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

Chin Yu Hsiao, Rung Tzung Huang, Xiaoshan Li, Guokuan Shao

Research output: Contribution to journalArticlepeer-review

Abstract

Let M be a complex manifold of dimension n with smooth connected boundary X. Assume that M‾ admits a holomorphic S1-action preserving the boundary X and the S1-action is transversal 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 Hm q(M‾) is finite dimensional, for every m∈Z and every q=0,1,…,n. This enables us to define ∑j=0 n(−1)jdimHm j(M‾) the m-th Fourier component of the Euler characteristic on M and to study large m-behavior of Hm q(M‾). In this paper, we establish an index formula for ∑j=0 n(−1)jdimHm j(M‾) and Morse inequalities for Hm q(M‾).

Original languageEnglish
Article number108558
JournalJournal of Functional Analysis
Volume279
Issue number3
DOIs
StatePublished - 15 Aug 2020

Keywords

  • Index theorem
  • Morse inequalities
  • Pseudodifferential operators
  • ∂‾-Neumann problem

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