Rigidity and Quasi-rigidity of Extremal Cycles in Hermitian Symmetric Spaces

Let M be a compact Hermitian symmetric space and let W 6= ∅ be a compact complex subvariety of M of codimension p. There exists a nontrivial holomorphic exterior differential system I on M with the property that any compact complex subvariety V ⊂ M of dimension p that satisfies [V ]∩ [W ] = 0 is necessarily an integral variety of I. The system I is almost never involutive. However, its integral varieties (when they exist) can sometimes be described explicitly by taking advantage of this non-involutivity. In this article, several of these ideals I will be analyzed, particularly in the case where M is a Grassmannian, and the results applied to prove various results about the rigidity of algebraic cycles with certain specified homology classes. These rigidity results have implications for the classification of certain holomorphic bundles over compact Kähler manifolds that are generated by their global sections. For example, if F → M is generated by its global sections and M is compact and Kähler, then, as is well-known, c2(F ) ≥ 0. If equality holds, then either F is the pullback to M of a holomorphic bundle F ′ → C over a curve C via a holomorphic map κ : M → C or else F = L⊕ T where L is a line bundle and T is trivial. There is a similar (though more complicated) characterization when c3(F ) = 0.