Abstract. Let X be a compact connected strongly pseudoconvex CR manifold of dimension 2n + 1, n ≥ 1 with a transversal CR S 1 -action on X. We introduce the Fourier components of the Ray-Singer analytic torsion on X with respect to the S 1 -action. We establish an asymptotic formula for the Fourier components of the analytic torsion with respect to the S 1 -action. This generalizes the asymptotic formula of Bismut and Vasserot on the holomorphic Ray-Singer torsion associated with high powers of a positive line bundle to strongly pseudoconvex CR manifolds with a transversal CR S 1 -action. IntroductionIn [ In orbifold geometry, we have Kawasaki's Hirzebruch-Riemann-Roch formula [12] and also general index theorem [16]. To study further geometric problem for orbifolds (for example, local family index theorem), it is important to know the corresponding heat kernel asymptotics and to have the concept of analytic torsion. The difficulty comes from the fact that in general the heat kernel expansion involves strata contribution (see [6]) and it is difficult to study local index theorem and analytic torsion on general orbifolds. Thus, it is natural to first attack such problems on some class of orbifolds. In complex orbifold geometry, we usually consider an orbifold ample line bundle L over a a projective orbifold M . An important case is when Tot (L * ) the space of all non-zero vectors in the dual bundle of L is smooth, where Tot (L * ) is considered as a complex manifold. In this case, the circle bundle C(L * ) is a smooth manifold.2000 Mathematics Subject Classification. Primary: 58J52, 58J28; Secondary: 57Q10.

In this paper, we give an explicit formula for the Szegő kernel for (0, q) forms on the Heisenberg group H n+1 .

Let M be a complex manifold of dimension n with smooth connected boundary X. Assume that M admits a holomorphic S 1 -action preserving the boundary X and the S 1 -action is transversal and CR on X. We show that the ∂-Neumann Laplacian on M is transversally elliptic and as a consequence, the m-th Fourier component of the q-th Dolbeault cohomology group H q m (M ) is finite dimensional, for every m ∈ Z and every q = 0, 1, . . . , n. This enables us to define n j=0 (−1) j dim H q m (M ) the m-th Fourier component of the Euler characteristic on M and to study large m-behavior of H q m (M ). In this paper, we establish an index formula for n j=0 (−1) j dim H q m (M ) and Morse inequalities for H q m (M ). CONTENTS 1. Introduction and statement of the main results 1 2. Preliminaries 7 2.1. Some standard notations 7 2.2. Set up 7 3. S 1 -equivariant ∂-Neumann problem 9 4. The operators ∂ β , (q) β and reduction to the boundary 13 5. Index theorem 24 6. The scaling technique 26 7. Holomorphic Morse inequalities on complex manifolds with boundary 28 References 34and dρ(x) = 0 at every point x ∈ X. Then the manifold X is a CR manifold with a natural CR structure T 1,0 X :It means that the S 1 -action preserves the complex structure J of M ′ . In this work, we assume that Assumption 1.1. The S 1 -action preserves the boundary X, that is, we can find a defining function ρ ∈ C ∞ (M ′ , R) of X such that ρ(e iθ • x) = ρ(x), for every x ∈ M ′ and every θ ∈ [0, 2π].The S 1 -action e iθ induces a S 1 -action e iθ on X. PutIn this work, we assume that (1.5) X reg is non-empty.Since X is connected, X reg is an open subset of X and X \ X reg is of measure zero. Let T ∈ C ∞ (M ′ , T M ′ ) be the global real vector field induced by e iθ , that is (T u)(x) = ∂ ∂θ u(e iθ • x)| θ=0 , for every u ∈ C ∞ (M ′ ). In this work, we assume thatThe Assumption 1.2 implies that the S 1 -action on M ′ induces a locally free S 1 -action on X. Let ρ be the defining function of M given in the Assumption 1.1. Since X is connected 4 S 1 -EQUIVARIANT INDEX THEOREMS AND MORSE INEQUALITIES ON COMPLEX MANIFOLDS WITH BOUNDARY † β v = γ∂ ⋆ fP v = 0. Combining (4.37), (4.38) with γ(∂ ∂ ⋆ f + ∂ ⋆ f ∂)P v = 0, we have γ∂ ⋆ fP γ∂P v = −γ∂P γ∂ ⋆ fP v = 0 and (4.39) γ∂ ⋆ fP (I − Q)γ∂P v = γ∂ ⋆ fP γ∂P v − γ∂ ⋆ fP Qγ∂P v = 0. Combining (4.39) with (4.12), we get γ∂ ⋆ fP (I − Q)γ∂P v = γ∂ ⋆ fP (P ⋆P ) −1 (∂ρ) ∧ Sγ∂P v = 0. Thus, (4.40) γ(∂ρ) ∧ ∂ ⋆ fP (P ⋆P ) −1 (∂ρ) ∧ Sγ∂P v = 0. In view of Theorem 4.9, we see that for every m ≥m 0 , m ∈ N, the operator

We construct contact forms with constant Q^\prime -curvature on compact three-dimensional CR manifolds which admit a pseudo-Einstein contact form and satisfy some natural positivity conditions. These contact forms are obtained by minimizing the CR analogue of the II -functional from conformal geometry. Two crucial steps are to show that the P^\prime -operator can be regarded as an elliptic pseudodifferential operator and to compute the leading order terms of the asymptotic expansion of the Green function for \sqrt{P^\prime} .