A proof of the stratified Morse inequalities for singular complex algebraic curves using  the Witten deformation

Ludwig, Ursula

doi:10.5802/aif.2657

A proof of the stratified Morse inequalities for singular complex algebraic curves using the Witten deformation

Ludwig, Ursula

Annales de l'Institut Fourier, Volume 61 (2011) no. 5, p. 1749-1777

Abstract
Résumé

The Witten deformation is an analytic method proposed by Witten which, given a Morse function $f : M \to R$ 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.

Soit $M$ une variété Riemannienne compacte et soit $f : M \to R$ 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.

MR 2961839 | Zbl 1242.32006

DOI : https://doi.org/10.5802/aif.2657
Classification: 58AXX, 58E05
Keywords: Morse theory, Witten deformation, Cone-like Singularities

@article{AIF_2011__61_5_1749_0,
     author = {Ludwig, Ursula},
     title = {A proof of the stratified Morse inequalities for singular complex algebraic curves using  the Witten deformation},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {5},
     year = {2011},
     pages = {1749-1777},
     doi = {10.5802/aif.2657},
     mrnumber = {2961839},
     zbl = {1242.32006},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2011__61_5_1749_0}
}

Ludwig, Ursula. A proof of the stratified Morse inequalities for singular complex algebraic curves using  the Witten deformation. Annales de l'Institut Fourier, Volume 61 (2011) no. 5, pp. 1749-1777. doi : 10.5802/aif.2657. http://www.numdam.org/item/AIF_2011__61_5_1749_0/

References

[1] Agmon, Shmuel Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of $N$ -body Schrödinger operators, Princeton University Press, Princeton, NJ, Mathematical Notes, Tome 29 (1982) | MR 745286 | Zbl 0503.35001

[2] Bismut, Jean-Michel; Lebeau, Gilles Complex immersions and Quillen metrics, Publ. Math. Inst. Hautes Étud. Sci., Tome 74 (1991), pp. 1-197 | Article | Numdam | MR 1188532 | Zbl 0784.32010

[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) | MR 1185803 | Zbl 0781.58039

[4] Brüning, Jochen $L^{2}$ -index theorems on certain complete manifolds, J. Differential Geom., Tome 32 (1990) no. 2, pp. 491-532 http://projecteuclid.org/getRecord?id=euclid.jdg/1214445317 | MR 1072916 | Zbl 0722.58043

[5] Brüning, Jochen; Lesch, Matthias Hilbert complexes, J. Funct. Anal., Tome 108 (1992) no. 1, pp. 88-132 | Article | MR 1174159 | Zbl 0826.46065

[6] Brüning, Jochen; Lesch, Matthias Kähler-Hodge theory for conformal complex cones, Geom. Funct. Anal., Tome 3 (1993) no. 5, pp. 439-473 | Article | MR 1233862 | Zbl 0795.58003

[7] Brüning, Jochen; Lesch, Matthias On the spectral geometry of algebraic curves, J. Reine Angew. Math., Tome 474 (1996), pp. 25-66 | MR 1390691 | Zbl 0846.14018

[8] Brüning, Jochen; Peyerimhoff, Norbert; Schröder, Herbert The $\bar{\partial}$ -operator on algebraic curves, Comm. Math. Phys., Tome 129 (1990) no. 3, pp. 525-534 http://projecteuclid.org/getRecord?id=euclid.cmp/1104180850 | Article | MR 1051503 | Zbl 0708.32007

[9] Brüning, Jochen; Seeley, Robert An index theorem for first order regular singular operators, Amer. J. Math., Tome 110 (1988) no. 4, pp. 659-714 | Article | MR 955293 | Zbl 0664.58035

[10] Cheeger, Jeff On the Hodge theory of Riemannian pseudomanifolds, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Amer. Math. Soc., Providence, R.I. (Proc. Sympos. Pure Math., XXXVI) (1980), pp. 91-146 | MR 573430 | Zbl 0461.58002

[11] Goresky, Mark; Macpherson, Robert Morse theory and intersection homology theory, Analysis and topology on singular spaces, II, III (Luminy, 1981), Soc. Math. France, Paris (Astérisque) Tome 101 (1983), pp. 135-192 | MR 737930 | Zbl 0524.57022

[12] Goresky, Mark; Macpherson, Robert Stratified Morse theory, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Tome 14 (1988) | MR 932724 | Zbl 0639.14012

[13] Grieser, Daniel; Lesch, Matthias On the $L^{2}$ -Stokes theorem and Hodge theory for singular algebraic varieties, Math. Nachr., Tome 246-247 (2002), pp. 68-82 | Article | MR 1944550 | Zbl 1034.58019

[14] Helffer, B. Semi-classical analysis for the Schrödinger operator and applications, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1336 (1988) | MR 960278 | Zbl 0647.35002

[15] Helffer, B.; Sjöstrand, J. Puits multiples en mécanique semi-classique. IV. Étude du complexe de Witten, Comm. Partial Differential Equations, Tome 10 (1985) no. 3, pp. 245-340 | Article | MR 780068 | Zbl 0597.35024

[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 1215.58004

[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 2766222 | Zbl 1207.58014

[19] Nagase, Masayoshi Gauss-Bonnet operator on singular algebraic curves, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Tome 39 (1992) no. 1, pp. 77-86 | MR 1157978 | Zbl 0765.58031

[20] Pardon, William; Stern, Mark Pure Hodge structure on the $L_{2}$ -cohomology of varieties with isolated singularities, J. Reine Angew. Math., Tome 533 (2001), pp. 55-80 | Article | MR 1823864 | Zbl 0960.14009

[21] Witten, Edward Supersymmetry and Morse theory, J. Differential Geom., Tome 17 (1982) no. 4, p. 661-692 (1983) | MR 683171 | Zbl 0499.53056