Semi-Local Exotic Lagrangian Tori in Dimension Four
[Constructions semi-locales de tores lagrangiens exotiques en dimension quatre]
Brendel, Joé1  ; Hauber, Johannes2  ; Schmitz, Joel2
1ETH Zürich, D-MATH, Rämistrasse 101, 8092 Zürich (Switzerland)
2Faculté des sciences, Institut de mathématiques, Université de Neuchâtel (Switzerland)
Annales de l'Institut Fourier, Online first, 48 p.

We study exotic Lagrangian tori in dimension four. In certain Stein domains $B_{dpq}$ (which naturally appear in almost toric fibrations) we find $d+1$ families of monotone Lagrangian tori which are mutually distinct, up to symplectomorphisms. We prove that these remain distinct under embeddings of $B_{dpq}$ into geometrically bounded symplectic four-manifolds. We show that there are infinitely many different such embeddings when $X$ is compact and (almost) toric and hence conclude that $X$ contains arbitrarily many Lagrangian tori which are distinct up to symplectomorphisms of $X$. In dimension four arbitrarily many different Lagrangian tori were previously known only in del Pezzo surfaces.

Neither the embedded tori, nor the ambient space $X$ needs to be monotone for our methods to work.

Nous étudions des tores lagrangiens exotiques en dimension quatre. Dans certains domaines de Stein $B_{dpq}$ (qui apparaissent naturellement dans les fibrations presque toriques), nous trouvons $d+1$ familles de tores lagrangiens monotones qui sont mutuellement distincts à symplectomorphismes près. Nous prouvons que ces familles restent distinctes sous plongements de $B_{dpq}$ dans des variétés symplectiques géométriquement bornés de dimension quatre. Nous montrons qu’il existe une infinité de tels plongements lorsque $X$ est compact et (presque) torique et concluons donc que $X$ contient un nombre infini de tores lagrangiens distincts à symplectomorphismes de $X$ près. En dimension quatre, ce résultat n’était connu que dans les surfaces del Pezzo, auparavant.

Ni les tores plongés, ni l’espace ambiant $X$ n’ont besoin d’être monotones pour que nos méthodes s’appliquent.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3789
Classification : 53D12, 53D20, 14J17
Keywords: Lagrangian submanifolds, Lagrangian knots, exotic Lagrangians, almost toric fibrations, displacement energy
Mots-clés : sous-variétés lagrangien, nœuds lagrangien, lagrangiens exotique, fibrations presque-torique, énergie de déplacement

Brendel, Joé  1   ; Hauber, Johannes  2   ; Schmitz, Joel  2

1 ETH Zürich, D-MATH, Rämistrasse 101, 8092 Zürich (Switzerland)
2 Faculté des sciences, Institut de mathématiques, Université de Neuchâtel (Switzerland) 
Brendel, Joé; Hauber, Johannes; Schmitz, Joel. Semi-Local Exotic Lagrangian Tori in Dimension Four. Annales de l'Institut Fourier, Online first, 48 p.
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