Classification of pluriharmonic maps from compact complex manifolds with positive first Chern class into complex Grassmann manifolds

Abstract

We prove that any pluriharmonic map from a compact complex manifold with positive first Chern class (defined outside a certain singularity set of codimension at least two) into a complex Grassmann manifold of rank two is explicitly constructed from a rational map into a complex projective space. Under some restrictions on dimension and rank of the domain manifold and the target manifold, respectively, we also prove that similar results hold for other complex Grassmann manifolds as targets. Introduction. Let φ: M -> N be a smooth map from a complex manifold into a Riemannian manifold. Then, φ is said to be pluriharmonic if the (0, l)-exterior covariant derivative Ddφ of the (1, O)-differential dφ of φ vanishes identically. Let V be the pull-back connection on the pull-back bundle φ~^TN. We have (0.1) (Pdφ){X, Y) = V$dφ(Y)-dφ(dχY), X, where Γ M 1 0 is the holomorphic tangent bundle of M. If φ~TN has the Koszul-Malgrange holomorphic structure, that is, the (0, l)-part of V coincides with the δ-operator, we may say that φ is pluriharmonic if and only if φ sends any holomorphic section of TM to a holomorphic section of φ~TN. It is easily seen that if φ is holomorphic and TV is a Kahler manifold then φ~TN' has the Koszul-Malgrange holomorphic structure, hence any holomorphic map is pluriharmonic. Note that an anti-holomorphic map is also pluriharmonic if N is a Kahler manifold. Conversely, the existence of the Koszul-Malgrange holomorphic structure OIK/) TN is ensured if φ is pluriharmonic and Nhas nonnegative or nonpositive curvature operator. In this case, if N is a Kahler manifold, then φ~TN' has the Koszul-Malgrange holomorphic structure (cf. [O-U2]). From the point of view of Riemannian geometry, the most interesting property of pluriharmonic maps is that it Partially supported by the Grants-in-Aid for Scientific Research, The Ministry of Education, Science and Culture, Japan. 1991 Mathematics Subject Classification. Primary 58E20; Secondary 53C42.