Hoering's conjecture: tangent-bundle summands on rationally connected manifolds are integrable

Theorem 1.1: on a smooth connected rationally connected projective manifold of dimension at least two, both summands of every holomorphic splitting $T_X=E_1\oplus E_2$ into positive-rank subbundles are integrable. Corollary 1.2 (with Hoering's 2007 product theorem): $X\cong X_1\times X_2$ with $E_i=\mathrm{pr}_i^*T_{X_i}$. Smoothness is needed (Hoering 2026 gives a singular counterexample); splittings with more than two summands are not treated.

Theorem 1.1: on a smooth connected rationally connected projective manifold of dimension at least two, both summands of every holomorphic splitting $T_X=E_1\oplus E_2$ into positive-rank subbundles are integrable. Corollary 1.2 (with Hoering's 2007 product theorem): $X\cong X_1\times X_2$ with $E_i=\mathrm{pr}_i^*T_{X_i}$. Smoothness is needed (Hoering 2026 gives a singular counterexample); splittings with more than two summands are not treated.