The Universal Connection for Principal Bundles over Homogeneous Spaces and Twistor Space of Coadjoint Orbits

Biswas, Indranil ; Wong, Michael Lennox

Documenta mathematica, Tome 23 (2018), pp. 77-115.

Résumé

Given a holomorphic principal bundle $Q\longrightarrow X$, the universal space of holomorphic connections is a torsor $C_1(Q)$ for ${ad}\, Q\otimes T^\ast X$ such that the pullback of $Q$ to $C_1(Q)$ has a tautological holomorphic connection. When $X= G/P$, where $P$ is a parabolic subgroup of a complex simple group $G$, and $Q$ is the frame bundle of an ample line bundle, we show that $C_1(Q)$ may be identified with $G/L$, where $L \subset P$ is a Levi factor. We use this identification to construct the twistor space associated to a natural hyper-Kähler metric on $T^\ast(G/P)$, recovering Biquard's description of this twistor space, but employing only finite-dimensional, Lie-theoretic means.

Détail

Classification : 14M17, 32L25, 32L10
Keywords: $\lambda$-connection, rational homogeneous space, twistor space, complexification, Levi subgroup