  We introduce Morse-type inequalities for a holomorphic circle action on a
holomorphic vector bundle over a compact Kaehler manifold. Our inequalities
produce bounds on the multiplicities of weights occurring in the twisted
Dolbeault cohomology in terms of the data of the fixed points and of the
symplectic reduction. This result generalizes both Wu-Zhang extension of
Witten's holomorphic Morse inequalities and Tian-Zhang Morse-type inequalities
for symplectic reduction.
  As an application we get a new proof of the Tian-Zhang relative index theorem
for symplectic quotients.
