Durfee's conjecture on the signature of smoothings of surface singularities

Kollár, János; Némethi, András

doi:10.24033/asens.2332

Durfee's conjecture on the signature of smoothings of surface singularities
[La conjecture de Durfee sur la signature des lissages des singularités des surfaces (avec un appendice de Tommaso de Fernex)]

Kollár, János ; Némethi, András

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 3, pp. 787-798.

Résumé
Abstract

En 1978 Durfee a conjecturé plusieurs inégalités entre la signature $σ$ et le genre géométrique $p_{g}$ d'une singularité normale de surface. Depuis, quelques contre-exemples ont été trouvés et des résultats positifs établis dans des cas particuliers.

Nous montrons ici une inégalité `forte' de type Durfee pour toute lissification d'une singularité de Gorenstein, sous la condition que la forme d'intersection de la résolution est unimodulaire. Nous prouvons aussi l'inégalité `faible' pour toute singularité d'hypersurface et pour les intersections complètes strictes de multiplicité suffisamment grande. Les preuves établissent des inégalités générales valables pour toute singularité normale et numériquement Gorenstein de surface.

In 1978 Durfee conjectured various inequalities between the signature $σ$ and the geometric genus $p_{g}$ of a normal surface singularity. Since then a few counter examples have been found and positive results established in some special cases.

We prove a `strong' Durfee-type inequality for any smoothing of a Gorenstein singularity, provided that the intersection form of the resolution is unimodular. We also prove the conjectured `weak' inequality for all hypersurface singularities and for sufficiently large multiplicity strict complete intersections. The proofs establish general inequalities valid for any numerically Gorenstein normal surface singularity.

MR Zbl

DOI : 10.24033/asens.2332

Classification : 32S05, 32S25, 15A63; 14Bxx, 32Sxx.
Keywords: Durfee's Conjecture, surface singularities, signature of smoothing, geometric genus, resolution, unimodular intersection form.
Mot clés : Conjecture de Durfee, singularités des surfaces, signature des lissages, genre géométrique, résolution, forme d'intersection unimodulaire.

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     author = {Koll\'ar, J\'anos and N\'emethi, Andr\'as},
     title = {Durfee's conjecture on the signature of smoothings of surface singularities},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {787--798},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
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Kollár, János; Némethi, András. Durfee's conjecture on the signature of smoothings of surface singularities. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 3, pp. 787-798. doi : 10.24033/asens.2332. http://www.numdam.org/articles/10.24033/asens.2332/

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