Product Formula for Analytic Torsion on CR manifolds with S^1-action

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陳雋介(Chun-Chien Chen) 查詢紙本館藏

畢業系所

數學系

論文名稱

(Product Formula for Analytic Torsion on CR manifolds with S^1-action)

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摘要(中)

在這篇論文中，我們先回顧Ray 和Singer 所引進的全純撓率的定義和一些性質，包括了乘積公式。接著，我們回顧了黃榮宗老師和蕭欽玉老師所引進的在有S^1-action 的柯西黎曼流形上解析橈率的傅立葉分量的定義。最後，我們得到了一些在有S^1-action 的柯西黎曼流形上解析橈率的性質，包括了乘積公式。

摘要(英)

In this thesis we first recall the definition and some properties, including product formula, of holomorphic torsion of Ray and Singer. Then we recall the definition of the Fourier components of the analytic torsion on CR manifolds with S^1-action of Huang and Hsiao. Finally, we obtain some properties, including product formula, of the Fourier components of the analytic torsion on CR manifolds with S^1-action.

關鍵字(中)

★ 柯西黎曼流形
★ 分析橈率
★ 全純橈率

關鍵字(英)

★ analytic torsion
★ CR manifold
★ holomorphic torsion

論文目次

摘要 i
Abstract ii
1. Introduction 1
2. Product formula for Holomorphic Torsion 2
2.1. The ∂ operator 2
2.2. The Kodaira Laplacian 4
2.3. Mellin transformation 5
2.4. Analytic Torsion on Complex Manifold 6
3. Product formula for CR Manifolds with S1action 11
3.1. CR manifolds 11
3.2. ∂b-complex 12
3.3. The Kohn Laplacian 13
3.4. CR manifolds with circle action 15
3.5. Heat Kernels of The Kohn Laplacians 17
3.6. Fourier Component of The Analytic Torsion 18
3.7. Analytic Torsion on CR manifolds with S^1 Action 20
Bibliography 23

參考文獻

[1] R. Beals & P.C. Greiner, Calculus on Heisenberg manifolds, Annals of Math. Studies ver.119, Princeton Univ. Press, 1988.
[2] N. Berline, E. Getzler, & M. Vergne, Heat kernels and Dirac operators. Grundlehren der Mathematischen Wissenschaften 298, Springer Verlag, Berlin 1992.
[3] A. Banyaga, & D. F. Houenou, A Brief Introduction to Symplectic and Contact Manifolds, Singapore: World Scientific, (2017)
[4] M.-S. Baouendi, L.-P. Rothschild & F.-Treves, CR structures with group action and extendability of CR functions, Invent. Math., 83 (1985), 359–396.
[5] J.-H. Cheng, C.-Y. Hsiao & I.-H. Tsai, Heat Kernel Asymptotics, Local Index Theorem and Trace Integrals for Cauchy-Riemann Manifolds with S1 Action, Mémoires de la Société Mathématique de France (N.S.), (2019), No.162, 139pp.
[6] S. Dragomir & G. Tomassini, Differential Geometry and Analysis on CR Manifolds, Progress in Mathematics, 246, Birkhäuser Boston, Inc., Boston, MA, 2006.
[7] S.Finski, On the full asymptotics of analytic torsion Journal of Functional analysis, 275(2018) 3457-3503.
[8] C.-Y. Hsiao, Szegö kernel asymptotics for high power of CR line bundles and Kodaira embedding theorems on CR manifolds, Mem. Amer. Math. Soc., 254(1217),1-146.
[9] C.-Y. Hsiao & R.-T. Huang. The asymptotics of the analytic torsion on CR manifolds with S1 action. Communications in Contemporary Mathematics. 2019 Jun 1;21(4):1750094.
[10] C.-Y. Hsiao & X. Li, Morse inequalities for Fourier components of Kohn-Rossi cohomology of CR manifolds with S1-action, Math.Z.284 (2016), 441-468.
[11] X. Ma & G. Marinescu, Holomorphic Morse inequalities and Bergman kernels, Progress in Mathematics, 254, Birkhäuser Verlag, Basel, (2007).
[12] D. B. Ray & I. M. Singer, R-torsion and the Laplacian on Riemannian manifolds Adv. in Math. 7 (1971), 145-210.
[13] D. B. Ray & I. M. Singer, Analytic Torsion for Complex Manifolds, Annals of Mathematics Second Series, Vol. 98, No. 1 (Jul., 1973), pp. 154-177
[14] L. W. Tu, An Introduction to Manifolds. 2nd ed. 2011, New York, NY: Springer New York, (2011)
[15] H. Wang, The Bochner-Kodaira-Nakano for line bundles on CR manifolds with transversal CR ℝ-action, Bulletin of the Institute of Mathematics, Academia Sinica (New Series), Vol. 16 (2021), No. 2, pp. 127-164. 23

指導教授

黃榮宗(Rung-Tzung Huang)

審核日期

2024-7-26

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