HOLOMORPHIC MORSE INEQUALITIES AND SYMPLECTIC REDUCTION

We introduce Morse-type inequalities for a holomorphic circle action on a holomorphic vector bundle over a compact Kähler 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 the Wu–Zhang extension of Witten’s holomorphic Morse inequalities and the 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.