Automorphism group of principal bundles, Levi reduction and invariant connections

TL;DR

This paper establishes a bijective correspondence between torus subbundles of the adjoint bundle of a principal G-bundle over a complex manifold and certain quadruples involving reductions of structure group and actions, also providing conditions for connections to be induced by reductions.

Contribution

It introduces a natural bijective correspondence between torus subbundles and quadruples involving reductions and actions, and characterizes when a connection on E_G is induced by a reduction to E_N.

Findings

Bijection between torus subbundles and quadruples of reductions and actions.

Necessary and sufficient conditions for connections to be induced by reductions.

Relation between Hermitian--Einstein connections on E_G and E'_C.

Abstract

Let $M$ be a compact connected complex manifold and $G$ a connected reductive complex affine algebraic group. Let $E_G$ be a holomorphic principal $G$--bundle over $M$ and $T\, \subset\, G$ a torus containing the connected component of the center of $G$. Let $N$ (respectively, $C$) be the normalizer (respectively, centralizer) of $T$ in $G$, and let $W$ be the Weyl group $N/C$ for $T$. We prove that there is a natural bijective correspondence between the following two: Torus subbundles $\mathbb T$ of ${\rm Ad}(E_G)$ such that for some (hence every) $x\, \in\, M$, the fiber ${\mathbb T}_x$ lies in the conjugacy class of tori in ${\rm Ad}(E_G)$ determined by $T$. Quadruples of the form $(E_W,\, \phi,\, E'_C,\, \tau)$, where $\phi\, :\, E_W\, \longrightarrow\, M$ is a principal $W$--bundle, $\phi^*E_G\, \supset\, E'_C\, \stackrel{\psi}{\longrightarrow}\, E_W$ is a holomorphic reduction of structure group of $\phi^* E_G$ to $C$, and $$ \tau\,:\, E'_C\times N \, \longrightarrow\, E'_C $$ is a holomorphic action of $N$ on $E'_C$ extending the natural action of $C$ on $E'_C$, such that the composition $\psi\circ\tau$ coincides with the composition of the quotient map $E'_C\times N\, \longrightarrow\, (E'_C/C)\times (N/C)\,=\, (E'_C\times N)/(C\times C)$ with the natural map $(E'_C/C)\times (N/C)\, \longrightarrow\, E_W$. The composition of maps $E'_C\, \longrightarrow\, E_W \, \longrightarrow\, M$ defines a principal $N$--bundle on $M$. This principal $N$--bundle $E_N$ is a reduction of structure group of $E_G$ to $N$. Given a complex connection $\nabla$ on $E_G$, we give a necessary and sufficient condition for $\nabla$ to be induced by a connection on $E_N$. This criterion relates Hermitian--Einstein connections on $E_G$ and $E'_C$ in a very precise manner.

Equations

$$\tau :E_C^{\prime }\times N⟶E_C^{\prime }$$

$$\begin{matrix}E^{\prime}_{C}\times N&\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}&E^{\prime}_{C}\\ \Big{\downarrow}&&~{}\Big{\downarrow}\psi\\ E_{W}\times W:=(E^{\prime}_{C}/C)\times(N/C)&\longrightarrow&E_{W}\end{matrix}$$

$$E_N=E_C^{\prime }\hookrightarrow {\varphi }^{\ast }{E}_G⟶E_G,$$

$$L(P)\hookrightarrow P⟶P/R_u(P)$$

$$p_0:E_G⟶M$$

$$q_0:E_G\times G⟶E_G$$

$$E_G\times G⟶E_G{\times }_M{E}_G,(y,z)⟶(y,q_0(y,z))$$

$$f_0:E_G\times G⟶\text{Ad}(E_G):=(E_G\times G)/\sim$$

$$p:\text{Ad}(E_G)⟶M,(y,z)⟼p_0(y),(y,z)\in E_G\times G$$

$$\mathbb{T}\subset \text{Ad}(E_G)\stackrel{p}{⟶}M$$

$${\mathbb{T}}_x:=\mathbb{T}\cap \text{Ad}(E_G{)}_x\subset \text{Ad}(E_G{)}_x$$

$$T\subset T_G\subset G$$

$$Z_0(G)\subset T,$$

$$N:=N_G(T)\subset G$$

$$C:=N_0\subset N$$

$$E_N\subset E_G$$

$$E_N\times T\subset f_0^{-1}(\mathbb{T})\subset E_G,$$

$$p_0:E_N⟶M$$

$$f_0(y{g}_0,t)=f_0(y,g_{0}t{g}_0^{-1})$$

$$E_W:=E_N/C=E_N{\times }^{N}W\stackrel{\varphi }{⟶}M$$

$$\psi :E_C^{\prime }=E_N⟶E_W:=E_N/C$$

$$\begin{matrix}E^{\prime}_{C}\times N&\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}&E^{\prime}_{C}\\ \Big{\downarrow}&&~{}\Big{\downarrow}\psi\\ E_{W}\times W:=(E^{\prime}_{C}/C)\times(N/C)&\longrightarrow&E_{W}\end{matrix}$$

$$T\subset T_G\subset B_G$$

$$E_C^{\prime }\subset {\varphi }^{\ast }{E}_G$$

$$\tau :E_C^{\prime }\times N⟶E_C^{\prime }$$

$$\begin{matrix}E^{\prime}_{C}\times N&\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}&E^{\prime}_{C}\\ \Big{\downarrow}&&~{}\Big{\downarrow}\psi\\ E_{W}\times W:=(E^{\prime}_{C}/C)\times(N/C)&\longrightarrow&E_{W}\end{matrix}$$

$$\text{Ad}(E_C^{\prime })\supset {\mathbb{T}}^{\prime }⟶E_W$$

$$\mathbb{T}:={\mathbb{T}}^{\prime }/W\subset \mathrm{Ad}(E_C^{\prime })/W\subset \mathrm{Ad}({\varphi }^{\ast }{E}_G)/W=\text{Ad}(E_G).$$

$$\tau :E_C^{\prime }\times N⟶E_C^{\prime }$$

Full text

1. (1)

Torus subbundles $\mathbb{T}$ of ${\rm Ad}(E_{G})$ such that for some (hence every) $x\,\in\,M$, the fiber ${\mathbb{T}}_{x}$ lies in the conjugacy class of tori in ${\rm Ad}(E_{G})$ determined by $T$. 2. (2)

Quadruples of the form $(E_{W},\,\phi,\,E^{\prime}_{C},\,\tau)$, where $\phi\,:\,E_{W}\,\longrightarrow\,M$ is a principal $W$–bundle, $\phi^{*}E_{G}\,\supset\,E^{\prime}_{C}\,\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}\,E_{W}$ is a holomorphic reduction of structure group of $\phi^{*}E_{G}$ to $C$, and

[TABLE]

is a holomorphic action of $N$ on $E^{\prime}_{C}$ extending the natural action of $C$ on $E^{\prime}_{C}$, such that the composition $\psi\circ\tau$ coincides with the composition of the quotient map $E^{\prime}_{C}\times N\,\longrightarrow\,(E^{\prime}_{C}/C)\times(N/C)\,=\,(E^{\prime}_{C}\times N)/(C\times C)$ with the natural map $(E^{\prime}_{C}/C)\times(N/C)\,\longrightarrow\,E_{W}$.

The composition of maps $E^{\prime}_{C}\,\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}\,E_{W}\,\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}\,M$ defines a principal $N$–bundle on $M$. This principal $N$–bundle $E_{N}$ is a reduction of structure group of $E_{G}$ to $N$. Given a complex connection $\nabla$ on $E_{G}$, we give a necessary and sufficient condition for $\nabla$ to be induced by a connection on $E_{N}$. This criterion relates Hermitian–Einstein connections on $E_{G}$ and $E^{\prime}_{C}$ in a very precise manner.

1. 1 Introduction

2. 2 Torus subbundle and Levi reduction of structure group of a principal bundle

3. 2.1 A Levi reduction from a torus subbundle

4. 2.2 A torus subbundle from a Levi reduction

5. 2.3 A tautological connection on a torus bundle

6. 3 Torus subbundles and connection

7. 3.1 Connections on a principal bundle

8. 3.2 Criterion for induced connection

9. 3.3 Hermitian–Einstein connection and Levi reduction

10. 4 Higgs bundles and Levi reduction

11. 4.1 $G$-Higgs bundles

12. 5 Torus for equivariant bundles

1. Introduction

Let $M$ be a compact connected complex manifold. Take a holomorphic vector bundle $E$ on $M$. In [BCW] the following question was addressed: When is the vector bundle $E$ the direct image of a vector bundle over an étale cover of $M$? The main result of [BCW] described all possible way $E$ is realized as the direct image of a vector bundle over an étale cover of $M$. The main result of [BCW] says that they are parametrized by the subbundles of the adjoint bundle $\text{Ad}(E)\,\longrightarrow\,M$ whose fibers are tori. To explain this with more details, given any triple $(Y,\,\beta,\,F)$, where $\beta\,:\,Y\,\longrightarrow\,M$ is an étale covering ($Y$ need not be connected) and $F$ is a holomorphic vector bundle on $Y$ such that $E\,=\,\beta_{*}F$, we construct a torus sub-bundle of $\text{Ad}(E)$; this sub-bundle is in fact the invertible part of $\beta_{*}{\mathcal{O}}_{Y}\,\subset\,\text{End}(\beta_{*}F)$. Conversely, given a sub-bundle of $\text{Ad}(E)$ with the typical fiber being a torus, we construct a triple $(Y,\,\beta,\,F)$ of the above form such that $E\,=\,\beta_{*}F$. In [BM], these results were generalized to the context of parabolic (orbifold) vector bundles over any Riemann surface; see [DP] for a somewhat related question.

Our aim here is to formulate and address the question in the context of principal bundles. Since direct image of a principal bundle does not quite make sense, a reformulation is warranted.

Let $G$ be a connected reductive complex affine algebraic group. Fix a complex torus $T\,\subset\,G$ that contains the connected component, containing the identity element, of the center of $G$. Denote the normalizer (respectively, centralizer) of $T$ in $G$ by $N$ (respectively, $C$). This $C$ is a Levi factor of a parabolic subgroup of $G$. The quotient $N/C$ is a finite group, which we shall denote by $W$.

We prove the following (see Theorem 2.5):

Theorem 1.1**.**

Take a holomorphic principal $G$–bundle $E_{G}$ over $M$. There is a natural bijective correspondence between the following two:

1. (1)

Torus subbundles $\mathbb{T}$ of ${\rm Ad}(E_{G})$ such that for some (hence every) $x\,\in\,M$, the fiber ${\mathbb{T}}_{x}$ lies in the conjugacy class of tori in ${\rm Ad}(E_{G})$ determined by $T$. 2. (2)

Quadruples of the form $(E_{W},\,\phi,\,E^{\prime}_{C},\,\tau)$, where $\phi\,:\,E_{W}\,\longrightarrow\,M$ is a principal $W$–bundle, $\phi^{*}E_{G}\,\supset\,E^{\prime}_{C}\,\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}\,E_{W}$ is a holomorphic reduction of structure group of the principal $G$–bundle $\phi^{*}E_{G}$ to the subgroup $C$, and

[TABLE]

is a holomorphic action of $N$ on $E^{\prime}_{C}$ extending the natural action of $C$ on the principal $C$–bundle $E^{\prime}_{C}$, such that the diagram of maps

[TABLE]

is commutative, where $E^{\prime}_{C}\times N\,\longrightarrow\,(E^{\prime}_{C}/C)\times(N/C)\,=\,(E^{\prime}_{C}\times N)/(C\times C)$ is the quotient map, and $\psi$ is the natural projection.

If we set $G\,=\,\text{GL}(r,{\mathbb{C}})$ in Theorem 1.1, then the above mentioned result of [BCW] is obtained.

Consider the composition of maps $E^{\prime}_{C}\,\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}\,E_{W}\,\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}\,M$ in (2). The action of $N$ on $E^{\prime}_{C}$ and this composition of maps together produce a holomorphic principal $N$–bundle over $M$. This principal $N$–bundle over $M$ will be denoted by $E_{N}$. Since $\phi^{*}E_{G}\,=\,E_{W}\times_{M}E_{G}$, we have a natural projection $\phi^{*}E_{G}\,\longrightarrow\,E_{G}$. Now using the composition of maps

[TABLE]

the principal $N$–bundle $E_{N}$ is a holomorphic reduction of structure group of the principal $G$–bundle $E_{G}$ to the subgroup $N\,\subset\,G$. Therefore, a complex connection on $E_{N}$ induces a complex connection on $E_{G}$.

Given a complex connection on $E_{G}$, it is natural to ask whether it is induced by a complex connection on $E_{N}$.

We prove the following criterion for it (see Theorem 3.1 and Remark 3.3):

Theorem 1.2**.**

Given a complex connection $D_{0}$ on $E_{G}$, let $D_{1}$ be the complex connection on the associated adjoint bundle ${\rm Ad}(E_{G})$ induced by $D_{0}$. The connection $D_{0}$ on $E_{G}$ is induced by a complex connection on the principal $N$–bundle $E_{N}$ if and only if the induced connection $D_{1}$ on ${\rm Ad}(E_{G})$ preserves the corresponding torus subbundle ${\mathbb{T}}\,\subset\,{\rm Ad}(E_{G})$ in (1) of Theorem 1.1.

When the connection $D_{1}$ on ${\rm Ad}(E_{G})$ induced by $D_{0}$ preserves the torus subbundle ${\mathbb{T}}\,\subset\,{\rm Ad}(E_{G})$, the connection $D_{0}$ is holomorphic if and only if the connection on $E_{N}$ inducing $D_{0}$ is holomorphic.

A complex connection on $E_{N}$ defines a complex connection on the principal $C$–bundle $E^{\prime}_{C}$ on $E_{W}$, because $E_{N}\,=\,E^{\prime}_{C}$ (see (1.1)) and the map $\phi$ is étale.

Now assume that $M$ is Kähler; fix a Kähler form $\omega$ on $M$ in order to define degree of a torsionfree coherent analytic sheaf on $M$. This enables us to define stable and polystable principal $G$–bundles on $M$. Fix a maximal compact subgroup $K_{G}\,\subset\,G$ to define the Hermitian–Einstein equation for principal $G$–bundles. So $K_{C}\,:=\,C\bigcap K_{G}$ is a maximal compact subgroup of $C$. A holomorphic principal $G$–bundle on $M$ admits a Hermitian–Einstein connection if and only if it is polystable [UY], [Do1], [RS], [AB]. The pulled back form $\phi^{*}\omega$ on $E_{W}$ is Kähler. However, $E_{W}$ need not be connected. Polystable bundles and Hermitian–Einstein connections on bundles over $E_{W}$ are defined in a suitable way.

We prove the following (see Proposition 3.4):

Proposition 1.3**.**

Take $E_{G}$ and $E^{\prime}_{C}$ as in Theorem 1.1. Assume that the principal $G$–bundle $E_{G}$ on $M$ is polystable. Let $\nabla$ be the Hermitian–Einstein connection on $E_{G}$. Then the following two hold:

1. (1)

The principal $C$–bundle $E^{\prime}_{C}$ on $E_{W}$ is polystable. 2. (2)

The Hermitian–Einstein connection $\phi^{*}\nabla$ on $\phi^{*}E_{G}$ preserves the reduction $E^{\prime}_{C}$ of structure group of the principal $G$–bundle $\phi^{*}E_{G}$ to the subgroup $C\,\subset\,G$. Furthermore, the connection on $E^{\prime}_{C}$ given by $\phi^{*}\nabla$ is Hermitian–Einstein.

Let $z({\mathfrak{k}})$ denote the center of the Lie algebra of $K_{G}$. Let $z({\mathfrak{k}}_{c})$ be the center of the Lie algebra of the maximal compact subgroup $K_{C}\,=\,K_{G}\bigcap C$ of $C$. We have $z({\mathfrak{k}})\,\subset\,z({\mathfrak{k}}_{c})$.

We also prove the following (see Proposition 3.5):

Proposition 1.4**.**

Take $E_{G}$ and $E^{\prime}_{C}$ as in Theorem 1.1. Assume that the principal $C$–bundle $E^{\prime}_{C}$ over $E_{W}$ is polystable. Let $\nabla$ be the Hermitian–Einstein connection on $E^{\prime}_{C}$. Assume that the element of $z({\mathfrak{k}}_{c})$ given by the curvature of $\nabla$ lies in the subspace $z({\mathfrak{k}})$. Then the following two hold:

1. (1)

The principal $G$–bundle $E_{G}$ on $M$ is polystable. 2. (2)

The Hermitian–Einstein connection on $E_{G}$ is given by $\nabla$.

As indicated in Section 5, all the above results extend to the equivariant set-up. This means that the results of [BM] extend to the context of complex reductive affine algebraic groups.

2. Torus subbundle and Levi reduction of structure group of a principal bundle

2.1. A Levi reduction from a torus subbundle

Let $G$ be a connected complex reductive affine algebraic group. A parabolic subgroup of $G$ is a Zariski closed connected subgroup $P\,\subset\,G$ such that the quotient variety $G/P$ is projective. The unipotent radical of a parabolic subgroup $P\,\subset\,G$ is denoted by $R_{u}(P)$. The quotient $P/R_{u}(P)$ is a reductive affine complex algebraic group. A connected reductive complex algebraic subgroup $L(P)\,\subset\,P$ is called a Levi factor of $P$ if the composition of maps

[TABLE]

is an isomorphism [Bo, p. 158, § 11.22]. There are Levi factors of $P$; any two Levi factors of $P$ differ by the inner automorphism of $P$ produced by an element of $R_{u}(P)$ [Bo, p. 158, § 11.23], [Hu, § 30.2, p. 184].

Fix a Borel subgroup $B_{G}\,\subset\,G$, and also fix a maximal torus $T_{G}\,\subset\,B_{G}$. Given any parabolic subgroup $P\,\subset\,G$, there is some element $g_{0}\,\in\,G$ such that we have $B_{G}\,\subset\,g^{-1}_{0}Pg_{0}$ [Hu, p. 134, Theorem 21.3]. Henceforth, whenever we consider a parabolic subgroup of $G$, we would assume that $P\,\supset\,B_{G}$. The connected component of the center of $G$ containing the identity element will be denoted by $Z_{0}(G)$; this $Z_{0}(G)$ is isomorphic to a product of copies of the multiplicative group ${\mathbb{C}}^{*}\,:=\,{\mathbb{C}}\setminus\{0\}$.

Let $M$ be a compact connected complex manifold. Let

[TABLE]

be a holomorphic principal $G$–bundle over $M$; this means that $E_{G}$ is a holomorphic fiber bundle over $M$ equipped with a holomorphic right-action of $G$

[TABLE]

such that the above projection $p_{0}$ is $G$–invariant and, furthermore, the resulting map to the fiber product

[TABLE]

is a biholomorphism. For notational convenience, for any $(y,\,z)\,\in\,E_{G}\times G$, the point $q_{0}(y,\,z)\,\in\,E_{G}$ will be denoted by $yz$. Given a holomorphic principal $G$–bundle $E_{G}$ as above, consider the quotient of $E_{G}\times G$, where two points $(y,\,z)$ and $(y_{1},\,z_{1})$ of $E_{G}\times G$ are identified if there is some $g_{0}\,\in\,G$ such that $y_{1}\,=\,yg_{0}$ and $z_{1}\,=\,g^{-1}_{0}zg_{0}$. Let

[TABLE]

be this quotient. Each fiber of the projection

[TABLE]

is a group isomorphic to $G$, where the group operation is given by $(y,\,z)\cdot(y,\,z^{\prime})\,=\,(y,\,zz^{\prime})$ (it is straightforward to check that the group operation is well-defined, meaning it is independent of $y$); note that the map $(y,\,z)\,\longmapsto\,z$ is an isomorphism of $\text{Ad}(E_{G})_{x}\,:=\,p^{-1}(x)$ with $G$. For any $x\,\in\,M$, the above isomorphism between $\text{Ad}(E_{G})_{x}$ and $G$ depends on the choice of the point $y\,\in\,p^{-1}_{0}(x)$. However, for two different choices of $y$, the corresponding isomorphisms differ by an inner automorphism of $G$. In other words, $\text{Ad}(E_{G})_{x}$ and $G$ are identified uniquely up to an inner automorphism. This $\text{Ad}(E_{G})$ is called the adjoint bundle for $E_{G}$.

The adjoint action of any $z\,\in\,G$ on $G$ fixes the subgroup $Z_{0}(G)$ pointwise. Therefore, $M\times Z_{0}(G)\,\longrightarrow\,M$ is a subbundle of $\text{Ad}(E_{G})$.

Let

[TABLE]

be a holomorphic sub-fiber bundle such that

• •

$M\times Z_{0}(G)\,\subset\,{\mathbb{T}}$, and

• •

for every point $x\,\in\,M$, the fiber

[TABLE]

is a torus (it need not be a maximal torus of $\text{Ad}(E_{G})_{x}$).

Take any point $x\,\in\,M$. Since $\text{Ad}(E_{G})_{x}$ is identified with $G$ up to an inner automorphism, the torus ${\mathbb{T}}_{x}\,\subset\,\text{Ad}(E_{G})_{x}$ determines a conjugacy class of tori in $G$. From the rigidity of tori in $G$ it follows that this conjugacy class of tori in $G$ is independent of the choice of the point $x\,\in\,M$. Note that any torus in $G$ is conjugate to a sub-torus of $T_{G}$, and the space of sub-tori in $T_{G}$ is a discrete (countable) set. Fix a torus

[TABLE]

in the conjugacy class determined by $\mathbb{T}$. Note that we have

[TABLE]

because $M\times Z_{0}(G)\,\subset\,{\mathbb{T}}$.

Let

[TABLE]

be the normalizer in $G$ of the subgroup $T$ in (2.4). The connected component of $N$

[TABLE]

containing the identity element is in fact the centralizer of $T$ in $G$. This component $C$ of $N$ is a Levi factor of a parabolic subgroup of $G$ [SpSt, § 3] (see also [Sp, p. 110, Theorem 6.4.7(i)]). The quotient $N/C$ is the Weyl group associated to $T\,\subset\,G$; it is a finite group. The normalizer of $C$ in $G$ actually coincides with $N$.

Now let

[TABLE]

be the unique largest subset such that

[TABLE]

where $f_{0}$, $\mathbb{T}$ and $T$ are as in (2.2), (2.3) and (2.4) respectively. Let

[TABLE]

be the restriction of the projection in (2.1); this repetition of notation should not cause any confusion. Clearly, $E_{N}$ is a complex manifold, and the projection $p_{0}$ in (2.9) is holomorphic because the projection in (2.1) is holomorphic.

We will prove that $E_{N}$ is a holomorphic principal $N$–bundle over $M$, where $N$ is constructed in (2.5). For this first note that for any $y\,\in\,E_{N}$ and $g_{0}\,\in\,G$, we have

[TABLE]

(see the construction of $f_{0}$ in (2.2)). Using (2.10) we will show that $yg_{0}\,\in\,E_{N}$ if and only if $g_{0}\,\in\,N$, which would imply that $E_{N}$ is a holomorphic principal $N$–bundle over $M$. To see that $yg_{0}\,\in\,E_{N}$ for all $g_{0}\,\in\,N$, first note that if $g_{0}\,\in\,N$, then $g_{0}tg^{-1}_{0}\,\in\,T$ whenever $t\,\in\,T$. Therefore, in view of (2.10), the given condition that $\{y\}\times T\,\subset\,f^{-1}_{0}({\mathbb{T}})$ immediately implies that $\{yg_{0}\}\times T\,\subset\,f^{-1}_{0}({\mathbb{T}})$ if $g_{0}\,\in\,N$. So we have $yg_{0}\,\in\,E_{N}$ for all $g_{0}\,\in\,N$.

To prove the converse, note that the given condition that $\{y\}\times T\,\subset\,f^{-1}_{0}({\mathbb{T}})$ implies that $f_{0}(\{y\}\times T)\,=\,{\mathbb{T}}_{p_{0}(y)}$, because $T$ is identified with ${\mathbb{T}}_{p_{0}(y)}$ up to an inner automorphism. Therefore, if $yg_{0}\,\in\,E_{N}$, then from (2.10) it follows that $g_{0}tg^{-1}_{0}\,\in\,T$ for all $t\,\in\,T$. This implies that $g_{0}\,\in\,N$ if $yg_{0}\,\in\,E_{N}$.

From (2.7) we conclude that the principal $N$–bundle $E_{N}$ is a holomorphic reduction of structure group of $E_{G}$ to the subgroup $N$ in (2.5). Equivalently, $E_{G}$ coincides with the holomorphic principal $G$–bundle on $M$ obtained by extending the structure group of the holomorphic principal $N$–bundle $E_{N}$ using the inclusion of $N$ in $G$.

For notational convenience, the Weyl group $N/C$ in (2.6) will be denoted by $W$. Let

[TABLE]

be the principal $W$–bundle over $M$ obtained by extending the structure group of the principal $N$–bundle $E_{N}$ using the quotient map $N\,\longrightarrow\,N/C\,=\,W$. So $\phi$ in (2.11) is an étale Galois covering with Galois group $\text{Gal}(\phi)\,=\,W$. This $E_{W}$ need not be connected.

Lemma 2.1**.**

The pulled back principal $N$–bundle $\phi^{*}E_{N}\,\longrightarrow\,E_{W}$ has a tautological reduction of structure group to the subgroup $C\,\subset\,N$ in (2.6), where $\phi$ is the projection in (2.11).

Proof.

Consider the quotient map $E_{N}\,\longrightarrow\,E_{N}/C\,=\,E_{W}$. This makes $E_{N}$ a holomorphic principal $C$–bundle over $E_{W}$. We will denote by $E^{\prime}_{C}$ this holomorphic principal $C$–bundle over $E_{W}$. The identity map $E_{N}\,=\,E^{\prime}_{C}\longrightarrow\,E_{N}$ is $C$–equivariant. For any $z\,\in\,E_{W}$, the $C$–equivariant map $(E^{\prime}_{C})_{z}\,\longrightarrow\,(E_{N})_{\phi(z)}$, obtained by restricting the above identity map of $E_{N}$, is evidently an embedding. Therefore, $E^{\prime}_{C}$ is a holomorphic reduction of structure group of the principal $N$–bundle $\phi^{*}E_{N}$ to the subgroup $C\,\subset\,N$. ∎

Since $E_{N}$ is a holomorphic reduction of structure group of $E_{G}$ to $N$, Lemma 2.1 implies that $E^{\prime}_{C}$ is also a holomorphic reduction of structure group of $\phi^{*}E_{G}$ to the subgroup $C\,\subset\,G$.

Let

[TABLE]

be the quotient map.

We note that the group $N$ acts holomorphically on $E^{\prime}_{C}$, because $E^{\prime}_{C}\,=\,E_{N}$; for this action $\tau\,:\,E^{\prime}_{C}\times N\,\longrightarrow\,E^{\prime}_{C}$ of $N$ on $E^{\prime}_{C}$, the following diagram of maps is evidently commutative:

[TABLE]

where $E^{\prime}_{C}\times N\,\longrightarrow\,(E^{\prime}_{C}/C)\times(N/C)\,=\,(E^{\prime}_{C}\times N)/(C\times C)$ is the natural quotient map, and $E_{W}\times W\,\longrightarrow\,E_{W}$ is the action of $W$ on the principal $W$–bundle $E_{W}$, while $\psi$ is the map in (2.12).

2.2. A torus subbundle from a Levi reduction

We will now describe a reverse of the construction done in Section 2.1.

As before, let $E_{G}$ be a holomorphic principal $G$–bundle over $X$. Take a torus

[TABLE]

such that $Z_{0}(G)\,\subset\,T$. As before, the normalizer (respectively, centralizer) of $T$ in $G$ will be denoted by $N$ (respectively, $C$). Denote the Weyl group $N/C$ by $W$.

Let $\phi\,:\,E_{W}\,\longrightarrow\,W$ be a principal $W$–bundle. Assume that the holomorphic principal $G$–bundle $\phi^{*}E_{G}\,\longrightarrow\,E_{W}$ has a holomorphic reduction of structure group to the subgroup $C\,\subset\,G$

[TABLE]

such that

1. (1)

the principal $C$–bundle $E^{\prime}_{C}$ is equipped with a holomorphic action of the complex group $N$

[TABLE]

that extends the natural action of the subgroup $C$ on the principal $C$–bundle $E^{\prime}_{C}$, and 2. (2)

the diagram of maps

[TABLE]

is commutative, where $E^{\prime}_{C}\times N\,\longrightarrow\,(E^{\prime}_{C}/C)\times(N/C)\,=\,(E^{\prime}_{C}\times N)/(C\times C)$ is the natural quotient map, and $\psi$ is the projection of $E^{\prime}_{C}$ to $E_{W}$, so $\psi$ coincides with the restriction to $E^{\prime}_{C}$ of the pullback $\phi^{*}p_{0}$, where $p_{0}$ is the projection in (2.1); note that (2.15) is similar to (2.13).

Lemma 2.2**.**

The action of $W$ on the principal $W$–bundle $E_{W}$ has a canonical lift to an action of $W$ on the adjoint bundle ${\rm Ad}(E^{\prime}_{C})\,=\,E^{\prime}_{C}\times^{C}C\,\longrightarrow\,E_{W}$ for $E^{\prime}_{C}$. This action of $W$ on ${\rm Ad}(E^{\prime}_{C})$ preserves the group structure of the fibers of ${\rm Ad}(E^{\prime}_{C})$.

Proof.

The adjoint action of $N$ on itself preserves $C$ because it is the connected component of $N$ containing the identity element. We recall that $\text{Ad}(E^{\prime}_{C})$ is the quotient of $E^{\prime}_{C}\times C$ where two elements $(y,\,z),\,(y_{1},\,z_{1})\,\in\,E^{\prime}_{C}\times C$ are identified if there is an element $c\,\in\,C$ such that $y_{1}\,=\,yc$ and $z_{1}\,=\,c^{-1}zc$.

Consider the following action of $N$ on $E^{\prime}_{C}\times C$: the action of any $n\,\in\,N$ sends any $(y,\,z)\,\in\,E^{\prime}_{C}\times C$ to $(\tau(y,\,n),\,n^{-1}zn)$, where $\tau$ is the action of $N$ in (2.15) that extends the action of $C$ on $E^{\prime}_{C}$. So for any $c\,\in\,C$, the action of $n$ sends $(yc,\,c^{-1}zc)$ to $(\tau(yc,\,n),\,n^{-1}c^{-1}zcn)\,=\,(\tau(y,\,cn),\,(cn)^{-1}zcn)$ (recall that the restriction of the action $\tau$ to $C\,\subset\,N$ coincides with the natural action of $C$ on $E^{\prime}_{C}$). The image of $(yc,\,c^{-1}zc)$ (respectively, $(\tau(y,\,cn),\,(cn)^{-1}zcn)$) in the quotient space $\text{Ad}(E^{\prime}_{C})$ of $E^{\prime}_{C}\times C$ coincides with the image of $(y,\,z)$ (respectively, $(yc,\,c^{-1}zc)$). Therefore, the above action of $N$ on $E^{\prime}_{C}\times C$ produces an action of $N$ on the quotient manifold $\text{Ad}(E^{\prime}_{C})$. Next note that if $n\,\in\,C$, then the image of $(\tau(y,\,n),\,n^{-1}zn)$ in $\text{Ad}(E^{\prime}_{C})$ coincides with the image of $(y,\,z)$ in $\text{Ad}(E^{\prime}_{C})$. Consequently, the above action of $N$ on $\text{Ad}(E^{\prime}_{C})$ produces an action of $W\,=\,N/C$ on $\text{Ad}(E^{\prime}_{C})$.

The above action of $W$ on $\text{Ad}(E^{\prime}_{C})$ preserves the group structure of the fibers of ${\rm Ad}(E^{\prime}_{C})$, because the adjoint action of a group on itself preserves the group structure. ∎

Consider the action $\tau$ of $N$ on $E^{\prime}_{C}$ in (2.15). From (2.15) it follows that this action and the composition of maps $E^{\prime}_{C}\,\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}\,E_{W}\,\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}\,M$ together define a holomorphic principal $N$–bundle over $M$. This holomorphic principal $N$–bundle over $M$ will be denoted by $E_{N}$.

Lemma 2.3**.**

The holomorphic principal $G$–bundle over $M$, obtained by extending the structure group of the above defined principal $N$–bundle $E_{N}$ using the inclusion of $N$ in $G$, is identified with $E_{G}$.

Proof.

Let $E_{N}\,\longrightarrow\,E_{G}$ be the composition of the natural map $\phi^{*}E_{G}\,\longrightarrow\,E_{G}$ with the inclusion $E_{N}\,=\,E^{\prime}_{C}\,\hookrightarrow\,\phi^{*}E_{G}$ in (2.14). This map $E_{N}\,\longrightarrow\,E_{G}$ is clearly $N$–equivariant. This implies that $E_{N}$ is a reduction of structure group of the principal $G$–bundle $E_{G}$ to the subgroup $N$. In other words, $E_{G}$ is identified with the holomorphic principal $G$–bundle over $M$ obtained by extending the structure group of the principal $N$–bundle $E_{N}$ using the inclusion of $N$ in $G$. ∎

Let

[TABLE]

be the bundle of connected components of centers containing identity element; so for any $y\,\in\,E_{W}$, the fiber $({\mathbb{T}}^{\prime})_{y}$ of $\mathbb{T}$ is the connected component of the center of the group $\text{Ad}(E^{\prime}_{C})_{y}$ containing the identity element. Note that ${\mathbb{T}}^{\prime}$ is identified with the trivial bundle $E_{W}\times T\,\longrightarrow\,E_{W}$, because $T$ is the connected component, containing the identity element, of the center of $C$; the identification between $E_{W}\times T$ and ${\mathbb{T}}^{\prime}$ sends any $(y,\,z)\,\in\,E_{W}\times T$ to the equivalence class of $(y^{\prime},\,z)$ in the quotient $\text{Ad}(E^{\prime}_{C})$ of $E^{\prime}_{C}\times C$, where $y^{\prime}$ is any element of the fiber $(E^{\prime}_{C})_{y}$ (this equivalence class does not depend on the choice of the point $y^{\prime}$ in $(E^{\prime}_{C})_{y}$). The action of $W$ on $\text{Ad}(E^{\prime}_{C})$ in Lemma 2.2 preserves ${\mathbb{T}}^{\prime}$ because the action of $W$ preserves the group structure of the fibers of ${\rm Ad}(E^{\prime}_{C})$ (hence the centers of fibers are preserved implying that their connected components containing identity element are also preserved). Consequently, we have a torus subbundle

[TABLE]

The construction of ${\mathbb{T}}$ in (2.17) from $E^{\prime}_{C}$ in (2.14) is the inverse of the construction of $E^{\prime}_{C}$ in Lemma 2.1 from ${\mathbb{T}}$ in (2.3). More precisely, starting with ${\mathbb{T}}$ in (2.3), construct $E^{\prime}_{C}$ as in Lemma 2.1. Now set $E^{\prime}_{C}$ in (2.14) to be this holomorphic principal $C$–bundle constructed from ${\mathbb{T}}$ in (2.3). Then the torus bundle ${\mathbb{T}}$ constructed in (2.17) coincides with the torus bundle in (2.3) that we started with.

Conversely, start with $E^{\prime}_{C}$ in (2.14) and construct $\mathbb{T}$ as in (2.17). Setting this to be the torus bundle in (2.3), the principal $C$–bundle constructed in Lemma 2.1 coincides with $E^{\prime}_{C}$ in (2.14) that we started with.

Remark 2.4*.*

Although the principal $C$–bundle $E^{\prime}_{C}$ in Lemma 2.2 does not, in general, descend to $M$, note that the adjoint bundle $\text{Ad}(E^{\prime}_{C})$, being $W$ equivariant (see Lemma 2.2), descends to $M$ as a subbundle of the adjoint bundle $\text{Ad}(E_{N})$ of $E_{N}$ in Lemma 2.3. For every point $x\,\in\,M$, the fiber of $\text{Ad}(E^{\prime}_{C})/W$ over $x$ is the connected component of $\text{Ad}(E_{N})_{x}$ containing the identity element. Recall that for any $z\,\in\,E_{W}$, the fiber ${\mathbb{T}}^{\prime}_{z}$ in (2.16) is the connected component of the center of the group $\text{Ad}(E^{\prime}_{C})_{z}$ containing the identity element. Consequently, for any $x\,\in\,X$, the fiber ${\mathbb{T}}_{x}$ in (2.17) is the connected component, containing the identity element, of the center of the fiber of $\text{Ad}(E^{\prime}_{C})/W$ over $x$.

Combining the results of Section 2.1 and Section 2.2, we have the following:

Theorem 2.5**.**

Let $E_{G}$ be a holomorphic principal $G$–bundle over $M$ and $T\,\subset\,G$ a torus containing $Z_{0}(G)$. The normalizer (respectively, centralizer) of $T$ in $G$ will be denoted by $N$ (respectively, $C$), while the Weyl group $N/C$ will be denoted by $W$. There is a natural bijective correspondence between the following two:

1. (1)

Torus subbundles $\mathbb{T}$ of ${\rm Ad}(E_{G})$ such that for some (hence every) $x\,\in\,M$, the fiber ${\mathbb{T}}_{x}$ lies in the conjugacy class of tori in ${\rm Ad}(E_{G})$ determined by $T$. 2. (2)

Quadruples of the form $(E_{W},\,\phi,\,E^{\prime}_{C},\,\tau)$, where $\phi\,:\,E_{W}\,\longrightarrow\,M$ is a principal $W$–bundle, $\phi^{*}E_{G}\,\supset\,E^{\prime}_{C}\,\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}\,E_{W}$ is a holomorphic reduction of structure group of $\phi^{*}E_{G}$ to $C$, and

[TABLE]

is a holomorphic action of $N$ on $E^{\prime}_{C}$ extending the natural action of $C$ on $E^{\prime}_{C}$, such that the diagram of maps

[TABLE]

is commutative, where $E^{\prime}_{C}\times N\,\longrightarrow\,(E^{\prime}_{C}/C)\times(N/C)\,=\,(E^{\prime}_{C}\times N)/(C\times C)$ is the quotient map.

2.3. A tautological connection on a torus bundle

We return to the set-up of Section 2.1. As in (2.3), let

[TABLE]

be a holomorphic sub-fiber bundle containing $M\times Z_{0}(G)$ such that for every point $x\,\in\,M$, the fiber

[TABLE]

is a torus.

A flat connection on the fiber bundle $\mathbb{T}$ is said to be compatible with the group structure of the fibers of $\mathbb{T}$ if for any two locally defined flat sections $s$ and $t$ of $\mathbb{T}$, defined over an open subset $U\,\subset\,M$, the section $s\cdot t$ of ${\mathbb{T}}|_{U}$ is again flat.

Proposition 2.6**.**

There is a tautological flat holomorphic connection on $\mathbb{T}$ which is compatible with the group structure of the fibers of $\mathbb{T}$.

Proof.

Consider the étale Galois covering $\phi\,:\,E_{W}\,\longrightarrow\,M$ is (2.11). Let

[TABLE]

be the holomorphic reduction of structure group constructed in Lemma 2.1. The adjoint action of $C$ on the torus $T$ in (2.4) is trivial, because $T$ is contained in the center of $C$. Therefore, $T$ defines a trivial subbundle

[TABLE]

where $E_{N}$ is the principal $N$–bundle constructed in (2.7). Note that $E_{W}\times T\,\subset\,\text{Ad}(E^{\prime}_{C})$ is the bundle consisting of connected components of the centers, containing the identity element, of the fibers of $\text{Ad}(E^{\prime}_{C})$, and $\text{Ad}(E^{\prime}_{C})$ is the connected component of $\text{Ad}(\phi^{*}E_{N})\,=\,\phi^{*}\text{Ad}(E_{N})$ containing the section given by the identity elements of the fibers.

Let $D_{0}$ denote the trivial (flat) connection on the trivial bundle $E_{W}\times T\,\longrightarrow\,E_{W}$ in (2.18). The tautological action of the Galois group $W\,=\,\text{Gal}(\phi)$ on $\phi^{*}\text{Ad}(E_{N})$ evidently preserves the subbundle $E_{W}\times T\,\subset\,\phi^{*}\text{Ad}(E_{N})$. The resulting action of $W$ on $E_{W}\times T$ clearly preserves the above trivial connection $D_{0}$ on $E_{W}\times T\,\longrightarrow\,E_{W}$.

Since the connection $D_{0}$ is preserved by the action of $W\,=\,\text{Gal}(\phi)$, it produces a flat holomorphic connection on the subbundle $(E_{W}\times T)/W$

[TABLE]

of $\text{Ad}(E_{N})$. But this subbundle $(E_{W}\times T)/W$ is identified with $\mathbb{T}$, because $E_{W}\times T\,\subset\,\phi^{*}\text{Ad}(E_{G})$ in (2.18) coincides with $\phi^{*}{\mathbb{T}}\,\subset\,\phi^{*}\text{Ad}(E_{G})$ (see (2.17); recall that ${\mathbb{T}}^{\prime}$ in (2.17) is identified with $E_{W}\times T$). ∎

3. Torus subbundles and connection

3.1. Connections on a principal bundle

Let $H$ be a complex Lie group. The Lie algebra of $H$ will be denoted by $\mathfrak{h}$. Let

[TABLE]

be a holomorphic principal $H$–bundle on $M$. The holomorphic tangent bundles of $M$ and $E_{H}$ will be denoted by $TM$ and $TE_{H}$ respectively. The quotient

[TABLE]

is a holomorphic vector bundle which is called the Atiyah bundle for $E_{H}$. The differential $d\beta\,:\,TE_{H}\,\longrightarrow\,\beta^{*}TM$ for the above projection $\beta$, being $H$–invariant, produces a surjective homomorphism

[TABLE]

The kernel of $d^{\prime}\beta$ is the relative tangent bundle for $\beta$ and it is identified with the adjoint vector bundle $\text{ad}(E_{H})\,=\,\text{kernel}(d\beta)/H$. We recall that $\text{ad}(E_{H})$ is the quotient of $E_{H}\times\mathfrak{h}$, where two points $(y,\,v)$ and $(y_{1},\,v_{1})$ of $E_{H}\times\mathfrak{h}$ are identified if there is some $h_{0}\,\in\,H$ such that $y_{1}\,=\,yh_{0}$ and $v_{1}\,=\,\text{Ad}(h^{-1}_{0})(v)$. Therefore, $\text{ad}(E_{H})\,\longrightarrow\,M$ is the Lie algebra bundle for the bundle $\text{Ad}(E_{H})$ of Lie groups on $M$. We have the short exact sequence of holomorphic vector bundles on $M$

[TABLE]

which is know as the Atiyah exact sequence. A complex connection on $E_{H}$ is a $C^{\infty}$ homomorphism of vector bundles

[TABLE]

such that $(d^{\prime}\beta)\circ D_{0}\,=\,\text{Id}_{TM}$ (see [At]). A holomorphic connection on $E_{H}$ is a holomorphic homomorphism of vector bundles

[TABLE]

such that $(d^{\prime}\beta)\circ D_{0}\,=\,\text{Id}_{TM}$.

Let $H^{\prime}$ be a complex Lie group and $\eta\,:\,H\,\longrightarrow\,H^{\prime}$ a holomorphic homomorphism of Lie groups. Let

[TABLE]

be the holomorphic principal $H^{\prime}$–bundle over $M$ obtained by extending the structure group of the holomorphic principal $H$–bundle $E_{H}$ using the above homomorphism $\eta$. Recall that $E_{H^{\prime}}$ is the quotient of $E_{H}\times H^{\prime}$ where any two points $(y,\,z)$ and $(y_{1},\,z_{1})$ of $E_{H}\times H^{\prime}$ are identified if there is an element $h\,\in\,H$ such that $y_{1}\,=\,yh$ and $z_{1}\,=\,\eta(h^{-1})z$. So sending any $y\,\in\,E_{H}$ to the equivalence class of $(y,\,e)$, where $e\,\in\,H^{\prime}$ is the identity element, we get a holomorphic map

[TABLE]

which satisfies the equation

[TABLE]

for all $y\,\in\,E_{H}$ and $h\,\in\,H$. The action of $H$ on $E_{H}$ produces an action of $H$ on the tangent bundle $TE_{H}$; using $\eta$, the action of $H^{\prime}$ on the tangent bundle $TE_{H^{\prime}}$ produces an action of $H$ on $TE_{H^{\prime}}$. From (3.1) it follows immediately that the differential $d\eta_{0}$ of the map $\eta_{0}$ is $H$–equivariant. Therefore, $d\eta_{0}$ induces a homomorphism

[TABLE]

This homomorphism $\eta_{1}$ satisfies the equation $\eta_{1}(\text{ad}(E_{H}))\,\subset\,\text{ad}(E_{H^{\prime}})$; in other words, we have a commutative diagram of holomorphic homomorphisms of vector bundles

[TABLE]

where the horizontal sequences are the Atiyah exact sequences. Therefore, if $D_{0}\,:\,TM\,\longrightarrow\,\text{At}(E_{H})$ is a complex connection on $E_{H}$, then

[TABLE]

is a complex connection on $E_{H^{\prime}}$; this $\eta_{1}\circ D_{0}$ is called the connection on $E_{H^{\prime}}$ induced by the connection $D_{0}$ on $E_{H}$. If the connection $D_{0}$ is holomorphic, then the induced connection $\eta_{1}\circ D_{0}$ is clearly holomorphic also.

3.2. Criterion for induced connection

We continue with the set-up of Section 2.1. Take a torus subbundle

[TABLE]

as in (2.3). Fix $T$ as in (2.4), and construct $N$ as in (2.5). Let $E_{N}\,\subset\,E_{G}$ be the holomorphic reduction of structure group to $N$ constructed in (2.7).

Let $D_{0}$ be a complex connection on $E_{G}$. Our aim in this section is to establish a necessary and sufficient condition for $D_{0}$ to be induced by a complex connection on $E_{N}$.

The connection $D_{0}$ on $E_{G}$ induces a complex connection on every holomorphic fiber bundle $E_{G}(F)$ associated to $E_{G}$ for a holomorphic action of $G$ on a complex manifold $F$. In particular, $D_{0}$ induces a complex connection on the bundle $\text{Ad}(E_{G})$ associated to $E_{G}$ for the adjoint action of $G$ on itself. Let $D_{1}$ denote the complex connection on $\text{Ad}(E_{G})$ induced by $D_{0}$.

Theorem 3.1**.**

The connection $D_{0}$ on $E_{G}$ is induced by a complex connection on the principal $N$–bundle $E_{N}$ if and only if the induced connection $D_{1}$ on ${\rm Ad}(E_{G})$ preserves the torus subbundle ${\mathbb{T}}\,\subset\,{\rm Ad}(E_{G})$.

Proof.

First assume that there is a complex connection $D_{2}$ on the principal $N$–bundle $E_{N}$ such that the connection on $E_{G}$ induced by $D_{2}$ coincides with $D_{0}$. Let $D_{3}$ be the complex connection on ${\rm Ad}(E_{N})$ induced by $D_{2}$. Let ${\mathcal{G}}\,\subset\,{\rm Ad}(E_{N})$ be the sub-fiber bundle whose fiber over any $x\,\in\,M$ is the connected component of ${\rm Ad}(E_{N})_{x}$ containing the identity element. So $\mathcal{G}$ coincides with $\text{Ad}(E^{\prime}_{C})/W$ in Remark 2.4. The connection $D_{3}$ on ${\rm Ad}(E_{N})$ clearly preserves ${\mathcal{G}}$. Let $D_{4}$ denote the complex connection on ${\mathcal{G}}$ given by $D_{3}$.

Since $D_{0}$ is induced by $D_{2}$, it follows that the connection $D_{1}$ on ${\rm Ad}(E_{G})$ preserves the torus subbundle ${\mathbb{T}}$ if and only if the connection $D_{4}$ on $\mathcal{G}$ preserves $\mathbb{T}$.

We recall that the torus subbundle ${\mathbb{T}}\,\subset\,\mathcal{G}$ is the bundle of connected components of centers, containing the identity element, of the fibers of $\mathcal{G}$ (see Remark 2.4). From this it can be deduced that the connection on $D_{4}$ on $\mathcal{G}$ preserves $\mathbb{T}$. To prove this, it is convenient to switch to the Lie algebra bundles from the Lie group bundles, because it is easier the work with connections on vector bundles.

Let $\widetilde{\mathbb{T}}$ (respectively, $\widetilde{\mathcal{G}}$) be the Lie algebra bundle over $M$ corresponding to the Lie group bundle ${\mathbb{T}}$ (respectively, ${\mathcal{G}}$). Note that $\widetilde{\mathcal{G}}$ coincides with $\text{ad}(E_{N})$, because $N/C$ is a finite group so $\text{Lie}(N)\,=\,\text{Lie}(C)$. Since ${\mathbb{T}}\,\subset\,\mathcal{G}$ is the bundle of connected components of centers, containing the identity element, of the fibers of $\mathcal{G}$, it follows immediately that for any $x\,\in\,M$, the fiber $\widetilde{\mathbb{T}}_{x}$ is the center of the Lie algebra $\widetilde{\mathcal{G}}_{x}$. Let $D^{\prime}_{4}$ be the complex connection on the vector bundle $\widetilde{\mathcal{G}}$ given by the connection $D_{4}$ on $\mathcal{G}$; note that $D^{\prime}_{4}$ coincides with the connection on $\text{ad}(E_{N})$ induced by the connection $D_{2}$ on $E_{N}$. If $s$ and $t$ are locally defined holomorphic sections of $\widetilde{\mathcal{G}}$ defined over an open subset $U\,\subset\,M$, then we have

[TABLE]

because $D^{\prime}_{4}$ is compatible with the Lie algebra structure of the fibers of $\widetilde{\mathcal{G}}$. Now if $s$ is a section of the subbundle $\widetilde{\mathbb{T}}$, then

[TABLE]

Therefore, we conclude that $[D^{\prime}_{4}(s),\,t]\,=\,0$ if $s$ is a section of $\widetilde{\mathbb{T}}|_{U}$. Hence $D^{\prime}_{4}(s)$ is a $C^{\infty}$ section of $\widetilde{\mathbb{T}}|_{U}\otimes\Omega^{1}_{U}$ if $s$ is a holomorphic section of $\widetilde{\mathbb{T}}|_{U}$. Consequently, the connection $D^{\prime}_{4}$ on $\widetilde{\mathcal{G}}$ preserves the subbundle $\widetilde{\mathbb{T}}$. Hence the connection $D_{4}$ on $\mathcal{G}$ preserves $\mathbb{T}$.

To prove the converse, assume that the connection $D_{0}$ on $E_{G}$ has the following property: the connection $D_{1}$ on ${\rm Ad}(E_{G})$ induced by $D_{0}$ preserves the torus subbundle ${\mathbb{T}}\,\subset\,{\rm Ad}(E_{G})$.

Recall that ${\rm ad}(E_{G})$ is the Lie algebra bundle on $M$ corresponding to the bundle ${\rm Ad}(E_{G})$ of groups. Let $D^{\prime}_{1}$ be the complex connection on the vector bundle ${\rm ad}(E_{G})$ induced by $D_{1}$. As before, $\widetilde{\mathbb{T}}$ denotes the Lie algebra bundle on $M$ corresponding to $\mathbb{T}$, so $\widetilde{\mathbb{T}}$ is an abelian subalgebra bundle of ${\rm ad}(E_{G})$. The given condition that the connection $D_{1}$ on ${\rm Ad}(E_{G})$ preserves $\mathbb{T}$ immediately implies that the connection $D^{\prime}_{1}$ preserves the subbundle $\widetilde{\mathbb{T}}\,\subset\,{\rm ad}(E_{G})$.

If $s$ and $t$ are locally defined holomorphic sections of ${\rm ad}(E_{G})$ defined over an open subset $U\,\subset\,M$, then we have

[TABLE]

because $D^{\prime}_{1}$ is compatible with the Lie algebra structure of the fibers of ${\rm ad}(E_{G})$. Now if $s$ is a section of the subbundle $\widetilde{\mathbb{T}}|_{U}$, and $t$ is a section of $\widetilde{\mathcal{G}}|_{U}$ (defined earlier), then we have

[TABLE]

indeed, $[s,\,t]\,=\,0$ because $\widetilde{\mathbb{T}}$ is the bundle the centers of $\widetilde{\mathcal{G}}$, and $[D^{\prime}_{1}(s),\,t]\,=\,0$ because $D^{\prime}_{1}$ preserves $\widetilde{\mathcal{T}}$. Therefore, we have

[TABLE]

if $s$ is a holomorphic section of the subbundle $\widetilde{\mathbb{T}}|_{U}$ and $t$ is a holomorphic section of $\widetilde{\mathcal{G}}|_{U}$. From this it follows that the subbundle $\widetilde{\mathcal{G}}\,\subset\,\text{ad}(E_{G})$ is preserved by the connection $D^{\prime}_{1}$ on $\text{ad}(E_{G})$. Indeed, as noted before, ${\mathcal{G}}\,=\,\text{ad}(E_{N})\,\subset\,\text{ad}(E_{G})$, because the Lie algebras $\text{Lie}(N)$ and $\text{Lie}(C)$ coincide (the quotient $N/C$ is a finite group). Hence for any $x\,\in\,M$, the subalgebra

[TABLE]

is the centralizer of $\widetilde{\mathbb{T}}_{x}$.

The normalizer of $\text{Lie}(C)$ in $\text{Lie}(G)$ is $\text{Lie}(C)$ itself; this is because the normalizer of $C$ in $G$ is $N$, and $\text{Lie}(C)\,=\,\text{Lie}(N)$. Therefore, from the above observation that $\widetilde{\mathcal{G}}\,=\,\text{ad}(E_{N})$ is preserved by the connection $D^{\prime}_{1}$ on $\text{ad}(E_{G})$ it follows that the connection $D_{0}$ on $E_{G}$ preserves $E_{N}$. In other words, the connection $D_{0}$ on $E_{G}$ is induced by a connection on $E_{N}$. This completes the proof. ∎

Proposition 3.2**.**

Let $D_{0}$ be a complex connection on $E_{G}$ such that the induced connection $D_{1}$ on ${\rm Ad}(E_{G})$ preserves the torus subbundle ${\mathbb{T}}\,\subset\,{\rm Ad}(E_{G})$. Then the connection on $\mathbb{T}$ given by $D_{1}$ coincides with the tautological connection on $\mathbb{T}$ in Proposition 2.6.

Proof.

From Theorem 3.1 we know that there is a complex connection $D_{2}$ on the principal $N$–bundle $E_{N}$ that induces $D_{0}$. Consider the pulled back connection $\phi^{*}D_{2}$ on $\phi^{*}E_{N}$, where $\phi$ is the projection from $E_{W}$ in (2.11). This connection $\phi^{*}D_{2}$ gives a connection on the principal $C$–bundle $E^{\prime}_{C}$ in Lemma 2.1, because $E^{\prime}_{C}$ is a union of some connected component of $\phi^{*}E_{N}$. Let $D^{\prime\prime}$ be the complex connection on $\text{Ad}(E^{\prime}_{C})$ induced by this connection on $E^{\prime}_{C}$ given by $\phi^{*}D_{2}$.

The adjoint action of $C$ on $T$ is trivial because $T$ is contained in the center of $T$. Therefore,

• •

the subbundle

[TABLE]

in (2.18) is preserved by the connection $D^{\prime\prime}$ on $\text{Ad}(E^{\prime}_{C})$, and

• •

the connection on $E_{W}\times T$ given by $D^{\prime\prime}$ coincides with the trivial connection of the trivial bundle.

Now from the construction of the tautological connection on $\mathbb{T}$ in Proposition 2.6 it follows that it coincides with the connection on $\mathbb{T}$ given by $D_{1}$. ∎

Remark 3.3*.*

We use the notation in Theorem 3.1. Let $D_{0}$ be a complex connection on $E_{G}$ such that the induced connection $D_{1}$ on ${\rm Ad}(E_{G})$ preserves the torus subbundle ${\mathbb{T}}\,\subset\,{\rm Ad}(E_{G})$. Let $D_{2}$ be the complex connection on $E_{N}$ inducing $D_{0}$. If the connection $D_{2}$ is holomorphic, then $D$ is holomorphic, because holomorphic connections induce holomorphic connection, as noted in Section 3.1. Since $N$ is a subgroup of $G$, and the principal $G$–bundle $E_{G}$ is the extension of structure group of the principal $N$–bundle $E_{N}$ for the inclusion of $N$ in $G$, it follows that the Atiyah bundle $\text{At}(E_{N})$ is a holomorphic subbundle of $\text{At}(E_{G})$. Therefore, if the connection $D$ is holomorphic then $D_{2}$ is also holomorphic.

3.3. Hermitian–Einstein connection and Levi reduction

In the subsection we assume that $M$ is Kähler and it is equipped with a Kähler form $\omega$. For any torsionfree coherent analytic sheaf $W$ on $X$, define

[TABLE]

where $d\,=\,\dim_{\mathbb{C}}M$ and the determinant line bundle $\det W$ is constructed as in [Ko, Ch. V, § 6]. A torsionfree coherent analytic sheaf $W$ is called stable (respectively, semistable) if every coherent analytic subsheaf $F\,\subset\,W$ with $0\,<\,\text{rank}(F)\,<\,\text{rank}(W)$, we have

[TABLE]

(see [Ko, p. 168]). Also, $W$ is called polystable if it is semistable and direct sum stable sheaves.

We shall now recall a generalization of these notions to the context of principal bundles [Ra], [RS].

An open dense subset $U$ of $M$ will be called big if the complement $M\setminus U$ is a complex analytic subspace of $M$ of complex codimension at least two. For a holomorphic line bundle $L$ on a big open subset $\iota\,:\,U\,\hookrightarrow\,M$, the degree of $L$ is defined to the degree of the direct image $\iota_{*}L$.

A character $\chi$ of a parabolic subgroup $P\,\subset\,G$ is called anti-dominant if the holomorphic line bundle on $G/P$ associated to $\chi$ is nef. Moreover, if the associated line bundle is ample, then $\chi$ is called strictly anti-dominant.

Let $E_{G}$ be a holomorphic principal $G$–bundle over $M$, where $G$, as before, is a connected reductive affine complex algebraic group. It is called stable (respectively, semistable) if for all triples of the form $(P,\,E_{P},\,\chi)$, where

• •

$P\,\subset\,G$ is proper (not necessarily maximal) parabolic subgroup,

• •

$E_{P}\,\subset\,E_{G}$ is a holomorphic reduction of structure group of $E_{G}$ to $P$ over a big open subset $U\,\subset\,M$, and

• •

$\chi$ is a strictly anti-dominant character of $P$ which is trivial on the center of $G$,

the following holds:

[TABLE]

(respectively, $\text{degree}(E_{P}(\chi))\,\geq\,0$), where $E_{P}(\chi)$ is the holomorphic line bundle on $U$ associated to the principal $P$–bundle $E_{P}$ for the character $\chi$. (See [Ra], [RS], [AB].)

Let $P$ be a parabolic subgroup of $G$ and $E_{P}\,\subset\,E_{G}$ a holomorphic reduction of structure group over $M$ of $E_{G}$ to the subgroup $P$. Such a reduction of structure group is called admissible if for every character $\chi$ of $P$ trivial on the center of $G$, the associated holomorphic line bundle $E_{P}(\chi)$ on $M$ is of degree zero.

A holomorphic principal $G$–bundle $E_{G}$ on $M$ is called polystable if either $E_{G}$ is stable or there is parabolic subgroup $P\,\subset\,G$ and a holomorphic reduction of structure group $E_{L(P)}\,\subset\,E_{G}$ over $M$ to a Levi factor $L(P)$ of $P$, such that

• •

the principal $L(P)$–bundle $E_{L(P)}$ is stable, and

• •

the reduction of structure group of $E_{G}$ to $P$ given by the extension of the structure group of $E_{L(P)}$ to $P$, using the inclusion of $L(P)$ in $P$, is admissible.

(See [RS], [AB].)

Fix a maximal compact subgroup

[TABLE]

Let $E_{K_{G}}\,\subset\,E_{G}$ be a $C^{\infty}$ reduction of structure group over $M$ of $E_{G}$ to the subgroup $K_{G}$. Then there is a unique $C^{\infty}$ connection $\nabla$ on $E_{K_{G}}$ such that the connection on $E_{G}$ induced by $\nabla$ is a complex connection [At, pp. 191–192, Proposition 5].

Let $z({\mathfrak{k}})$ denote the center of the Lie algebra of $K_{G}$. A $C^{\infty}$ reduction of structure group over $M$

[TABLE]

is called a Hermitian–Einstein reduction if the corresponding connection $\nabla$ has the property that the curvature ${\mathcal{K}}(\nabla)$ of $\nabla$ satisfies the equation

[TABLE]

for some $c\,\in\,z({\mathfrak{k}})$, where $d\,=\,\dim_{\mathbb{C}}M$. If $E_{K_{G}}$ is a Hermitian–Einstein reduction, then the connection on $E_{G}$ induced by the corresponding connection $\nabla$ on $E_{K_{G}}$ is called a Hermitian–Einstein connection.

A holomorphic principal $G$–bundle $E_{G}$ on $M$ admits a Hermitian–Einstein connection if and only if $E_{G}$ is polystable, and furthermore, if $E_{G}$ is polystable, then it has a unique Hermitian–Einstein connection. When $M$ is a complex projective manifold and $G\,=\,\text{GL}(r,{\mathbb{C}})$, this was proved in [Do1], [Do2]; when $M$ is Kähler and $G\,=\,\text{GL}(r,{\mathbb{C}})$, this was proved in [UY]; when $M$ is a complex projective manifold and $G$ is an arbitrary complex reductive group, this was proved in [RS]; when $M$ is Kähler and $G$ is an arbitrary complex reductive group, this was proved in [AB].

Now consider the set-up of Theorem 2.5; let $T$, $C$, $N$ and $W$ be as in Theorem 2.5.

Let $E_{G}$ be a holomorphic principal $G$–bundle over $M$. Let $\mathbb{T}$ be a torus subbundles of ${\rm Ad}(E_{G})$ such that for some (hence every) $x\,\in\,M$, the fiber ${\mathbb{T}}_{x}$ lies in the conjugacy class of tori in ${\rm Ad}(E_{G})$ determined by $T$. Let $(E_{W},\,\phi,\,E^{\prime}_{C},\,\tau)$ be the corresponding quadruple in Theorem 2.5.

Equip $E_{W}$ with the Kähler form $\phi^{*}\omega$. Note that $E_{W}$ need not be connected. Take a holomorphic principal $H$–bundle $E_{H}$ on $E_{W}$, where $H$ is a connected complex reductive affine algebraic group (we do not use the notation $G$ because it may create confusion as $G$ is used above). We will call $E_{H}$ to be polystable if the following two conditions hold:

1. (1)

the restriction of $E_{H}$ to each connected component of $E_{W}$ is polystable, and 2. (2)

for each character $\chi$ of $H$, the associated holomorphic line bundle $E_{H}(\chi)$ on $E_{W}$ has the property that degrees of its restriction to the connected components of $E_{W}$ coincide. (If $E_{W}$ is connected then this condition is vacuously satisfied.)

Fix a maximal compact subgroup $K_{H}\,\subset\,H$. Let $z({\mathfrak{k}}_{h})$ denote the center of the Lie algebra of $K_{H}$. A $C^{\infty}$ reduction of structure group over $E_{W}$

[TABLE]

will be called a Hermitian–Einstein reduction if the corresponding connection $\nabla$ has the property that there is an element $c\,\in\,z({\mathfrak{k}}_{h})$ such that the curvature ${\mathcal{K}}(\nabla)$ of $\nabla$ satisfies the equation

[TABLE]

If $E_{K_{H}}$ is a Hermitian–Einstein reduction, then the connection on $E_{H}$ induced by the corresponding connection $\nabla$ on $E_{K_{H}}$ will be called a Hermitian–Einstein connection.

If $E_{H}$ is polystable, then the Hermitian–Einstein connections on the restrictions of $E_{H}$ to the connected components of $E_{W}$ together produce a Hermitian–Einstein connection on $E_{H}$. The Hermitian–Einstein connection for the restriction of $E_{H}$ to each component of $E_{W}$ produces an element of $z({\mathfrak{k}}_{h})$ (the element $c$ in (3.3)). The second condition in the definition of polystability for $E_{H}$ ensures that this element of $z({\mathfrak{k}}_{h})$ is independent of the component of $E_{W}$; the elements of $z({\mathfrak{k}}_{h})$ for different connected components of $E_{W}$ coincide. Furthermore, the Hermitian–Einstein connection on $E_{H}$ is unique. Conversely, if $E_{H}$ admits a Hermitian–Einstein connection, then the restriction of $E_{H}$ to each connected components of $E_{W}$ is polystable. Since the element of $z({\mathfrak{k}}_{h})$ for the Hermitian–Einstein connection (the element $c$ in (3.4)) does not depend on the component of $E_{W}$, the second condition in the definition of polystability is satisfied for $E_{H}$.

The intersection $K_{C}\,:=\,C\bigcap K_{G}$ is a maximal compact subgroup of $C$. It will be used for defining the Hermitian–Einstein equation for holomorphic principal $C$–bundles on $E_{W}$.

Proposition 3.4**.**

Assume that the principal $G$–bundle $E_{G}$ on $M$ is polystable. Let $\nabla$ be the Hermitian–Einstein connection on $E_{G}$. Then the following two hold:

1. (1)

The principal $C$–bundle $E^{\prime}_{C}$ on $E_{W}$ is polystable. 2. (2)

The Hermitian–Einstein connection $\phi^{*}\nabla$ on $\phi^{*}E_{G}$ preserves the reduction $E^{\prime}_{C}$ of structure group of $\phi^{*}E_{G}$ to $C$. Furthermore, the connection on $E^{\prime}_{C}$ given by $\phi^{*}\nabla$ is Hermitian–Einstein.

Proof.

Recall that $T$ is contained in the center of $C$. So the natural action of $C$ on the principal $C$–bundle $E^{\prime}_{C}$ commutes with the action of $T\,\subset\,C$ on $E^{\prime}_{C}$. So $T$ acts on the total space of $E^{\prime}_{C}$ preserving its principal $C$–bundle structure. Recall that ${\mathbb{T}}^{\prime}$ in (2.16) is identified with $E_{W}\times T$. The above action of $T$ on $E^{\prime}_{C}$ produces the identification of $E_{W}\times T$ with ${\mathbb{T}}^{\prime}$. Since $\phi^{*}E_{G}$ is identified with the principal $G$–bundle on $E_{W}$ obtained by extending the structure group $E^{\prime}_{C}$ to the group $G$, the action of $T$ on $E^{\prime}_{C}$ produces an action of $T$ on $\phi^{*}E_{G}$. Indeed, $\phi^{*}E_{G}$ is the quotient of $E^{\prime}_{C}\times G$ where two elements $(y,\,z)$ and $(y^{\prime},\,z^{\prime})$ of $E^{\prime}_{C}\times G$ are identified if there is a $g\,\in\,C$ such that $y^{\prime}\,=\,yg$ and $z^{\prime}\,=\,g^{-1}z$. Consider the diagonal action of $T$ on $E^{\prime}_{C}\times G$ given by the above action of $T$ on $E^{\prime}_{C}$ and the trivial action of $T$ on $G$. This diagonal action of $T$ on $E^{\prime}_{C}\times G$ descends to an action of $T$ on the quotient space $\phi^{*}E_{G}$.

Consider the Hermitian–Einstein connection $\phi^{*}\nabla$ on $\phi^{*}E_{G}$. From the uniqueness of a Hermitian–Einstein connection it follows immediately that the above action of $T$ on $\phi^{*}E_{G}$ preserves the Hermitian–Einstein connection $\phi^{*}\nabla$. Since ${\mathbb{T}}\,=\,{\mathbb{T}}^{\prime}/W\,=\,(E_{W}\times T)/W$ (see (2.17)), this implies that the subbundle ${\mathbb{T}}\,\subset\,\text{Ad}(E_{G})$ is preserved by the connection on $\text{Ad}(E_{G})$ induced by $\nabla$. Hence from Theorem 3.1 if follows that the connection $\nabla$ is induced by a complex connection $\nabla^{\prime}$ on the principal $N$–bundle $E_{N}$.

A connection on $E_{N}$ gives a connection on the principal $C$–bundle $E^{\prime}_{C}$; recall that the total spaces of $E_{N}$ and $E^{\prime}_{C}$ coincide. The connection on $E^{\prime}_{C}$ given by the above connection $\nabla^{\prime}$ on the principal $N$–bundle $E_{N}$ will be denoted by $\nabla^{\prime\prime}$. Since the connection $\nabla^{\prime}$ on $E_{N}$ induces the connection $\nabla$ on $E_{G}$, from the definition of $\nabla^{\prime\prime}$ it follows that the connection $\phi^{*}\nabla$ on $\phi^{*}E_{G}$ is induced by $\nabla^{\prime\prime}$. As the connection $\phi^{*}\nabla$ is Hermitian–Einstein, it follows immediately that the inducing connection $\nabla^{\prime\prime}$ is also Hermitian–Einstein.

Since $\nabla^{\prime\prime}$ is a Hermitian–Einstein connection on $E^{\prime}_{C}$, the principal $C$–bundle $E^{\prime}_{C}$ is polystable. ∎

We will now prove a converse of Proposition 3.4. Consider the maximal compact subgroup $K_{C}\,:=\,C\bigcap K_{G}$ of $C$. Let $z({\mathfrak{k}}_{c})$ be the center of the Lie algebra of $K_{C}$. Note that $z({\mathfrak{k}})\,\subset\,z({\mathfrak{k}}_{c})$, because the connected component of the center of $K_{G}$, containing the identity element, is contained in $C$.

Proposition 3.5**.**

Assume that the principal $C$–bundle $E^{\prime}_{C}$ over $E_{W}$ is polystable. Let $\nabla$ be the Hermitian–Einstein connection on $E^{\prime}_{C}$. Assume that the element of $z({\mathfrak{k}}_{c})$ given by the curvature of $\nabla$ (the element $c$ in (3.4)) lies in the subspace $z({\mathfrak{k}})$. Then the following two hold:

1. (1)

The principal $G$–bundle $E_{G}$ on $M$ is polystable. 2. (2)

The Hermitian–Einstein connection on $E_{G}$ is given by $\nabla$.

Proof.

Since $\phi^{*}E_{G}$ is the extension of structure group of $E^{\prime}_{C}$ to $G$ using the inclusion of $C$ in $G$, a connection on $E^{\prime}_{C}$ induces a connection on $\phi^{*}E_{G}$. Let $\nabla^{\prime}$ be the connection of $\phi^{*}E_{G}$ induced by the connection $\nabla$ on $E^{\prime}_{C}$. Since $\nabla$ satisfies the Hermitian–Einstein equation with the element of $z({\mathfrak{k}}_{c})$, given by the curvature of $\nabla$, lying in the subspace $z({\mathfrak{k}})$, it follows immediately that $\nabla^{\prime}$ satisfies the Hermitian–Einstein equation for $\phi^{*}E_{G}$.

Consider the natural action of the Galois group $\text{Gal}(\phi)\,=\,W$ on the pulled back bundle $\phi^{*}E_{G}$. From the uniqueness of the Hermitian–Einstein connection on $\phi^{*}E_{G}$ it follows immediately that this action of $W$ on $\phi^{*}E_{G}$ preserves the Hermitian–Einstein connection $\nabla^{\prime}$. Hence $\nabla^{\prime}$ descends to a connection on $E_{G}$; this connection on $E_{G}$ given by $\nabla^{\prime}$ will be denoted by $\nabla^{\prime\prime}$. Since $\nabla^{\prime}$ satisfies the Hermitian–Einstein equation, it follows immediately that $\nabla^{\prime\prime}$ also satisfies the Hermitian–Einstein equation.

Since $E_{G}$ admits a Hermitian–Einstein connection, namely $\nabla^{\prime\prime}$, the principal $G$–bundle $E_{G}$ is polystable. ∎

4. Higgs bundles and Levi reduction

In this section we work with the set-up of Section 2.1. Take a torus bundle

[TABLE]

as in (2.3). Fix $T$ as in (2.4), and construct $N$ as in (2.5). Let $E_{N}\,\subset\,E_{G}$ be the holomorphic reduction of structure group to $N$ constructed in (2.7).

Let $\iota\,:\,\text{ad}(E_{N})\,\longrightarrow\,\text{ad}(E_{G})$ be the inclusion map. Take a holomorphic vector bundle $V$ on $M$. Let

[TABLE]

be the natural homomorphism. For any $x\,\in\,M$, the adjoint action of $\text{Ad}(E_{G})_{x}$ on $\text{ad}(E_{G})_{x}$ and the trivial action of $\text{Ad}(E_{G})_{x}$ on $V_{x}$ together produce a diagonal action of $\text{Ad}(E_{G})_{x}$ on $(\text{ad}(E_{G})\otimes V)_{x}$. A section

[TABLE]

is said to be fixed by $\mathbb{T}$ if for every $x\,\in\,M$ and $g\,\in\,{\mathbb{T}}_{x}$, the above action of $g$ on $(\text{ad}(E_{G})\otimes V)_{x}$ fixes $\theta(x)$.

Lemma 4.1**.**

A section

[TABLE]

lies in the image of the above homomorphism $(\iota\otimes{\rm Id}_{V})_{*}$ if and only if $\theta$ is fixed by $\mathbb{T}$.

Proof.

For any $x\,\in\,M$, the fixed point locus $(\text{ad}(E_{G})_{x})^{{\mathbb{T}}_{x}}$ for the adjoint action of ${\mathbb{T}}_{x}$ on $\text{ad}(E_{G})_{x}$ is $\text{ad}(E_{N})_{x}$. The lemma follows from this fact. ∎

4.1. $G$-Higgs bundles

In this subsection we set $V\,=\,\Omega^{1}_{M}$. The Lie algebra structure of fibers of $\text{ad}(E_{G})$ and the natural projection $\bigotimes^{2}\Omega^{1}_{M}\,\longrightarrow\,\Omega^{2}_{M}$ together define a homomorphism

[TABLE]

which we shall denote by $\bigwedge$.

We recall that a Higgs field on $E_{G}$ is a holomorphic section

[TABLE]

such that $\theta\bigwedge\theta\,=\,0$ [Si1], [Si2], [Hi]. A $G$–Higgs bundle on $M$ is a pair of the form $(E_{G},\,\theta)$, where $E_{G}$ is a holomorphic principal $G$–bundle over $M$ and $\theta$ is a Higgs field on $E_{G}$.

As in Section 3.3, we assume that $M$ is Kähler and it is equipped with a Kähler form $\omega$.

A $G$–Higgs bundle $(E_{G},\,\theta)$ is called stable (respectively, semistable) if for all triples of the form $(P,\,E_{P},\,\chi)$, where

• •

$P\,\subset\,G$ is proper (not necessarily maximal) parabolic subgroup,

• •

$E_{P}\,\subset\,E_{G}$ is a holomorphic reduction of structure group of $E_{G}$ to $P$ over a big open subset $U\,\subset\,M$ such that

[TABLE]

and

• •

$\chi$ is a strictly anti-dominant character of $P$ which is trivial on the center of $G$,

the following holds:

[TABLE]

(respectively, $\text{degree}(E_{P}(\chi))\,\geq\,0$), where $E_{P}(\chi)$ is the holomorphic line bundle over $U$ associated to the principal $P$–bundle $E_{P}$ for the character $\chi$. (See [Si2], [BS].)

A $G$–Higgs bundle $(E_{G},\,\theta)$ over $M$ is called polystable if either $(E_{G},\,theta)$ is stable or there is parabolic subgroup $P\,\subset\,G$ and a holomorphic reduction of structure group $E_{L(P)}\,\subset\,E_{G}$ over $M$ to a Levi factor $L(P)$ of $P$, such that

• •

$\theta\,in\,H^{0}(M,\,\text{ad}(E_{L(P)})\otimes\Omega^{1}_{M})$,

• •

the $L(P)$–Higgs bundle $(E_{L(P)},\,\theta)$ is stable, and

• •

the reduction of structure group of $E_{G}$ to $P$ given by the extension of the structure group of $E_{L(P)}$ to $P$, using the inclusion of $L(P)$ in $P$, is admissible.

Fix a maximal compact subgroup $K_{G}$ as in (3.2). Let $(E_{G},\,\theta)$ be a $G$–Higgs bundle on $M$. A $C^{\infty}$ reduction of structure group over $M$

[TABLE]

is called a Hermitian–Yang–Mills reduction if the corresponding connection $\nabla$ on $E_{K_{G}}$ has the property that the curvature ${\mathcal{K}}(\nabla)$ of $\nabla$ satisfies the equation

[TABLE]

where $c$ and $d$ are as in (3.3), and $\theta^{*}$ is the adjoint of $\theta$ (the $\mathbb{R}$–vector space $\text{Lie}(G)$ has the decomposition $\text{Lie}(G)\,=\,\text{Lie}(K_{G})\bigoplus\mathfrak{p}$; the adjoint is the conjugate linear automorphism of the vector space $\text{Lie}(G)$ that acts on $\text{Lie}(K_{G})$ as multiplication by $-1$ and acts on $\mathfrak{p}$ as the identity map). If $E_{K_{G}}$ is a Hermitian–Yang–Mills reduction, then the connection on $E_{G}$ induced by the corresponding connection $\nabla$ on $E_{K_{G}}$ is called a Hermitian–Yang–Mills connection [Si1], [Si2], [BS].

A $G$–Higgs bundle $(E_{G},\,\theta)$ admits a Hermitian–Yang–Mills connection if and only if $(E_{G},\,\theta)$ is polystable, and furthermore, if $(E_{G},\,\theta)$ is polystable, then it has a unique Hermitian–Yang–Mills connection [Si1], [Hi], [BS].

Let $T$, $C$, $N$ and $W$ be as in Theorem 2.5. Let $(E_{G},\,\theta)$ be a $G$–Higgs bundle on $M$. Let $\mathbb{T}$ be a torus subbundles of ${\rm Ad}(E_{G})$ such that

• •

for some (hence every) $x\,\in\,M$, the fiber ${\mathbb{T}}_{x}$ lies in the conjugacy class of tori in ${\rm Ad}(E_{G})$ determined by $T$, and

• •

for every $x\,\in\,M$, the action of ${\mathbb{T}}_{x}$ on $(\text{ad}(E_{G})\otimes\Omega^{1}_{M})_{x}$ fixes the element $\theta(x)$.

Let $(E_{W},\,\phi,\,E^{\prime}_{C},\,\tau)$ be the corresponding quadruple in Theorem 2.5. Equip $E_{W}$ with the Kähler form $\phi^{*}\omega$.

The notions of Higgs bundle, polystability and Hermitian–Yang–Mills connection extend to $E_{W}$ as before.

We have the following generalization of Proposition 3.4:

Proposition 4.2**.**

Assume that the $G$–Higgs bundle $(E_{G},\,\theta)$ is polystable. Let $\nabla$ be the Hermitian–Yang–Mills connection on $E_{G}$. Then the following three hold:

1. (1)

$\theta$* defines a Higgs on the principal $C$–bundle $E^{\prime}_{C}$ over $E_{W}$.* 2. (2)

The $C$–Higgs bundle $(E^{\prime}_{C},\,\theta)$ is polystable. 3. (3)

The Hermitian–Yang–Mills connection $\phi^{*}\nabla$ on $\phi^{*}E_{G}$ preserves the reduction $E^{\prime}_{C}$ of structure group of $\phi^{*}E_{G}$ to the subgroup $C$. Furthermore, the connection on $E^{\prime}_{C}$ given by $\phi^{*}\nabla$ is Hermitian–Yang–Mills connection for the $C$–Higgs bundle $(E^{\prime}_{C},\,\theta)$.

Proof.

In view of Lemma 4.1, and the uniqueness of the Hermitian–Yang–Mills connection on a polystable bundle, the proof works along the same line as the proof of Proposition 3.4. We omit the details. ∎

The following is a generalization of Proposition 3.5:

Proposition 4.3**.**

Assume that the $C$–Higgs bundle $(E^{\prime}_{C},\,\theta)$ is polystable. Let $\nabla$ be the Hermitian–Yang–Mills connection on $E^{\prime}_{C}$. Assume that the element of $z({\mathfrak{k}}_{c})$ given by the Hermitian–Yang–Mills equation for $(E^{\prime}_{C},\,\theta)$ lies in $z({\mathfrak{k}})$. Then the following two hold:

1. (1)

The $G$–Higgs bundle $(E_{G},\,\theta)$ on $M$ is polystable. 2. (2)

The Hermitian–Yang–Mills connection on $E_{G}$ is given by $\nabla$.

Proof.

The proof is similar to the proof of Proposition 3.5. ∎

5. Torus for equivariant bundles

Let $\Gamma$ be a group and

[TABLE]

a left–action such that the self–map $x\,\longmapsto\,\varphi_{0}(\gamma,x)$ of $M$ is a biholomorphism for every $\gamma\,\in\,\Gamma$. An equivariant holomorphic principal $G$–bundle on $M$ is a pair $(E_{G},\,\varphi)$, where $p_{0}\,:\,E_{G}\,\longrightarrow\,M$ is a holomorphic principal $G$–bundle and

[TABLE]

is an action of $\Gamma$ on $E_{G}$ such that

• •

for every $\gamma\,\in\,\Gamma$, the self–map $y\,\longmapsto(\gamma,y)$ of $E_{G}$ is a biholomorphism,

• •

$p_{0}\circ\varphi\,=\,\varphi_{0}\circ(\text{Id}_{\Gamma}\times p_{0})$, and

• •

the actions $G$ and $\Gamma$ on $E_{G}$ commute.

Let $(E_{G},\,\varphi)$ be an equivariant holomorphic principal $G$–bundle on $M$. For any complex manifold $F$ equipped with a holomorphic action of $G$, consider the diagonal action of $\Gamma$ on $E_{G}\times F$ given by the action $\varphi$ on $E_{G}$ and the trivial action of $\Gamma$ on $F$. This diagonal action of $\Gamma$ on $E_{G}\times F$ produces an action of $\Gamma$ on the quotient space of $E_{G}\times F$ defining the fiber bundle over $M$ associated to $E_{G}$ for $F$. In particular, $\Gamma$ acts on $\text{Ad}(E_{G})$; this action of $\Gamma$ on $\text{Ad}(E_{G})$ preserves the group structure of the fibers of $\text{Ad}(E_{G})$.

As in (2.3), let

[TABLE]

be a holomorphic sub-fiber bundle such that

• •

the above action of $\Gamma$ on $\text{Ad}(E_{G})$ preserves $\mathbb{T}$,

• •

$M\times Z_{0}(G)\,\subset\,{\mathbb{T}}$, and

• •

for every point $x\,\in\,M$, the fiber

[TABLE]

is a torus (it need not be a maximal torus of $\text{Ad}(E_{G})_{x}$).

Fix $T$ as in (2.4). Then the principal $N$–bundle $E_{N}\,\subset\,E_{G}$ in (2.7) is evidently preserved by the action $\varphi$ of $\Gamma$ on $E_{G}$. The resulting action of $\Gamma$ on $E_{N}$ produces an action of $\Gamma$ on the quotient $E_{W}$ of $E_{N}$ in (2.11). The projection $\phi$ in (2.11) clearly intertwines the action of $\Gamma$ on $E_{W}$ and $M$.

Consider the diagonal action $\Gamma$ on $E_{W}\times E_{G}$ constructed using the actions of $\Gamma$ on $E_{W}$ and $E_{G}$. Recall the $\phi^{*}E_{G}$ is the submanifold of $E_{W}\times E_{G}$ consisting of all $(x,\,y)\,\in\,E_{W}\times E_{G}$ such that $\phi(x)\,=\,p_{0}(y)$, where $p_{0}$ and $\phi$ are the maps in (2.1) and (2.11) respectively. This submanifold of $E_{W}\times E_{G}$ is preserved by the diagonal action of $\Gamma$. Therefore, this action of $\Gamma$ on $E_{W}\times E_{G}$ produces an action of $\Gamma$ on $\phi^{*}E_{G}$. For this action of $\Gamma$ on $\phi^{*}E_{G}$, the projection $\phi^{*}E_{G}\,\longrightarrow\,E_{W}$ is clearly $\Gamma$–equivariant. Also, the natural map $\phi^{*}E_{G}\,\longrightarrow\,E_{G}$ is also $\Gamma$–equivariant. The diagonal action of $\Gamma$ on $E_{W}\times E_{N}$ similarly produces an action of $\Gamma$ on $\phi^{*}E_{N}$, where $E_{N}$ is constructed in (2.7). Note that the inclusion map $\phi^{*}E_{N}\,\hookrightarrow\,\phi^{*}E_{G}$ (which is the pullback of the inclusion map in (2.7)) is $\Gamma$–equivariant. Since the principal $N$–bundle $E_{N}\,\subset\,E_{G}$ is preserved by the action $\varphi$ of $\Gamma$ on $E_{G}$, the principal $C$–bundle $E_{C}\,\subset\,\phi^{*}E_{N}\,\subset\,\phi^{*}E_{G}$ constructed in Lemma 2.1 is preserved by the above action of $\Gamma$ on $\phi^{*}E_{G}$.

Conversely, fix a torus $T\,\subset\,G$ a containing $Z_{0}(G)$. The normalizer (respectively, centralizer) of $T$ in $G$ will be denoted by $N$ (respectively, $C$), while the Weyl group $N/C$ will be denoted by $W$. Let

[TABLE]

be a principal $W$–bundle such that $E_{W}$ is equipped with an action of $\Gamma$ satisfying the following conditions:

• •

the map $\phi$ is $\Gamma$–equivariant, and

• •

the actions of $\Gamma$ and $W$ on $E_{W}$ commute.

Let $(E_{G},\,\varphi)$ be an equivariant holomorphic principal $G$–bundle on $M$. As before, the diagonal action of $\Gamma$ on $E_{W}\times E_{G}$ produces an action of $\Gamma$ on $\phi^{*}E_{G}$. Let

[TABLE]

be a holomorphic reduction of structure group of the principal $G$–bundle $\phi^{*}E_{G}$ to $C$ such that the action of $\Gamma$ on $\phi^{*}E_{G}$ preserves $E^{\prime}_{C}$. Assume that we are further given a holomorphic action of the complex Lie group $N$ on $E^{\prime}_{C}$

[TABLE]

such that

• •

the restriction of the map $\tau$ to $E^{\prime}_{C}\times C$ is the natural action of $C$ on the principal $C$–bundle $E^{\prime}_{C}$,

• •

the actions of $N$ and $\Gamma$ on $E^{\prime}_{C}$ commute, and

• •

the diagram of maps

[TABLE]

is commutative, where $E^{\prime}_{C}\times N\,\longrightarrow\,(E^{\prime}_{C}/C)\times(N/C)\,=\,(E^{\prime}_{C}\times N)/(C\times C)$ is the quotient map, and $\psi$ is the natural projection of the principal $C$–bundle.

Then the action of $\Gamma$ on $\text{Ad}(E_{G})$, given by the action $\varphi$ of $\Gamma$ on $E_{G}$, preserves the torus sub-bundle ${\mathbb{T}}\,\subset\,\text{Ad}(E_{G})$ constructed in (2.17).

Therefore, Theorem 2.5 has the following generalization:

Theorem 5.1**.**

Let $(E_{G},\,\varphi)$ be an equivariant holomorphic principal $G$–bundle on $M$ and $T\,\subset\,G$ a torus containing $Z_{0}(G)$. The normalizer (respectively, centralizer) of $T$ in $G$ will be denoted by $N$ (respectively, $C$), while the Weyl group $N/C$ will be denoted by $W$. There is a natural bijective correspondence between the following two:

1. (1)

$\Gamma$–invariant torus subbundles $\mathbb{T}$ of ${\rm Ad}(E_{G})$ such that for some (hence any) $x\,\in\,M$, the fiber ${\mathbb{T}}_{x}$ lies in the conjugacy class determined by $T$. 2. (2)

Quadruples of the form $(E_{W},\,\phi,\,E^{\prime}_{C},\,\tau)$, where $\phi\,:\,E_{W}\,\longrightarrow\,M$ is a $\Gamma$–equivariant principal $W$–bundle, $E^{\prime}_{C}\,\subset\,\phi^{*}E_{G}$ is a $\Gamma$–invariant holomorphic reduction of structure group of $\phi^{*}E_{G}$ to $C$, and

[TABLE]

is a holomorphic action of $N$ on $E^{\prime}_{C}$ extending the natural action of $C$ on the principal $C$–bundle $E^{\prime}_{C}$, such that the actions of $N$ and $\Gamma$ on $E^{\prime}_{C}$ commute, and the diagram of maps

[TABLE]

is commutative.

All the results in section 3 and Section 4 also extend to the equivariant set-up.