A product of complex manifolds X = Y1 x Y2 carries a canonical holomorphic splitting
  T_X = pr1^* T_Y1 (+) pr2^* T_Y2
with pr_i the projection onto the i-th factor. A holomorphic splitting of a general compact complex manifold is a fixed decomposition T_X = E1 (+) E2 into holomorphic subbundles of positive rank. A subbundle E of T_X is integrable when the Lie bracket of any two local holomorphic sections of E is again a section of E.

Holomorphic Frobenius gives integrability a concrete local meaning: at every point of X there are coordinates (z1, z2) in a product of polydiscs, with z_i in D^(r_i), in which E1 is the z1-coordinate direction and E2 is the z2-coordinate direction. The problem is not existence of local product charts but whether those charts assemble into a global product.

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Two obstructions keep an integrable splitting from producing a compatible product on X. The holonomy of one foliation along leaves of the other can fail to close up. And even when X is a product, a given splitting may not come from that product. The second obstruction forces integrability to appear as an explicit hypothesis in every statement below.

Beauville's example makes this concrete. Take X = A x P^1 with A a complex abelian surface. Let U, V be independent translation-invariant vector fields on A, and S, T vector fields on P^1 with [S, T] != 0, both viewed as vector fields on the product. Set E1 to be the rank-two span of U + S and V + T, and E2 = T_(P^1). Transversality gives T_X = E1 (+) E2, but
  [U + S, V + T] = [U, V] + [S, T] = [S, T] in T_(P^1)
is not in E1. The splitting exists, X is a product, and yet neither summand matches any product decomposition of X. Integrability of the specific summands is therefore a real constraint.

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Beauville stated the compatible universal-cover conjecture for compact Kaehler manifolds with integrable partial sums of the tangent summands (Section 2.3 of the above) and settled the Kaehler-Einstein and compact-Kaehler-surface cases there (Theorems A and C). For two summands the partial-sum hypothesis amounts to simultaneous integrability of E1 and E2. Hoering later proved automatic integrability on non-uniruled projective manifolds (Hoering 2007, Theorem 1.3), which turns the conjecture into a theorem in that class.

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Main theorem. Let X be a compact connected Kaehler manifold of complex dimension n >= 2, and T_X = E1 (+) E2 a holomorphic splitting into integrable subbundles of positive ranks r1, r2. Let pi : Xtilde -> X be the ordinary universal covering. There are connected simply connected complex manifolds Y1, Y2 and a biholomorphism Phi : Xtilde -> Y1 x Y2 satisfying
  dPhi(pi^* E_i) = pr_i^* T_Yi, i = 1, 2.
In particular dim_C Yi = ri.

The equation dPhi(pi^* E_i) = pr_i^* T_Yi asks more than existence of some product structure on Xtilde: it asks that the two specified subbundles become the two coordinate directions after lifting. The factors are the universal covers of the two foliation leaves through a chosen basepoint, and they can be noncompact. The conclusion concerns the ordinary universal cover and does not by itself describe finite covers of X.