A study on deformations of the complex structure of complex manifolds goes back to Riemann. After one hundred years from Riemann, Kodaira and Spencer developed the deformation theory of higher dimensional compact complex manifolds. They showed that an infinitesimal deformation of a compact complex manifold should be represented by the Kodaira-Spencer class, which is an element of the first cohomology group with coefficients in the sheaf of germs of holomorphic vector fields.
For a rational homogeneous manifold $$extract_itex$$G/P$$/extract_itex$$ for a complex simple Lie group $$extract_itex$$G$$/extract_itex$$ and a parabolic subgroup $$extract_itex$$P subset G$$/extract_itex$$, the Bott-Borel-Weil theorem and Kodaira-Spencer deformation theory imply the local deformation rigidity of rational homogeneous manifolds. Furthermore, Hwang and Mok have proved the global deformation rigidity of a rational homogeneous manifold of Picard number 1 different from the orthogonal Grassmannian $$extract_itex$$Gr_q(2, 7)$$/extract_itex$$.
It is then natural to have questions about the deformation rigidity of quasi-homogeneous manifolds.
Let $$extract_itex$$V$$/extract_itex$$ be a complex vector space endowed with a skew-symmetric bilinear form $$extract_itex$$omega$$/extract_itex$$ of maximal rank. When $$extract_itex$$dim V$$/extract_itex$$ is odd, say, $$extract_itex$$2n+1$$/extract_itex$$, we call the variety $$extract_itex$$Gr_{omega}(k, 2n+1)$$/extract_itex$$ of all $$extract_itex$$k$$/extract_itex$$-dimensional isotropic subspaces of $$extract_itex$$V$$/extract_itex$$ as the odd symplectic Grassmannian, which is not homogeneous and has two orbits under the action of its automorphism group if $$extract_itex$$2 leq k leq n$$/extract_itex$$.
I explain the rigidity under K"{a}hler deformation of the complex structure of odd Lagrangian Grassmannians, i.e., the Lagrangian case $$extract_itex$$Gr_{omega}(n, 2n+1)$$/extract_itex$$ of odd symplectic Grassmannians. To obtain the global deformation rigidity of the odd Lagrangian Grassmannian, we use results about the automorphism group of this manifold, the Lie algebra of infinitesimal automorphisms of the affine cone of the variety of minimal rational tangents and its prolongations.