Morse inequalities for Fourier components of Kohn–Rossi cohomology of CR manifolds with $$S^1$$ S 1 -action

Hsiao, Chin-Yu; Li, Xiaoshan

doi:10.1007/s00209-016-1661-6

Cited by 25 publications

(45 citation statements)

References 19 publications

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“…When m is small, we can expect that n j=0 (−1) j dim H q m (M ) should involve some integration of some characteristic forms over the domain M . In Section 7, we generalize the technique developed in [9] to∂ β case (pseudodifferential operator case) and by using the identification (1.2), we successfully establish Morse inequalities for H q m (M ). We now formulate our main results.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…b,m is self-adjoint with respect to ( · | · ) X . In order to apply our analysis about (q) b,m (see [9]), we will introduce another operatorˆ (q) β,m which is similar to (q) β,m but self-adjoint with respect to ( · | · ) X . LetQ : L 2 (X, T * 0,q M ′ ) → L 2 (0,q) (X) be the orthogonal projection with respect to ( · | · ) X .…”

Section: Set Up Let M Be a Relatively Compact Open Subset With Smootmentioning

confidence: 99%

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

Hsiao

Huang

et al. 2020

Journal of Functional Analysis

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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

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Section: Set Up Let M Be a Relatively Compact Open Subset With Smootmentioning

confidence: 99%

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

Hsiao

Huang

et al. 2020

Journal of Functional Analysis

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“…Moreover, C(L * ) is a smooth CR manifold with a transversal CR S 1 action. The point is that since C(L * ) is a smooth manifold, if we work directly on C(L * ), we should get better results than general orbifolds case (see [9], [10], [11] and [5]). For example, in [5], we can reinterpret Kawasaki's Hirzebruch-Riemann-Roch formula as an index theorem obtained by an integral over a smooth CR manifold which is essentially the circle bundle of this line bundle.…”

Section: Introductionmentioning

confidence: 99%

The asymptotics of the analytic torsion on CR manifolds with S1 action

Hsiao

Huang

2019

Commun. Contemp. Math.

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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.

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“…Thus, we should consider such problems on some class of CR manifolds. It turns out that Kohn's b operator on CR manifolds with S 1 action including Sasakian manifolds of interest in String Theory (see [23]) is a natural one of geometric significance among those transversally elliptic operators initiated by Atiyah and Singer (see [14], [16], [17] and [9]). In [15], Hsiao and the author considered a compact connected strongly pseudoconvex CR manifold X and we introduced the Fourier components of the Ray-Singer analytic torsion on X with respect to a transversal CR S 1 -action.…”

Section: Introductionmentioning

confidence: 99%

The Anomaly Formula of the Analytic Torsion on CR Manifolds with S^1 action

Huang¹

2017

Bulletin of the Inst. of Math. Academia Sinica(N.S.)

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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. In this paper we introduce the Quillen metric on the determinant line of the Fourier components of the Kohn-Rossi cohomology on X with respect to the S 1 -action. We study the behavior of the Quillen metric under the change of the metrics on the manifold X and on the vector bundle over X. We obtain an anomaly formula for the Quillen metric on X with respect to the S 1 -action.

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Morse inequalities for Fourier components of Kohn–Rossi cohomology of CR manifolds with $$S^1$$ S 1 -action

Cited by 25 publications

References 19 publications

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

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

The asymptotics of the analytic torsion on CR manifolds with S1 action

The Anomaly Formula of the Analytic Torsion on CR Manifolds with S^1 action

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