A proof of the stratified Morse inequalities for singular complex algebraic curves using the Witten deformation
[Une preuve des inegalités de Morse pour les courbes algébriques singulières utilisant la déformation de Witten]
Annales de l'Institut Fourier, Tome 61 (2011) no. 5, pp. 1749-1777.

Soit M une variété Riemannienne compacte et soit f:MR une fonction de Morse sur M. La méthode de Witten utilise une déformation du complexe de de Rham pour démontrer les inegalités de Morse. Le but de cette note est d’étendre cette méthode au cas des courbes algébriques singulières et aux fonctions de Morse stratifiées au sens de la théorie de Goresky/MacPherson.

Dans une note précédente, l’auteur a donné une généralisation de la méthode de Witten pour le cas modèle d’une courbe à singularités coniques et des fonctions de Morse admissibles. Ici on présente les méthodes et arguments nécessaires pour étendre la théorie au courbes équipées de la métrique induite par la métrique de Fubini-Study de l’espace ambiant et à toutes les fonctions de Morse stratifiées.

The Witten deformation is an analytic method proposed by Witten which, given a Morse function f:MR on a smooth compact manifold M, allows to prove the Morse inequalities. The aim of this article is to generalise the Witten deformation to stratified Morse functions (in the sense of stratified Morse theory as developed by Goresky and MacPherson) on a singular complex algebraic curve. In a previous article the author developed the Witten deformation for the model of an algebraic curve with cone-like singularities and a certain class of functions called admissible Morse functions. The perturbation arguments needed to understand the Witten deformation on the curve with its metric induced from the Fubini-Study metric of the ambient projective space and for any stratified Morse function are presented here.

DOI : 10.5802/aif.2657
Classification : 58AXX, 58E05
Keywords: Morse theory, Witten deformation, Cone-like Singularities
Mot clés : théorie de Morse, déformation de Witten, singularités coniques
Ludwig, Ursula 1

1 Universität Freiburg Mathematisches Institut Eckerstrasse 1 79104 Freiburg (Allemagne)
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Ludwig, Ursula. A proof of the stratified Morse inequalities for singular complex algebraic curves using  the Witten deformation. Annales de l'Institut Fourier, Tome 61 (2011) no. 5, pp. 1749-1777. doi : 10.5802/aif.2657. http://www.numdam.org/articles/10.5802/aif.2657/

[1] Agmon, Shmuel Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N-body Schrödinger operators, Mathematical Notes, 29, Princeton University Press, Princeton, NJ, 1982 | MR | Zbl

[2] Bismut, Jean-Michel; Lebeau, Gilles Complex immersions and Quillen metrics, Publ. Math. Inst. Hautes Étud. Sci., Volume 74 (1991), pp. 1-197 | DOI | EuDML | Numdam | MR | Zbl

[3] Bismut, Jean-Michel; Zhang, Weiping An extension of a theorem by Cheeger and Müller, Astérisque (1992) no. 205, pp. 235 (With an appendix by François Laudenbach) | Numdam | MR | Zbl

[4] Brüning, Jochen L 2 -index theorems on certain complete manifolds, J. Differential Geom., Volume 32 (1990) no. 2, pp. 491-532 http://projecteuclid.org/getRecord?id=euclid.jdg/1214445317 | MR | Zbl

[5] Brüning, Jochen; Lesch, Matthias Hilbert complexes, J. Funct. Anal., Volume 108 (1992) no. 1, pp. 88-132 | DOI | MR | Zbl

[6] Brüning, Jochen; Lesch, Matthias Kähler-Hodge theory for conformal complex cones, Geom. Funct. Anal., Volume 3 (1993) no. 5, pp. 439-473 | DOI | MR | Zbl

[7] Brüning, Jochen; Lesch, Matthias On the spectral geometry of algebraic curves, J. Reine Angew. Math., Volume 474 (1996), pp. 25-66 | MR | Zbl

[8] Brüning, Jochen; Peyerimhoff, Norbert; Schröder, Herbert The ¯-operator on algebraic curves, Comm. Math. Phys., Volume 129 (1990) no. 3, pp. 525-534 http://projecteuclid.org/getRecord?id=euclid.cmp/1104180850 | DOI | MR | Zbl

[9] Brüning, Jochen; Seeley, Robert An index theorem for first order regular singular operators, Amer. J. Math., Volume 110 (1988) no. 4, pp. 659-714 | DOI | MR | Zbl

[10] Cheeger, Jeff On the Hodge theory of Riemannian pseudomanifolds, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) (Proc. Sympos. Pure Math., XXXVI), Amer. Math. Soc., Providence, R.I., 1980, pp. 91-146 | MR | Zbl

[11] Goresky, Mark; MacPherson, Robert Morse theory and intersection homology theory, Analysis and topology on singular spaces, II, III (Luminy, 1981) (Astérisque), Volume 101, Soc. Math. France, Paris, 1983, pp. 135-192 | Numdam | MR | Zbl

[12] Goresky, Mark; MacPherson, Robert Stratified Morse theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 14, Springer-Verlag, Berlin, 1988 | MR | Zbl

[13] Grieser, Daniel; Lesch, Matthias On the L 2 -Stokes theorem and Hodge theory for singular algebraic varieties, Math. Nachr., Volume 246-247 (2002), pp. 68-82 | DOI | MR | Zbl

[14] Helffer, B. Semi-classical analysis for the Schrödinger operator and applications, Lecture Notes in Mathematics, 1336, Springer-Verlag, Berlin, 1988 | MR | Zbl

[15] Helffer, B.; Sjöstrand, J. Puits multiples en mécanique semi-classique. IV. Étude du complexe de Witten, Comm. Partial Differential Equations, Volume 10 (1985) no. 3, pp. 245-340 | DOI | MR | Zbl

[16] Ludwig, Ursula The Witten complex for singular spaces of dimension 2 with cone-like singularities accepted at Mathematische Nachrichten, see also C. R., Math., Acad. Sci. Paris, 347 (11-12):651-654 | Zbl

[17] Ludwig, Ursula The Witten deformation for conformal cones (in preparation)

[18] Ludwig, Ursula The geometric complex for algebraic curves with cone-like singularities and admissible Morse function (2010) (to appear in Ann. Inst. Fourier) | Numdam | MR | Zbl

[19] Nagase, Masayoshi Gauss-Bonnet operator on singular algebraic curves, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Volume 39 (1992) no. 1, pp. 77-86 | MR | Zbl

[20] Pardon, William; Stern, Mark Pure Hodge structure on the L 2 -cohomology of varieties with isolated singularities, J. Reine Angew. Math., Volume 533 (2001), pp. 55-80 | DOI | MR | Zbl

[21] Witten, Edward Supersymmetry and Morse theory, J. Differential Geom., Volume 17 (1982) no. 4, p. 661-692 (1983) | MR | Zbl

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