Holomorphic quantization formula in singular reduction

In this note, we prove a quantization formula for singular reductions. The main result is obtained as a simple application of an extended quantization formula proved in [TZ2].

In this note, we show various versions of holomorphic Morse inequalities tensoring with a coherent sheaf.

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 a...

In this paper, we establish strong holomorphic Morse inequalities on non-compact manifolds under the condition of optimal fundamental estimates. We show that optimal fundamental estimates are satisfied and then strong holomorphic Morse inequalities hold true in various settings.

The goal of this note is to present the potential relationships between certain Monge-Amp\`ere integrals appearing in holomorphic Morse inequalities, and asymptotic cohomology estimates for tensor powers of line bundles, as recently introduced by algebraic geometers. The expected most general statements are still conjectural and owe a debt to Riemann's pioneering work, which led to the concept of Hilbert polynomials and to the Hirzebruch-Riemann-Roch formula during the XX-th ...

We study asymptotic estimates of the dimension of cohomology on possibly non-compact complex manifolds for line bundles endowed with Hermitian metrics with algebraic singularities. We give a unified approach to establishing singular holomorphic Morse inequalities for hyperconcave manifolds, pseudoconvex domains, $q$-convex manifolds and $q$-concave manifolds, and we generalize related estimates of Berndtsson. We also consider the case of metrics with more general than algebra...

In this paper, we compute the Tian-Zhu invariant on hypersurfaces of complex projective spaces.

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 investi...

The aim of this paper is to provide a proof for a version of Morse inequality for manifolds with boundary. Our main results are certainly known to the experts on Morse theory, nevertheless it seems necessary to write down a complete proof for it. Our proof is analytic and is based on J. Roe's account of Witten's approach to Morse Theory.

We use the method of homological quantum reduction to construct a deformation quantization on singular symplectic quotients in the situation, where the coefficients of the moment map define a complete intersection. Several examples are discussed, among others one where the singularity type is worse than an orbifold singularity.