Two-orbit K\"ahler manifolds and Morse Theory

We deal with compact K"ahler manifolds $M$ acted on by a compact Lie group $K$ of isometries, whose complexification $K^{\text{C}}$ has exactly one open and one closed orbit in $M$. If the $K$-action is Hamiltonian, we obtain results on the cohomology and the $K$-equivariant cohomology of $M$.