Abstract
The semiclassical method in Lefschetz theory is presented and applied to the computation of Lefschetz numbers of endomorphisms of elliptic complexes on manifolds with singularities. Two distinct cases are considered, one in which the endomorphism is geometric and the other in which the endomorphism is specified by Fourier integral operators associated with a canonical transformation. In the latter case, the problem includes a small parameter and the formulas are (semiclassically) asymptotic. In the first case, the parameter is introduced artificially and the semiclassical method gives exact answers. In both cases, the Lefschetz number is the sum of contributions of interior fixed points given (in the case of geometric endomorphisms) by standard formulas plus the contribution of fixed singular points. The latter is expressed as a sum of residues in the lower or upper half-plane of a meromorphic operator expression constructed from the conormal symbols of the operators involved in the problem.
A preliminary version of the paper was published as a preprint at Chalmers University of Technology and supported by a grant from the Swedish Royal Academy of Sciences. We also acknowledge support from the RFBR under grants Nos.05-01-00466 and 05-01-00982.
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References
M.F. Atiyah and R. Bott, A Lefschetz Fixed Point Formula for Elliptic Complexes I. Ann. of Math. 86 (1967), 374–407.
S. Lefschetz, L’analysis situs et la géométrie algébrique. Gauthier-Villars, Paris, 1924.
V. Nazaikinskii, B.-W. Schulze, B. Sternin, and V. Shatalov, The Atiyah-Bott-Lefschetz Fixed Point Theorem for Manifolds with Conical Singularities. Ann. Global Anal. Geom. 17:5 (1999), 409–439.
V. Nazaikinskii, B.-W. Schulze, and B. Sternin, A Semiclassical Quantization on Manifolds with Singularities and the Lefschetz Formula for Elliptic Operators. Preprint No. 98/19, Univ. Potsdam, Institut für Mathematik, Potsdam, 1998.
V. Nazaikinskii, Semiclassical Lefschetz Formulas on Smooth and Singular Manifolds. Russ. J. Math. Phys. 6:2 (1999), 202–213.
B. Sternin and V. Shatalov, Atiyah-Bott-Lefschetz Fixed Point Theorem in Symplectic Geometry. Dokl. Akad. Nauk 348:2 (1996), 165–168.
B. Sternin and V. Shatalov, The Fixed Point Lefschetz Theorem for Quantized Canonical Transformations. Funktsional. Anal. i Prilozhen. 32:4 (1998), 35–48.
B. Sternin and V. Shatalov, Quantization of Symplectic Transformations and the Lefschetz Fixed Point Theorem. Preprint No. 92, Max-Planck-Institut für Mathematik, Bonn, 1994.
V.P. Maslov, Perturbation Theory and Asymptotic Methods. Moscow State University, Moscow, 1965.
V.P. Maslov, Operator Methods. Nauka, Moscow, 1973.
A. Mishchenko, V. Shatalov, and B. Sternin, Lagrangian Manifolds and the Maslov Operator. Springer-Verlag, Berlin-Heidelberg, 1990.
V. Nazaikinskii, B. Sternin, and V. Shatalov, Contact Geometry and Linear Differential Equations. Walter de Gruyter, Berlin-New York, 1992.
V. Nazaikinskii, B. Sternin, and V. Shatalov, Methods of Noncommutative Analysis. Theory and Applications. Walter de Gruyter, Berlin-New York, 1995.
B.V. Fedosov, Trace Formula for Schrödinger Operator. Russ. J. Math. Phys. 1:4 (1993), 447–463.
B. Sternin and V. Shatalov, Ten Lectures on Quantization. Preprint No. 22, Universit é de Nice-Sophia Antipolis, Nice, 1994.
B.-W. Schulze, Elliptic Complexes on Manifolds with Conical Singularities. In: Seminar Analysis of the Karl Weierstraß Institute 1986/87, number 106 in Teubner Texte zur Mathematik, Teubner, Leipzig, 1988. Pages 170–223.
B.-W. Schulze, Pseudodifferential Operators on Manifolds with Singularities. North-Holland, Amsterdam, 1991.
B.A. Plamenevskii, Algebras of Pseudodifferantial Operators. Kluwer, Dordrecht, 1989.
B.-W. Schulze, B. Sternin, and V. Shatalov, Differential Equations on Singular Manifolds. Semiclassical Theory and Operator Algebras. Wiley-VCH, Berlin-New York, 1998.
M. Agranovich and M. Vishik, Elliptic Problems with Parameter and Parabolic Problems of General Type. Uspekhi Mat. Nauk 19:3 (1964), 53–161.
M.A. Shubin, Pseudodifferential Operators and Spectral Theory. Springer-Verlag, Berlin-Heidelberg, 1985.
M.S. Agranovich, B.Z. Katsenelenbaum, A.N. Sivov, and N.N. Voitovich, Generalized Method of Eigenoscillations in Diffraction Theory. Wiley-VCH, Berlin, 1999.
A. Martinez, Microlocal Exponential Estimates and Applications to Tunneling. In: Microlocal Analysis and Spectral Theory, Kluwer, 1997. Pages 349–376.
L. Hörmander, The Spectral Function of an Elliptic Operator. Acta Math. 121 (1968), 193–218.
R. Melrose, Transformation of Boundary Problems. Acta Math. 147 (1981), 149–236.
V. Nazaikinskii, B.-W. Schulze, B. Sternin, and V. Shatalov, Quantization of Symplectic Transformations on Manifolds with Conical Singularities. Preprint No. 97/23, Univ. Potsdam, Institut für Mathematik, Potsdam, 1997.
V. Nazaikinskii, B.-W. Schulze, B. Sternin, and V. Shatalov, A Lefschetz Fixed Point Theorem on Manifolds with Conical Singularities. Preprint No. 97/20, Univ. Potsdam, Institut für Mathematik, Potsdam, 1997.
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Nazaikinskii, V., Sternin, B. (2006). Lefschetz Theory on Manifolds with Singularities. In: Bojarski, B., Mishchenko, A.S., Troitsky, E.V., Weber, A. (eds) C*-algebras and Elliptic Theory. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7687-1_8
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