The Newlander-Nirenberg Theorem for principal bundles

TL;DR

This paper extends the classical Newlander-Nirenberg theorem to principal bundles with complex structures, establishing conditions for integrability and holomorphic reductions in the context of complex Lie groups and bundles.

Contribution

It introduces a new framework for analyzing almost complex structures on principal bundles and characterizes their integrability via an obstruction form, generalizing classical results.

Findings

Characterization of integrability via the vanishing of an obstruction form

Existence of local pseudo-holomorphic sections under certain conditions

Holomorphic reduction of principal bundles when integrability holds

Abstract

Let $G$ be an arbitrary (not necessarily isomorphic to a closed subgroup of $\mathrm{GL}(r,\mathbb{C})$) complex Lie group, $U$ a complex manifold and $p:P\to U$ a $\mathcal{C}^\infty$ principal $G$-bundle on $U$. We introduce and study the space $\mathcal{J}^\kappa_P$ of bundle almost complex structures of H{\"o}lder class $\mathcal{C}^\kappa$ on $P$. To any $J\in \mathcal{J}^\kappa_P$ we associate an $\mathrm{Ad}(P)$-valued form $\mathfrak{f}_J$ of type (0,2) on $U$ which should be interpreted as the obstruction to the integrability of $J$. For $\kappa\geq 1$ we have $\mathfrak{f}_J\in\mathcal{C}^{\kappa-1}(U,\bigwedge\hspace{-3.5pt}^{0,2}_{\,\,U}\otimes\mathrm{Ad}(P))$ whereas, for $\kappa\in[0,1)$, $\mathfrak{f}_J$ is a form with distribution coefficients. Let $J\in \mathcal{J}^\kappa_P$ with $\kappa\in (0,+\infty]\setminus\mathbb{N}$. We prove that $J$ admits locally $J$-pseudo-holomorphic sections of class $\mathcal{C}^{\kappa+1}$ if and only if $\mathfrak{f}_J=0$. If this is the case, $J$ defines a holomorphic reduction of the underlying $\mathcal{C}^{\kappa+1}$-bundle of $P$ in the sense of the theory of principal bundles on complex manifolds. The proof is based on classical regularity results for the $\bar\partial$-Neumann operator on compact, strictly pseudo-convex complex manifolds with boundary.The result will be used in forthcoming articles dedicated to moduli spaces of holomorphic bundles (on a compact complex manifold $X$) framed along a real hypersurface $S\subset X$.