Projective Ranks of Compact Hermitian Symmetric Spaces

Let M be a compact irreducible Hermitian symmetric space and write M = G/K, with G the group of holomorphic isometries of M and K the stability group of the point of 0 ∈ M . We determine the maximal dimension of a complex projective space embedded in M as a totally geodesic submanifold. AMS Subject Classification: 14L35, 22F30, 20G05 Introduction Let M be a simply connected compact complex manifold carrying an Hermitian metric of everywhere nonnegative holomorphic bisectional curvature. In [11] Mok proved that if the second Betti number b2(M) = 1, then M is biholomorphic to an irreducible compact Hermitan symmetric space. It was earlier proved by Siu and Yau [14] and Mori [12] that when the above curvature is everywhere positive M is biholomorphic to a complex projective space. In each of the three papers cited above the existence of a minimally embedded projective line in M plays a crucial role. It turns out that such minimally embedded projective lines are totally geodesic in M . In this paper we study the maximal totally geodesic complex submanifolds of M which are biholomorphic to a complex projective space. We call the dimension of such a submanifold the projective rank of M . We calculate the projective ranks of each of the irreducible compact Hermitian symmetric spaces. The results are given in Section 5. In Sections 1 through 4 we develop the techniques used to make these calculations and in the last section we discuss the degrees of the holomorphic totally geodesic maps φ : P C → M where r = projective rank of M . We also discuss the conjugacy of these maximal totally geodesic complex projective spaces in M under the action of the group of isometries of M . We shall make use of the work of Chen and Nagano [5] on totally geodesic submanifolds of symmetric spaces. Several authors have studied the question of minimal or energy minimizing maps from S2 to a compact symmetric space. The interested reader may consult [2] and [3] for connections with the present paper. It is the recent work of Robert Bryant [18] which motivates the author to revive these results which were summarized without proofs in the article [19]. Unlike Bryant, the approach taken here is very much an algebraic one with only an occasional nod to the topological and analytic methods which underlie many of the foundational results. We thank the referee of an earlier version of this paper for pointing out an error in our original discussion of the case of the quadrics. 1 Preliminaries on Hermitian Symmetric Spaces. A complex manifold M0 (noncompact) with Hermitian metric h is an Hermitian symmetric space or H.S.S. if each point of M0 is an isolated fixed point of an involutive holomorphic isometry of M0. Let G0 denote the connected component of the group of holomorphic isometries of M0. Then G0 is a connected Lie group which acts transitively on M0. Fix p0 ∈ M0 and let K ⊂ G0 be the isotropy group of p0 so M0 = G0/K. We introduce the following notations: