Beauville's splitting conjecture for two integrable summands

Theorem 1.1: if a compact connected Kahler manifold has $T_X=E_1\oplus E_2$ with both summands integrable holomorphic subbundles of positive rank, its universal cover is biholomorphic to $Y_1\times Y_2$ (connected, simply connected) with $d\Phi(\pi^*E_i)=\mathrm{pr}_i^*T_{Y_i}$. Corollary: non-uniruled projective manifolds with a two-summand splitting (integrability by Hoering 2007). It does not treat three or more summands, gives no finite-cover product, and assumes integrability (automatic integrability is the companion entry, for rationally connected manifolds).

Theorem 1.1: if a compact connected Kahler manifold has $T_X=E_1\oplus E_2$ with both summands integrable holomorphic subbundles of positive rank, its universal cover is biholomorphic to $Y_1\times Y_2$ (connected, simply connected) with $d\Phi(\pi^*E_i)=\mathrm{pr}_i^*T_{Y_i}$. Corollary: non-uniruled projective manifolds with a two-summand splitting (integrability by Hoering 2007). It does not treat three or more summands, gives no finite-cover product, and assumes integrability (automatic integrability is the companion entry, for rationally connected manifolds).