The geometric complex for algebraic curves with cone-like singularities and admissible Morse functions

Ursula Ludwig[1]

  • [1] Universität Freiburg Mathematisches Institut Eckerstrasse 1 79104 Freiburg (Allemagne)

Annales de l’institut Fourier (2010)

  • Volume: 60, Issue: 5, page 1533-1560
  • ISSN: 0373-0956

Abstract

top
In a previous note the author gave a generalisation of Witten’s proof of the Morse inequalities to the model of a complex singular curve X and a stratified Morse function f . In this note a geometric interpretation of the complex of eigenforms of the Witten Laplacian corresponding to small eigenvalues is provided in terms of an appropriate subcomplex of the complex of unstable cells of critical points of f .

How to cite

top

Ludwig, Ursula. "The geometric complex for algebraic curves with cone-like singularities and admissible Morse functions." Annales de l’institut Fourier 60.5 (2010): 1533-1560. <http://eudml.org/doc/116313>.

@article{Ludwig2010,
abstract = {In a previous note the author gave a generalisation of Witten’s proof of the Morse inequalities to the model of a complex singular curve $X$ and a stratified Morse function $f$. In this note a geometric interpretation of the complex of eigenforms of the Witten Laplacian corresponding to small eigenvalues is provided in terms of an appropriate subcomplex of the complex of unstable cells of critical points of $f$.},
affiliation = {Universität Freiburg Mathematisches Institut Eckerstrasse 1 79104 Freiburg (Allemagne)},
author = {Ludwig, Ursula},
journal = {Annales de l’institut Fourier},
keywords = {Morse theory; Witten deformation; Cone-like Singularities; cone-like singularities},
language = {eng},
number = {5},
pages = {1533-1560},
publisher = {Association des Annales de l’institut Fourier},
title = {The geometric complex for algebraic curves with cone-like singularities and admissible Morse functions},
url = {http://eudml.org/doc/116313},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Ludwig, Ursula
TI - The geometric complex for algebraic curves with cone-like singularities and admissible Morse functions
JO - Annales de l’institut Fourier
PY - 2010
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 5
SP - 1533
EP - 1560
AB - In a previous note the author gave a generalisation of Witten’s proof of the Morse inequalities to the model of a complex singular curve $X$ and a stratified Morse function $f$. In this note a geometric interpretation of the complex of eigenforms of the Witten Laplacian corresponding to small eigenvalues is provided in terms of an appropriate subcomplex of the complex of unstable cells of critical points of $f$.
LA - eng
KW - Morse theory; Witten deformation; Cone-like Singularities; cone-like singularities
UR - http://eudml.org/doc/116313
ER -

References

top
  1. J.-M. Bismut, W. Zhang, Milnor and Ray-Singer metrics on the equivariant determinant of a flat vector bundle, Geom. Funct. Anal. 4 (1994), 136-212 Zbl0830.58030MR1262703
  2. Jean-Michel Bismut, Gilles Lebeau, Complex immersions and Quillen metrics., Publ.Math.Inst.Hautes Etud.Sci. 74 (1991), 1-197 Zbl0784.32010MR1188532
  3. Jean-Michel Bismut, Weiping Zhang, An extension of a theorem by Cheeger and Müller. With an appendix by François Laudenbach., (1992), Astérisque. 205. Paris. Zbl0781.58039MR1185803
  4. J. Brüning, M. Lesch, Hilbert complexes, J. Funct. Anal. 108 (1992), 88-132 Zbl0826.46065MR1174159
  5. Jeff Cheeger, On the Hodge theory of Riemannian pseudomanifolds, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) (1980), 91-146, Amer. Math. Soc., Providence, R.I. Zbl0461.58002MR573430
  6. Mark Goresky, Robert MacPherson, Intersection homology theory., Topology 19 (1980), 135-165 Zbl0448.55004MR572580
  7. Mark Goresky, Robert MacPherson, Stratified Morse theory, 14 (1988), Springer-Verlag, Berlin Zbl0639.14012MR932724
  8. B. Helffer, J. Sjöstrand, Puits multiples en mécanique semi-classique. IV. Étude du complexe de Witten, Comm. Partial Differential Equations 10 (1985), 245-340 Zbl0597.35024MR780068
  9. François Laudenbach, Appendix: On the Thom-Smale complex, (1992), Astérisque. 205. Paris: Société Mathématique de France MR1185803
  10. Ursula Ludwig, The Witten complex for spaces of dimension two with cone-like singularities Zbl1215.58004
  11. Ursula Ludwig, The geometric complex for algebraic curves with cone-like singularities and admissible Morse functions., C. R., Math., Acad. Sci. Paris 347 (2009), 801-804 Zbl1168.58016MR2543986
  12. Ursula Ludwig, The Witten complex for algebraic curves with cone-like singularities., C. R., Math., Acad. Sci. Paris 347 (2009), 651-654 Zbl1166.32015MR2532924
  13. Masayoshi Nagase, Hodge theory of singular algebraic curves., Proc. Am. Math. Soc. 108 (1990), 1095-1101 Zbl0686.58002MR1002162
  14. Jacob Palis, Welington de Melo, Geometric theory of dynamical systems, (1982), Springer-Verlag, New York Zbl0491.58001MR669541
  15. G.N. Watson, A treatise on the theory of Bessel functions. 2nd ed., (1966), London: Cambridge University Press. VII Zbl0174.36202MR1349110
  16. Edward Witten, Supersymmetry and Morse theory, J. Differential Geom. 17 (1982), 661-692 Zbl0499.53056MR683171

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.