity element, and $T\,\subset\,\text{Aut}(E_{G})^{0}$ is the fixed maximal torus. Since $T$ is abelian, from (5.4) it follows immediately that

[TABLE]

Therefore, the maximal torus $T\,\subset\,\text{Aut}(E_{G})^{0}$ (see (5.1)) is also a maximal torus of $\text{Aut}(E_{H})^{0}$. Since $T_{G}$ is the connected component, containing the identity element, of the center of $H$, and $T$ is the image of $T_{G}$ in $\text{Aut}(E_{H})^{0}$, it now follows that the maximal torus of $\text{Aut}(E_{S})^{0}$ is trivial. Hence every holomorphic section of $\text{ad}(E_{S})$ is nilpotent.

Take any nonzero element $\gamma\,\in\,H^{0}(X,\,{\rm ad}(E_{S}))$. Following the proof of [AB, Proposition 3.9], using $\gamma$ we construct a holomorphic reduction of the structure group of $E_{S}$ to a parabolic subgroup of $S$ as follows. For each $x\,\in\,X$, since $\gamma(x)\,\in\,{\rm ad}(E_{S})_{x}$ is nilpotent, there is a parabolic Lie subalgebra $\mathfrak{p}_{x}\,\subset\,{\rm ad}(E_{S})_{x}$ canonically associated to $\gamma(x)$ [AB, p. 340, Lemma 3.7]. Exponentiating $\mathfrak{p}_{x}$ we get a proper parabolic subgroup $P_{x}\,\subset\,{\rm Ad}(E_{S})_{x}$ associated to $\gamma(x)$. Since there are only finitely many conjugacy classes of nilpotent elements of $\mathfrak{s}$, and the algebraic subvariety of $\mathfrak{s}$ defined by the nilpotent elements has a natural filtration defined using the adjoint action of $S$, there is a finite subset $C\,\subset\,X$ such that the conjugacy classes of $P_{x}$, $x\,\in\,X\setminus C$, coincide. Fix a parabolic subgroup $P\,\subset\,S$ in this conjugacy class.

For any $x\,\in\,X\setminus C$, consider the projection map

[TABLE]

where $\widetilde{(z,\,s)}$ is the equivalence class of $(z,\,s)$. Define

[TABLE]

For the natural action of $S$ on $(E_{S})_{x}$, the action of $P\,\subset\,S$ preserves $(E_{P})_{x}$. Since $P$ is a parabolic subgroup of $S$, its normalizer $N_{S}(P)$ is $P$ itself [Hu, p. 143, Corollary B]. So the action of $P$ on $(E_{P})_{x}$ is transitive (and also free, since the $G$–action on $(E_{S})_{x}$ is free). Therefore, we have a holomorphic reduction of structure group $E_{P}\,\subset\,E_{S}$ to $P\,\subset\,S$ over $X\setminus C$. This holomorphic reduction defines a holomorphic section $\eta\,:\,X\setminus C\,\longrightarrow\,E_{S}/P$ which is meromorphic over $X$. Since $S/P$ is a projective variety, the above section $\eta$ extends holomorphically to a section $\widetilde{\eta}\,:\,X\,\longrightarrow\,E_{S}/P$. This defines a holomorphic reduction of structure group $E_{P}\,\subset\,E_{S}$ to $P$.

From Corollary 3.2 it follows that the cohomology class $\beta$ in (5.12) lies in the image of the natural homomorphism $H^{1}(X,\,{\rm ad}(E_{P})\otimes K_{X})\,\longrightarrow\,H^{1}(X,\,{\rm ad}(E_{S})\otimes K_{X})$. Therefore, $\beta$ is represented by an ${\rm ad}(E_{P})$–valued $(1,\,1)$–form $\widehat{\beta}$ on $X$.

Now for all $x\,\in\,X\setminus C$, the element $\gamma(x)\,\in\,{\rm ad}(E_{S})_{x}$ lies in the Lie algebra ${\mathfrak{u}}_{x}$ of the unipotent radical of ${\rm ad}(E_{P})_{x}$ [Hu, p. 186, Corollary A], and $\widehat{\beta}(x)\,\in\,\mathfrak{p}_{x}$. Therefore, $\kappa(\widehat{\beta}(x),\,\gamma(x))\,=\,0$, since $\mathfrak{u}_{x}$ is the orthogonal complement $({\rm ad}(E_{P})_{x})^{\perp}$ with respect to the Killing form on ${\rm ad}(E_{S})_{x}$. Hence from (5.14) we have $\beta^{\prime}(\gamma)\,=\,0$. This proves (5.13), and completes the proof of the proposition. ∎

As noted before, Proposition 5.2 completes the proof of Theorem 5.1. ∎

5.2. $T$–invariant logarithmic connections with given residue

The automorphism group ${\rm Aut}(E_{G})$ has a natural action on the space of all logarithmic connections on $E_{G}$ singular over $D$. Given a maximal torus $T\,\subset\,{\rm Aut}(E_{G})^{0}$, by a $T$–invariant logarithmic connection we mean a logarithmic connection on $E_{G}$ singular over $D$ which is fixed by the action of $T$.

Theorem 5.3**.**

Let $E_{G}$ be a holomorphic principal $G$–bundle on $X$, where $G$ is reductive. Fix $w_{x}\,\in\,{\rm ad}(E_{G})_{x}$ for each $x\,\in\,D$. Fix a maximal torus $T\,\subset\,{\rm Aut}(E_{G})^{0}$. The following two are equivalent:

1. (1)

There is a $T$–invariant logarithmic connection on $E_{G}$ singular over $D$ with residue $w_{x}$ at every $x\,\in\,D$. 2. (2)

The element $w_{x}$ is $T$–rigid for each $x\,\in\,D$, and (5.7) holds for every character $\chi$ of $H$.

Proof.

Let $\theta$ be a $T$–invariant logarithmic connection on $E_{G}$ singular over $D$ with residue $w_{x}$ at every $x\,\in\,D$. Since $\theta$ is $T$–invariant, its residues are also $T$–invariant. Hence $w_{x}$ is $T$–rigid for each $x\,\in\,D$. From Theorem 5.1 we know that (5.7) holds for every character $\chi$ of $H$.

Now assume that the second statement in the theorem holds. From Theorem 5.1 we know that there is a logarithmic connection on $E_{G}$ singular over $D$ with residue $w_{x}$ at every $x\,\in\,D$.

As noted in Section 4.2, for a holomorphic principal $M$–bundle $E_{M}$ on $X$, the center $Z_{M}$ of $M$ is contained in the automorphism group $\text{Aut}(E_{M})$. It is straight-forward to check that the action of $Z_{M}\,\subset\,\text{Aut}(E_{M})$ on the space of all logarithmic connections on $E_{M}$ is trivial.

Since $T_{G}$ is contained in the center of $H$ (see (5.2)), and $T$ is the image of $T_{G}$ in $\text{Aut}(E_{H})$, every logarithmic connection on the principal $H$–bundle $E_{H}$ in Proposition 5.2 is $T$–invariant. Consequently, from Proposition 5.2 it follows that $E_{G}$ admits a $T$–invariant logarithmic connection singular over $D$ with residue $w_{x}$ at every $x\,\in\,D$. ∎

Acknowledgements

The first author is supported by a J. C. Bose Fellowship. The second author is supported by ERCEA Consolidator Grant $615655$-NMST and also by the Basque Government through the BERC $2014-2017$ program and by Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-$2013-0323$.