Holomorphic quantization formula in singular reduction

We show that the holomorphic Morse inequalities proved by Tian and the author [TZ1, 2] are in effect equalities by refining the analytic arguments in [TZ1, 2]. §0. Introduction and the statement of main results Let (M,ω, J) be a compact Kähler manifold with the Kähler form ω and the complex structure J . Let g denote the corresponding Kähler metric. We make the assumption that there exists a Hermitian line bundle L over M admitting a Hermitian connection ∇L such that √ −1 2π (∇L)2 = ω. Then L admits a unique holomorphic structure so that ∇L is the associated Hermitian holomorphic connection. We call L the prequantum line bundle over M . Next, suppose that (M,ω, J) admits a holomorphic Hamiltonian action of a compact connected Lie group G with Lie algebra g. Let μ : M → g∗ be the corresponding moment map. Then a formula due to Kostant [K] (cf. [TZ1, (1.13)]) induces a natural g action on L. We make the assumption that this g action can be lifted to a holomorphic G action on L. Then this G action preserves ∇L. After an integration if necessary, we also assume that this G action preserves the Hermitian metric on L. Then for any integer p ≥ 0, the p-th Dolbeault cohomology H(M,L) is a G-representation. We denote its G-invariant part by H(M,L). Now let a ∈ g∗ be a regular value of μ. Let Oa ⊂ g∗ be the coadjoint orbit of a. For simplicity, we assume that G acts on μ(Oa) freely. Then the quotient space MG,a = μ (Oa)/G is smooth. Also, ω descends canonically to a symplectic form ωG,a onMG,a so that one gets the Marsden-Weinstein reduction (MG,a, ωG,a). Furthermore, the complex structure J descends canonically to a complex structure JG,a on TMG,a so that (MG,a, ωG,a, JG,a) is again Kähler. On the other hand, the pair (L,∇L) also descends canonically to a Hermitian holomorphic line bundle LG,a over M . ∗Partially supported by the NNSF, SEC of China and the Qiu Shi Foundation.