We show that an equivariantly embedded Hermitian symmetric space in a projective space which contains neither a projective space nor a hyperquadric as a component is characterized by its fundamental forms as a local submanifold of the projective space. Using some invariant-theoretic properties of the fundamental forms and Seashi's work on linear differential equations of finite type, we reduce the proof to the vanishing of certain Spencer cohomology groups. The vanishing is checked by Kostant's harmonic theory for Lie algebra cohomology.
Jun-Muk Hwang. Keizo Yamaguchi. "Characterization of Hermitian symmetric spaces by fundamental forms." Duke Math. J. 120 (3) 621 - 634, 1 December 2003. https://doi.org/10.1215/S0012-7094-03-12035-9