# ${L}^{2}$-estimates for the $d$-equation and Witten’s proof of the Morse inequalities

Bo Berndtsson^{[1]}

- [1] Department of Mathematics, Chalmers University of Technology and the University of Göteborg, S-412 96 Göteborg, Sweden

Annales de la faculté des sciences de Toulouse Mathématiques (2007)

- Volume: 16, Issue: 4, page 773-797
- ISSN: 0240-2963

## Access Full Article

top## Abstract

top## How to cite

topBerndtsson, Bo. "$L^2$-estimates for the $d$-equation and Witten’s proof of the Morse inequalities." Annales de la faculté des sciences de Toulouse Mathématiques 16.4 (2007): 773-797. <http://eudml.org/doc/10069>.

@article{Berndtsson2007,

abstract = {This is an introduction to Witten’s analytic proof of the Morse inequalities. The text is directed primarily to readers whose main interest is in complex analysis, and the similarities to Hörmander’s $L^2$-estimates for the $\bar\{\partial \}$-equation is used as motivation. We also use the method to prove $L^2$-estimates for the $d$-equation with a weight $e^\{-t\phi \}$ where $\phi $ is a nondegenerate Morse function.},

affiliation = {Department of Mathematics, Chalmers University of Technology and the University of Göteborg, S-412 96 Göteborg, Sweden},

author = {Berndtsson, Bo},

journal = {Annales de la faculté des sciences de Toulouse Mathématiques},

keywords = {-estimates; Morse inequalities},

language = {eng},

number = {4},

pages = {773-797},

publisher = {Université Paul Sabatier, Toulouse},

title = {$L^2$-estimates for the $d$-equation and Witten’s proof of the Morse inequalities},

url = {http://eudml.org/doc/10069},

volume = {16},

year = {2007},

}

TY - JOUR

AU - Berndtsson, Bo

TI - $L^2$-estimates for the $d$-equation and Witten’s proof of the Morse inequalities

JO - Annales de la faculté des sciences de Toulouse Mathématiques

PY - 2007

PB - Université Paul Sabatier, Toulouse

VL - 16

IS - 4

SP - 773

EP - 797

AB - This is an introduction to Witten’s analytic proof of the Morse inequalities. The text is directed primarily to readers whose main interest is in complex analysis, and the similarities to Hörmander’s $L^2$-estimates for the $\bar{\partial }$-equation is used as motivation. We also use the method to prove $L^2$-estimates for the $d$-equation with a weight $e^{-t\phi }$ where $\phi $ is a nondegenerate Morse function.

LA - eng

KW - -estimates; Morse inequalities

UR - http://eudml.org/doc/10069

ER -

## References

top- Berndtsson (B.).— Bergman kernels related to Hermitian line bundles over compact complex manifolds, Contemporary Mathematics, AMS, Volume 332, 2003. Zbl1038.32003MR2016088
- Berman (R.).— Bergman kernels and local holomorphic Morse inequalities, Math. Z. 248, no. 2, p. 325–344 (2004). Zbl1066.32002MR2088931
- Brascamp (H.J.), Lieb (E.H.).— On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation., J. Functional Analysis 22, no. 4, p.366–389 (1976). Zbl0334.26009MR450480
- Demailly (J.-P.).— Holomorphic Morse inequalities., Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, p.93–114 (1989). Zbl0755.32008MR1128538
- Helffer (B.).— Semi-classical analysis for the Schrödinger operator and applications., Lecture Notes in Mathematics (1336). Springer-Verlag, Berlin (1988). vi+107 pp. ISBN 3-540-50076-6. Zbl0647.35002MR960278
- Hörmander (L.).— An introduction to complex analysis in several variables. Third edition., North-Holland Mathematical Library, 7. North-Holland Publishing Co., Amsterdam-New York, 1990. Zbl0271.32001MR1045639
- Warner (F.).— Foundations of differentiable manifolds and Lie groups, pringer-Verlag, New York-Berlin, 1983. Zbl0516.58001MR722297
- Witten (E.).— Supersymmetry and Morse theory., J. Differential Geom. 17 (1982), no. 4, p. 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.