Weakly complex homogeneous spaces (Q2249345)

The article gives a classification of all simply connected compact homogeneous manifolds \(M=G/H\) with Euler characteristic \(\chi(M)\not=0\) which are weakly complex in the sense that their tangent bundle is weakly complex. The condition \(\chi(M)\not=0\) means that \(G\) and \(H\) are of equal rank. An application of results of \textit{A. Borel} and \textit{J. de Siebenthal} [Comment. Math. Helv. 23, 200--221 (1949; Zbl 0034.30701)] shows that \(M\) is weakly complex iff it can be represented as a product of irreducible weakly complex homogeneous spaces \(M_i=G_i/H_i\) with \(\chi(M_i)\not=0\) and \(G_i\) a compact simple Lie group. The authors present a detailed and complete list of the possibilities for the \(M_i\). The \(M_i\) carry an invariant almost complex structure or have stably trivial tangent bundle or are of one of the following types: \(F_4/(\)Spin\((4)\times T^2)\), \(F_4/(\)Spin\((4)\times U(2))\), SO\((2p+2q+1)/(\)SO\((2p)\times U)\), Sp\((p+q)/(\)Sp\((1)^p\times U)\). The article uses classificaton results of \textit{R. Hermann} [Trans. Am. Math. Soc. 83, 471-481 (1956; Zbl 0073.18404)] and of \textit{W. Singhof} and \textit{D. Wemmer} [Math. Ann. 274, 157--176 (1986; Zbl 0572.57014)]. It extends results of a joint paper of the authors with \textit{P. Gauduchon} [Invent. Math. 184, No. 2, 389--403 (2011; Zbl 1225.53028)].