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, p. 773-797

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 ¯-equation is used as motivation. We also use the method to prove L 2 -estimates for the d-equation with a weight e -tφ where φ 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 L 2 pour l’équation ¯ de Hörmander servent de motivation. La méthode est aussi appliquée pour donner des estimées L 2 pour l’équation d à poids e -tφ , où φ est une fonction de Morse non dégénérée.

@article{AFST_2007_6_16_4_773_0,
     author = {Berndtsson, Bo},
     title = {$L^2$-estimates for the $d$-equation and Witten's proof of the Morse inequalities},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 16},
     number = {4},
     year = {2007},
     pages = {773-797},
     doi = {10.5802/afst.1166},
     zbl = {pre05363339},
     language = {en},
     url = {http://www.numdam.org/item/AFST_2007_6_16_4_773_0}
}
Berndtsson, 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, Serie 6, Volume 16 (2007) no. 4, pp. 773-797. doi : 10.5802/afst.1166. http://www.numdam.org/item/AFST_2007_6_16_4_773_0/

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