OAI.UniversalCoverSplitting.main

The theorem states that, for a compact connected complex manifold X of complex dimension n ≥ 2 (Hausdorff and second countable, with charts into ℂⁿ and holomorphic transition maps) carrying a Kähler metric g, and a holomorphic splitting S of its holomorphic tangent bundle into complementary pieces of positive ranks r₁ and r₂, the following holds. Here g is a smooth real inner product on tangent spaces that is invariant under multiplication by i, and whose fundamental form is closed. The splitting S is a field of ℂ-linear idempotent endomorphisms P of the tangent spaces, holomorphic in charts, whose image has constant complex dimension r₁ and whose kernel has constant complex dimension r₂. Assume both P and the complementary projection id − P are integrable, meaning that in every chart, on every open set, the Lie bracket of any two holomorphic vector fields lying in the image of the projection again lies in it. Let Z be any complex manifold of dimension n with an ordinary universal cover π : Z → X, that is, a covering map with Z connected and simply connected whose lifted complex structure makes π a local biholomorphism. Then there exist complex manifolds Y₁ of dimension r₁ and Y₂ of dimension r₂, each connected and simply connected, and a homeomorphism Φ from Z to Y₁ × Y₂ that is holomorphic in both directions. Moreover, Φ's differential carries the π-pullback of the image of P onto the tangent directions of the first factor, namely the kernel of the projection to Y₂, and carries the π-pullback of the kernel of P onto the tangent directions of the second factor, namely the kernel of the projection to Y₁. The statement is admitted in the source with a placeholder proof.