Rigidity of irreducible Hermitian symmetric spaces of the compact type under K"ahler deformation

Abstract

We study deformations of irreducible Hermitian symmetric spaces $S$ of the compact type, known to be locally rigid, as projective-algberaic manifolds and prove that no jump of complex structures can occur. For each $S$ of rank $\ge 2$ there is an associated reductive linear group $G$ such that $S$ admits a holomorphic $G$-structure, corresponding to a reduction of the structure group of the tangent bundle. $S$ is characterized as the unique simply-connected compact complex manifold admitting such a $G$-structure which is at the same time integrable. To prove the deformation rigidity of $S$ it suffices that the corresponding integrable $G$-structures converge. We argue by contradiction using the deformation theory of rational curves. Assuming that a jump of complex structures occurs, cones of vectors tangent to degree-1 rational curves on the special fiber $X_0$ are linearly degenerate, thus defining a proper meromorphic distribution $W$ on $X_0$. We prove that such $W$ cannot possibly exist. On the one hand, integrability of $W$ would contradict the fact that $b_2(X)=1$. On the other hand, we prove that $W$ would be automatically integrable by producing families of integral complex surfaces of $W$ as pencils of degree-1 rational curves. For the verification that there are enough integral surfaces we need a description of generic cones on the special fiber. We show that they are in fact images of standard cones under linear projections. We achieve this by studying deformations of normalizations of Chow spaces of minimal rational curves marked at a point, which are themselves Hermitian symmetric, irreducible except in the case of Grassmannians.