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

Ursula Ludwig^{[1]}

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

Annales de l’institut Fourier (2011)

- Volume: 61, Issue: 5, page 1749-1777
- ISSN: 0373-0956

## Access Full Article

top## Abstract

top## How to cite

topLudwig, Ursula. "A proof of the stratified Morse inequalities for singular complex algebraic curves using the Witten deformation." Annales de l’institut Fourier 61.5 (2011): 1749-1777. <http://eudml.org/doc/219758>.

@article{Ludwig2011,

abstract = {The Witten deformation is an analytic method proposed by Witten which, given a Morse function $f : M \rightarrow \bf R$ 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.},

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 = {1749-1777},

publisher = {Association des Annales de l’institut Fourier},

title = {A proof of the stratified Morse inequalities for singular complex algebraic curves using the Witten deformation},

url = {http://eudml.org/doc/219758},

volume = {61},

year = {2011},

}

TY - JOUR

AU - Ludwig, Ursula

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

JO - Annales de l’institut Fourier

PY - 2011

PB - Association des Annales de l’institut Fourier

VL - 61

IS - 5

SP - 1749

EP - 1777

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

LA - eng

KW - Morse theory; Witten deformation; Cone-like Singularities; cone-like singularities

UR - http://eudml.org/doc/219758

ER -

## References

top- Shmuel Agmon, Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of $N$-body Schrödinger operators, 29 (1982), Princeton University Press, Princeton, NJ Zbl0503.35001MR745286
- Jean-Michel Bismut, Gilles Lebeau, Complex immersions and Quillen metrics, Publ. Math. Inst. Hautes Étud. Sci. 74 (1991), 1-197 Zbl0784.32010MR1188532
- Jean-Michel Bismut, Weiping Zhang, An extension of a theorem by Cheeger and Müller, Astérisque (1992) Zbl0781.58039MR1185803
- Jochen Brüning, ${L}^{2}$-index theorems on certain complete manifolds, J. Differential Geom. 32 (1990), 491-532 Zbl0722.58043MR1072916
- Jochen Brüning, Matthias Lesch, Hilbert complexes, J. Funct. Anal. 108 (1992), 88-132 Zbl0826.46065MR1174159
- Jochen Brüning, Matthias Lesch, Kähler-Hodge theory for conformal complex cones, Geom. Funct. Anal. 3 (1993), 439-473 Zbl0795.58003MR1233862
- Jochen Brüning, Matthias Lesch, On the spectral geometry of algebraic curves, J. Reine Angew. Math. 474 (1996), 25-66 Zbl0846.14018MR1390691
- Jochen Brüning, Norbert Peyerimhoff, Herbert Schröder, The $\overline{\partial}$-operator on algebraic curves, Comm. Math. Phys. 129 (1990), 525-534 Zbl0708.32007MR1051503
- Jochen Brüning, Robert Seeley, An index theorem for first order regular singular operators, Amer. J. Math. 110 (1988), 659-714 Zbl0664.58035MR955293
- 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
- Mark Goresky, Robert MacPherson, Morse theory and intersection homology theory, Analysis and topology on singular spaces, II, III (Luminy, 1981) 101 (1983), 135-192, Soc. Math. France, Paris Zbl0524.57022MR737930
- Mark Goresky, Robert MacPherson, Stratified Morse theory, 14 (1988), Springer-Verlag, Berlin Zbl0639.14012MR932724
- Daniel Grieser, Matthias Lesch, On the ${L}^{2}$-Stokes theorem and Hodge theory for singular algebraic varieties, Math. Nachr. 246-247 (2002), 68-82 Zbl1034.58019MR1944550
- B. Helffer, Semi-classical analysis for the Schrödinger operator and applications, 1336 (1988), Springer-Verlag, Berlin Zbl0647.35002MR960278
- 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
- Ursula Ludwig, The Witten complex for singular spaces of dimension 2 with cone-like singularities Zbl1215.58004
- Ursula Ludwig, The Witten deformation for conformal cones Zbl06154651
- Ursula Ludwig, The geometric complex for algebraic curves with cone-like singularities and admissible Morse function, (2010) Zbl1207.58014MR2766222
- Masayoshi Nagase, Gauss-Bonnet operator on singular algebraic curves, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 39 (1992), 77-86 Zbl0765.58031MR1157978
- William Pardon, Mark Stern, Pure Hodge structure on the ${L}_{2}$-cohomology of varieties with isolated singularities, J. Reine Angew. Math. 533 (2001), 55-80 Zbl0960.14009MR1823864
- Edward Witten, Supersymmetry and Morse theory, J. Differential Geom. 17 (1982), 661-692 (1983) Zbl0499.53056MR683171

## NotesEmbed ?

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