  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.
