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The geometric complex for algebraic curves with cone-like singularities and admissible Morse functions. (English) Zbl 1168.58016

In the present paper it is given a geometric interpretation of the complex of eigenforms of the Witten Laplace operator corresponding to small eigenvalues. The proof relies on the analysis of an appropriate subcomplex of the complex of unstable cells for the critical points of the admissible Morse function.

MSC:

58K45 Singularities of vector fields, topological aspects
32S65 Singularities of holomorphic vector fields and foliations
53A04 Curves in Euclidean and related spaces
58J50 Spectral problems; spectral geometry; scattering theory on manifolds

References:

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[2] Goresky, M.; MacPherson, R., Stratified Morse Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 14 (1988), Springer-Verlag: Springer-Verlag Berlin · Zbl 0639.14012
[3] Helffer, B.; Sjöstrand, J., Puits multiples en mécanique semi-classique. IV : Étude du complexe de Witten, Comm. Partial Differential Equations, 10, 3, 245-340 (1985) · Zbl 0597.35024
[4] F. Laudenbach, Appendix: On the Thom-Smale complex, Astérisque 205 (1992); F. Laudenbach, Appendix: On the Thom-Smale complex, Astérisque 205 (1992)
[5] Ludwig, U., The Witten complex for algebraic curves with cone-like singularities, C. R. Acad. Sci. Paris, Ser. I, 347, 11-12, 651-654 (2009) · Zbl 1166.32015
[6] Witten, E., Supersymmetry and Morse theory, J. Differential Geom., 17, 4, 661-692 (1982) · Zbl 0499.53056
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