Totally complex submanifolds of a complex Grassmann manifold of 2-planes

A complex Grassmann manifold G2(Cm+2)G2(Cm+2) of all 2-dimensional complex subspaces in Cm+2Cm+2 has two nice geometric structures – the Kähler structure and the quaternionic Kähler structure. We study totally complex submanifolds of G2(Cm+2)G2(Cm+2) with respect to the quaternionic Kähler structure. We show that the projective cotangent bundle P(T⁎CPm+1)P(T⁎CPm+1) of a complex projective space CPm+1CPm+1 is a twistor space of the quaternionic Kähler manifold G2(Cm+2)G2(Cm+2). Applying the twistor theory, we construct maximal totally complex submanifolds of G2(Cm+2)G2(Cm+2) from complex submanifolds of CPm+1CPm+1. Then we obtain many interesting examples. In particular we classify maximal homogeneous totally complex submanifolds. We show the relationship between the geometry of complex submanifolds of CPm+1CPm+1 and that of totally complex submanifolds of G2(Cm+2)G2(Cm+2).