Legendrian submanifolds from Bohr-Sommerfeld covers of monotone Lagrangian tori (Q6598520)

Prequantisation \(\mathbb{S}^{1}\)-bundles \(\pi : E\longrightarrow M^{2n}\) form an important class of contact manifolds \((E^{2n+1},\alpha)\) that have been well studied from many different points of view. By definition, the contact form \(\alpha\) is a connection one-form for the \(\mathbb{S}^{1}\)-bundle, and the curvature is a symplectic two-form \(\omega\) on \(M^{2n}.\) For that reason, there is a close relationship between the symplectic geometry of \((M^{2n},\omega)\) and the contact geometry of \((E, \ker\alpha).\) For instance, every Legendrian immersion inside \(E\) projects to a Lagrangian immersion inside \((M^{2n},\omega)\) which satisfies the Bohr-Sommerfeld condition.The connection between Bohr-Sommerfeld Lagrangian immersions in the projective plane and Legendrians in the standard contact sphere was also studied in recent work by \textit{S. Baldridge} et al. [Open Book Ser. 5, 43--79 (2022; Zbl 1526.57021)]. It is a standard fact that any prequantisation bundle admits symplectic fillings (which need not be exact). In the case of a complex line bundle over a Kähler manifold, this is the standard fact that the unit disc-bundle of any negative complex line bundle has a strictly pseudoconvex boundary; its boundary is a prequantisation bundle. A similar construction works in the symplectic case as well. \textit{K. Mohnke} [Ann. Math. (2) 154, No. 1, 219--222 (2001; Zbl 1007.53065)] has shown that any Legendrian in the boundary of a subcritically fillable contact manifold admits a Reeb chord. \N\textit{F. Ziltener} [Int. Math. Res. Not. 2016, No. 3, 795--826 (2016; Zbl 1344.53062)] later improved this result to a quantitative statement in the case of the round sphere. In fact, by a result due to F. Ziltener [loc. cit.], there exist no closed embedded Bohr-Sommerfeld Lagrangians inside \(\mathbb{C}P^{n}\) for the prequantisation bundle whose total space is the standard contact sphere. On the other hand, any embedded monotone Lagrangian torus has a canonical nontrivial cover which is a Bohr-Sommerfeld immersion. In this paper the authors study certain embedded Legendrians \(\Lambda\subset(E,\mathrm{Ker}\alpha)\) whose projection \(\pi(\Lambda)\subset M^{2n}\) again has an embedded image. They are mainly interested in the case \(M^{2n}=\mathbb{C}P^{n}\) and the line bundle \(\mathcal{O},\) which produces the standard round contact sphere \((E,\alpha)=(\mathbb{S}^{1},\alpha_{st}).\) They draw the front projections for the corresponding Legendrian lifts inside a contact Darboux ball of the threefold covers of both the two-dimensional Clifford and Chekanov tori (the former is the Legendrian link of the Harvey-Lawson special Lagrangian cone), and compute the associated Chekanov-Eliashberg algebras. Although these Legendrians are not loose, they show that they both admit exact Lagrangian cobordisms to the loose Legendrian sphere; they hence admit exact Lagrangian caps in the symplectisation, which are non-regular Lagrangian cobordisms. Along the way, they also compute the bilinearised Legendrian contact homology of a general Legendrian surface in the standard contact vector space when all Reeb chords are of positive degree, as well as the augmentation variety in the case of tori.