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 -estimates for the -equation is used as motivation. We also use the method to prove -estimates for the -equation with a weight where is a nondegenerate Morse function.
On donne une introduction à la preuve analytique de E. Witten des inégalités de Morse. Le texte s’adresse principalement aux lecteurs spécialistes en analyse complexe, et les similarités avec les estimées pour l’équation de Hörmander servent de motivation. La méthode est aussi appliquée pour donner des estimées pour l’équation à poids , où est une fonction de Morse non dégénérée.
@article{AFST_2007_6_16_4_773_0,
author = {Bo Berndtsson},
title = {$L^2$-estimates for the $d$-equation and {Witten{\textquoteright}s} proof of the {Morse} inequalities},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {773--797},
publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
address = {Toulouse},
volume = {Ser. 6, 16},
number = {4},
year = {2007},
doi = {10.5802/afst.1166},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1166/}
}
TY - JOUR AU - Bo Berndtsson 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 SP - 773 EP - 797 VL - 16 IS - 4 PB - Université Paul Sabatier, Institut de Mathématiques PP - Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1166/ DO - 10.5802/afst.1166 LA - en ID - AFST_2007_6_16_4_773_0 ER -
%0 Journal Article %A Bo Berndtsson %T $L^2$-estimates for the $d$-equation and Witten’s proof of the Morse inequalities %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2007 %P 773-797 %V 16 %N 4 %I Université Paul Sabatier, Institut de Mathématiques %C Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1166/ %R 10.5802/afst.1166 %G en %F AFST_2007_6_16_4_773_0
Bo Berndtsson. $L^2$-estimates for the $d$-equation and Witten’s proof of the Morse inequalities. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 16 (2007) no. 4, pp. 773-797. doi : 10.5802/afst.1166. https://afst.centre-mersenne.org/articles/10.5802/afst.1166/
[1] Berndtsson (B.).— Bergman kernels related to Hermitian line bundles over compact complex manifolds, Contemporary Mathematics, AMS, Volume 332, 2003. | MR | Zbl
[2] Berman (R.).— Bergman kernels and local holomorphic Morse inequalities, Math. Z. 248, no. 2, p. 325–344 (2004). | MR | Zbl
[3] 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). | MR | Zbl
[4] Demailly (J.-P.).— Holomorphic Morse inequalities., Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, p.93–114 (1989). | MR | Zbl
[5] 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. | MR | Zbl
[6] 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. | MR | Zbl
[7] Warner (F.).— Foundations of differentiable manifolds and Lie groups, pringer-Verlag, New York-Berlin, 1983. | MR | Zbl
[8] Witten (E.).— Supersymmetry and Morse theory., J. Differential Geom. 17 (1982), no. 4, p. 661–692 (1983). | MR | Zbl
Cited by Sources: