Rigidity and quasi-rigidity of extremal cycles in Hermitian symmetric spaces

Abstract

I use local differential geometric techniques to prove that the algebraic cycles in certain extremal homology classes in Hermitian symmetric spaces are either rigid (i.e., deformable only by ambient motions) or quasi-rigid (roughly speaking, foliated by rigid subvarieties in a nontrivial way). These rigidity results have a number of applications: First, they prove that many subvarieties in Grassmannians and other Hermitian symmetric spaces cannot be smoothed (i.e., are not homologous to a smooth subvariety). Second, they provide characterizations of holomorphic bundles over compact Kahler manifolds that are generated by their global sections but that have certain polynomials in their Chern classes vanish (for example, c_2 = 0, c_1c_2 - c_3 = 0, c_3 = 0, etc.).

arXiv comment113 pages, 6 figures, latex2e with packages hyperref, amsart, graphicx. For Version 2: Many typos corrected, important references added (esp. to Maria Walters' thesis), several proofs or statements improved and/or corrected

Cited by in corpus (5)

• Positivity of Chern classes of Schubert cells and varieties
• Fubini-Griffiths-Harris rigidity and Lie algebra cohomology
• Rigid Schubert varieties in compact Hermitian symmetric spaces
• Exterior differential systems, Lie algebra cohomology, and the rigidity of homogenous varieties
• On embeddings of the Grassmannian $Gr(2,m)$ into the Grassmannian $Gr(2,n)$