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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 on X. We show that the ∂‾Neumann Laplacian on M is transversally elliptic and as a consequence, the mth Fourier component of the qth Dolbeault cohomology group H_{m} ^{q}(M‾) is finite dimensional, for every m∈Z and every q=0,1,…,n. This enables us to define ∑_{j=0} ^{n}(−1)^{j}dimH_{m} ^{j}(M‾) the mth Fourier component of the Euler characteristic on M and to study large mbehavior of H_{m} ^{q}(M‾). In this paper, we establish an index formula for ∑_{j=0} ^{n}(−1)^{j}dimH_{m} ^{j}(M‾) and Morse inequalities for H_{m} ^{q}(M‾).
Original language  English 

Article number  108558 
Journal  Journal of Functional Analysis 
Volume  279 
Issue number  3 
DOIs  
State  Published  15 Aug 2020 
Keywords
 Index theorem
 Morse inequalities
 Pseudodifferential operators
 ∂‾Neumann problem
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Dive into the research topics of 'S^{1}equivariant Index theorems and Morse inequalities on complex manifolds with boundary'. Together they form a unique fingerprint.Projects
 2 Finished

Analytic Torsion and Geometric Quantization on Complex and Cr Manifolds(2/2)
1/08/19 → 31/07/21
Project: Research
