2023/08/25 by Teleman, Andrei
#32G15 #32L05 #32T15 #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2308.13239
Let G be an arbitrary (not necessarily isomorphic to a closed subgroup of GL(r,ℂ)) complex Lie group, U a complex manifold and p:P→ U a C^∞ principal G-bundle on U. We introduce and study the space JκP of bundle almost complex structures of Hölder class Cκ on P. To any J∈ JκP we associate an Ad(P)-valued form \mathfrakfJ of type (0,2) on U which should be interpreted as the obstruction to the integrability of J. For κ≥ 1 we have \mathfrakfJ\inCκ-1(U,\bigwedge\hspace-3.5pt0,2 U\otimesAd(P)) whereas, for κ∈[0,1), \mathfrakfJ is a form with distribution coefficients. Let J∈ JκP with κ∈ (0,+∞]∖ℕ. We prove that J admits locally J-pseudo-holomorphic sections of class Cκ+1 if and only if \mathfrakfJ=0. If this is the case, J defines a holomorphic reduction of the underlying Cκ+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 ∂-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⊂ X.