Proceedings of the American Mathematical Society

Author(s): Nikolai Tarkhanov
Journal: Proc. Amer. Math. Soc. 132 (2004), 2411-2419.
MSC (2000): Primary 32S50; Secondary 58J20
Posted: March 24, 2004
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Abstract: We show a Lefschetz fixed point formula for holomorphic functions in a bounded domain $\mathcal{D}$ with smooth boundary in the complex plane. To introduce the Lefschetz number for a holomorphic map of $\mathcal{D}$ , we make use of the Bergman kernel of this domain. The Lefschetz number is proved to be the sum of the usual contributions of fixed points of the map in $\mathcal{D}$ and contributions of boundary fixed points, these latter being different for attracting and repulsing fixed points.

1.: M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes. 1, Ann. Math. 86 (1967), no. 2, 374-407. MR 35:3701
2.: A. V. Brenner and M. A. Shubin, The Atiyah-Bott-Lefschetz theorem for manifolds with boundary, Funkts. Analiz 15 (1981), no. 4, 67-68; English transl., Funct. Anal. Appl. 15 (1981), 286-287. MR 83a:58083
3.: H. Donnelly and C. Fefferman, Fixed point formula for the Bergman kernel, Amer. J. Math. 108 (1986), no. 5, 1241-1258. MR 87k:32043
4.: M. Eichler, Eine Verallgemeinerung der Abelschen Integrale, Math. Z. 67 (1957), 267-298. MR 19:740a
5.: B. Fedosov, On G-trace and G-index in deformation quantization, Lett. Math. Phys. 52 (2000), 29-49. MR 2001k:53175
6.: R. Harvey and J. Polking, The $\bar{\partial}\,$ -Neumann kernel in the ball in $\mathbb{C}^{n}$ , Proc. Sympos. Pure Math. 41 (1984), 117-136. MR 85j:32034
7.: H. Hopf, Über die algebraische Anzahl von Fixpunkten, Math. Z. 29 (1929), 493-524.
8.: L. Hörmander, The analysis of linear partial differential operators. III. Pseudodifferential operators, Springer-Verlag, Berlin, 1985. MR 87d:35002a
9.: S. Lefschetz, Intersections and transformations of complexes and manifolds, Trans. Amer. Math. Soc. 28 (1926), 1-49.
10.: I. Lieb and R. M. Range, Integral representations on Hermitian manifolds: the $\bar{\partial}\,$ -Neumann solution of the Cauchy-Riemann equations, Bull. Amer. Math. Soc. 11 (1984), 355-358. MR 85h:32004
11.: V. K. Patodi, Holomorphic Lefschetz fixed point formula, Bull. Amer. Math. Soc. 79 (1973), 825-828. MR 47:5906
12.: N. Tarkhanov, Complexes of differential operators, Kluwer Academic Publishers, Dordrecht, 1995. MR 96k:58205
13.: D. Toledo and Y. L. Tong, Duality and intersection theory in complex manifolds. 2: The holomorphic Lefschetz formula, Ann. of Math. 108 (1978), 519-538. MR 80j:32045
14.: A. K. Tsikh, Multidimensional residues and their applications, Amer. Math. Soc., Providence, RI, 1992. MR 93g:32005

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