Lie Algebroid Connections on Principal Bundles
Samit Ghosh, Arjun Paul
TL;DR
This work extends the notion of Lie algebroid connections from holomorphic vector bundles to holomorphic principal $G$-bundles over complex projective curves, via a $\mathcal{V}$-valued Atiyah framework. It establishes an obstruction class $\Phi_{\mathcal{V}}(E_G)\in H^1(X,\mathrm{ad}(E_G)\otimes V^*)$ whose vanishing exactly characterizes the existence of a $\mathcal{V}$-valued connection, and defines a curvature $\kappa_{\nabla}$ measuring nonlinearity of the connection. The paper proves that, for genus $g\ge2$ curves and reductive $G$, a stability condition with $\mu(V)<2-2g$ guarantees existence of a $\mathcal{V}$-valued connection on any $G$-bundle, using canonical reductions to Levi subgroups and cohomological vanishing arguments. It also analyzes the behavior under extension and reduction of structure groups, showing how connections descend to associated bundles and highlighting special cases where connections reduce to holomorphic or flat connections. Overall, the results generalize prior vector-bundle statements to principal bundles and provide concrete obstruction-vanishing criteria for the existence of Lie algebroid connections in this broader setting.
Abstract
Let $X$ be an irreducible smooth complex projective variety. Let $G$ be a linear algebraic group over $\mathbb{C}$. We define the notion of Lie algebroid valued connection on holomorphic principal $G$--bundles on $X$, and study their basic properties under extension and reduction of structure group. Finally we investigate criterions for existence of a Lie algebroid connection on principal $G$--bundles over smooth complex projective curves.