Asymptotic estimate of a twisted Cauchy-Riemann operator with Neumann boundary condition

Abstract

For a holomorphic function f defined on a strongly pseudo-convex domain in {ℂ}^n such that it has only isolated critical points, we define a twisted Cauchy-Riemann operator {{\displaystyle \bar{\partial }}}_{\tau f}:\bar{\partial }+\tau \partial f\wedge. We will give an asymptotic estimate of the corresponding harmonic forms as τ tends to infinity. This asymptotic estimate is used to recover the residue pairing of the singularity defined by f.

Key words： Asymptotic estimate; residue pairing

References

1 BismutJ M,LebeauG. Complex immersions and Quillen metrics.Publ Math Inst Hautes ′Etudes Sci, 1991, 74: 1–291

2 ChangK C, LiuJ. A cohomology complex for manifolds with boundary. TopolMethods Nonlinear Anal, 1995, 5(2): 325–340

3 FanH J. Schr¨odinger equations, deformation theory and tt ∗-geometry. arXiv: 1107.1290

4 FollandG B, KohnJ J. The Neumann Problem for the Cauchy-Riemann Complex. Princeton: Princeton Univ Press and Univ of Tokyo Press, 1972

5 LiC Z, LiS, SaitoK. Primitive forms via polyvector fields.arXiv: 1311.1659

6 WenH, FanH J. A twisted ∂f-Neumann problem and Toeplitz n-tuples from singularity theory. Manuscripta Math (to appear)

7 WittenE. Supersymmetry and Morse theory. J Differential Geom, 1982, 17(4): 661–692

8 ZhangW P. Lectures on Chern-Weil Theory and Witten Deformations. Singapore: World Scientific Publishing Co Inc, 2001