S1-equivariant Index theorems and Morse inequalities on complex manifolds with boundary

Abstract

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

Morse Inequalities and Embeddings for CR Manifolds with Circle Action

“…In [23] we studied the Morse inequalities on CR manifolds which admit rigid CR line bundles. For Morse inequalities on non-compact complex manifolds which will be not focused on in this paper we refer the reader to [4,26,31,32] and the references therein.…”