  We prove the equivariant holomorphic Morse inequalities for a holomorphic
torus action on a holomorphic vector bundle over a compact Kahler manifold when
the fixed-point set is not necessarily discrete. Such inequalities bound the
twisted Dolbeault cohomologies of the Kahler manifold in terms of those of the
fixed-point set. We apply the inequalities to obtain relations of Hodge numbers
of the connected components of the fixed-point set and the whole manifold. We
also investigate the consequences in geometric quantization, especially in the
context of symplectic cutting.
