Constructing local models for Lagrangian torus fibrations

Evans, Jonathan David; Mauri, Mirko

doi:10.5802/ahl.80

Evans, Jonathan David ¹; Mauri, Mirko ²

¹ Department of Mathematics and Statistics, University of Lancaster, Bailrigg, LA1 4YW (United Kingdom)
² Max Planck Institute for Mathematics, Vivatgasse 7, office 407, 53111, Bonn (Germany)

Annales Henri Lebesgue, Volume 4 (2021), pp. 537-570.

Abstract
Résumé

We give a construction of Lagrangian torus fibrations with controlled discriminant locus on certain affine varieties. In particular, we apply our construction in the following ways:

We find a Lagrangian torus fibration on the 3-fold negative vertex whose discriminant locus has codimension 2; this provides a local model for finding torus fibrations on compact Calabi–Yau 3-folds with codimension 2 discriminant locus.
We find a Lagrangian torus fibration on a neighbourhood of the one-dimensional stratum of a simple normal crossing divisor (satisfying certain conditions) such that the base of the fibration is an open subset of the cone over the dual complex of the divisor. This can be used to construct an analogue of the non-archimedean SYZ fibration constructed by Nicaise, Xu and Yu.

On donne une construction de fibrations en tores Lagrangiens avec discriminant fixé sur des variétés affines. En particulier, on applique cette construction dans les cas suivants :

on décrit une fibration en tores Lagrangiens avec discriminant de codimension 2 sur le « sommet négatif » , c’est-à-dire un modèle local de fibrations en tores sur les variétés de Calabi–Yau de dimension 3 ;
on construit une fibration en tores Lagrangiens sur un voisinage de la strate de dimension 1 d’un diviseur à croisements normaux simples (satisfaisant certaines conditions) tel que la base de la fibration est un ouvert dans le cône sur le complexe dual du diviseur. Ceci constitue un analogue symplectique de la fibration SYZ non-archimédienne construite par Nicaise, Xu et Yu.

Received: 2019-06-06
Revised: 2020-04-23
Accepted: 2020-07-27
Published online: 2021-05-27

DOI: 10.5802/ahl.80

Classification: 53D37, 53D35, 14J33, 14G22
Mots-clés : Lagrangian torus, SYZ fibration, dual complex, negative vertex

Author's affiliations:

Evans, Jonathan David ¹; Mauri, Mirko ²

¹ Department of Mathematics and Statistics, University of Lancaster, Bailrigg, LA1 4YW (United Kingdom)
² Max Planck Institute for Mathematics, Vivatgasse 7, office 407, 53111, Bonn (Germany)

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     author = {Evans, Jonathan David and Mauri, Mirko},
     title = {Constructing local models for {Lagrangian} torus fibrations},
     journal = {Annales Henri Lebesgue},
     pages = {537--570},
     publisher = {\'ENS Rennes},
     volume = {4},
     year = {2021},
     doi = {10.5802/ahl.80},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/ahl.80/}
}

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Evans, Jonathan David; Mauri, Mirko. Constructing local models for Lagrangian torus fibrations. Annales Henri Lebesgue, Volume 4 (2021), pp. 537-570. doi : 10.5802/ahl.80. http://www.numdam.org/articles/10.5802/ahl.80/

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