The structure of uniruled manifolds with split tangent bundle (Q1001972)

The article under review is a contribution to the classification theory of compact Kähler manifolds \(X\) and focuses on the case where \(X\) is uniruled and its holomorphic tangent bundle \(T_X\) splits into a direct sum \(T_X\simeq V_1\oplus V_2\) of proper subbundles. The author studies the structure of the almost holomorphic MRC-fibration \(\phi: X\dashrightarrow Y\) of \(X\) onto a normal Kähler variety \(Y\) with the universal property that its general fiber \(Z\) is rationally connected and \(Y\) is not uniruled [cf. \textit{J. Kollár, Y. Miyaoka} and \textit{S. Mori}, J. Differ. Geom. 36, No. 3, 765--779 (1992; Zbl 0759.14032)]. The splitting of \(T_X\) induces a splitting of \(T_Z\simeq (T_Z\cap V_{1|Z})\oplus(T_Z\cap V_{2|Z})\). This observation leads to the following structure theorem: If \(1\leq\) rank \(( T_Z\cap V_{1|Z})\leq\) rank \(V_1=2\), and \(T_Z\cap V_{1|Z}\) or \(T_Z\cap V_{2_|Z}\) is integrable, then either rank \(( T_Z\cap V_{1|Z})=2\) and \(X\) admits the structure of an analytic fibre bundle with rationally connected general fiber and \(T_{X/Y}=V_1\), or \(X\) admits an equidimensional holomorphic map onto a Kähler variety with general fiber \(F\) a rational curve and \(T_F\subset V_{1|F}\). For the case rank \(V_1=1\) see \textit{M. Brunella, J. V. Pereira} and \textit{F. Touzet} [Bull. Soc. Math. Fr. 134, No. 2, 241--252 (2006; Zbl 1187.32018)]. If \(X\) is in addition projective, \(V_1\) and \(V_2\) integrable and \(T_Z\cap V_{1|Z}\not=\{0\}\), then the splitting of \(T_X\) induces a splitting \(\tilde{X}\simeq X_1\times X_2\) of the universal covering manifold \(\tilde{X}\) of \(X\). The proof of this fact uses foliation theory, the Ehresmann fibration theorem, and is based on former results of the author [Math. Z. 256, No. 3, 465-479 (2007; Zbl 1137.14030)], where he proved the splitting of \(X\) when it is rationally connected and \(V_1\) or \(V_2\) is integrable. Finally the author applies the theory of Mori contractions to study for projective \(X\) the situation where \(T_X=\bigoplus\limits_{j=1}^k V_j\) with rank \(V_j\leq 2\), \(1\leq j\leq k\). He proves that \(V_j\) is integrable if \(T_Z\cap V_{j_|Z}\not=\{0\}\) and that the MRC-fibration of \(X\) can be realized as a flat fibration onto a projective manifold \(Y\) with \(T_Y=\bigoplus\limits_{j=1}^k T\phi(V_j)\) where the \(T\phi(V_j)\) are specific reflexive subsheaves of \(T_Y\).