Automorphism group of principal bundles, Levi reduction and invariant connections

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 $\mathrm{Ad}(E_G)$ such that for some (hence every) $x\in M$, the fiber ${\mathbb{T}}_x$ lies in the conjugacy class of tori in $\mathrm{Ad}(E_G)$ determined by $T$. Quadruples of the form $(E_W,\varphi ,E_C^{\prime },\tau )$, where $\varphi :E_W⟶M$ is a principal $W$--bundle, ${\varphi }^{\ast }{E}_G\supset E_C^{\prime }\stackrel{\psi }{⟶}{E}_W$ is a holomorphic reduction of structure group of ${\varphi }^{\ast }{E}_G$ to $C$, and $$\tau :E_C^{\prime }\times N⟶E_C^{\prime }$$ is a holomorphic action of $N$ on $E_C^{\prime }$ extending the natural action of $C$ on $E_C^{\prime }$, such that the composition $\psi \circ \tau$ coincides with the composition of the quotient map $E_C^{\prime }\times N⟶(E_C^{\prime }/C)\times (N/C)=(E_C^{\prime }\times N)/(C\times C)$ with the natural map $(E_C^{\prime }/C)\times (N/C)⟶E_W$. The composition of maps $E_C^{\prime }⟶E_W⟶M$ 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 $E_C^{\prime }$ in a very precise manner.

Final version