I prove that any complex manifold that has a projective second fundmental form isomorphic to one of a rank two compact Hermitian symmetric space (other than a quadric hypersurface) at a general point must be an open subset of such a space. This contrasts the non-rigidity of all other compact Hermitian symmetric spaces observed in J.M. Landsberg and L. Manive's articles. A key step is the use of higher order Bertini type theorems that may be of interest in their own right.
"Griffiths-Harris rigidity of compact Hermitian symmetric spaces." J. Differential Geom. 74 (3) 395 - 405, November 2006. https://doi.org/10.4310/jdg/1175266232