Infinitely many exotic Lagrangian tori in higher projective spaces (Q6630791)

Let \((M, \omega )\) be a symplectic manifold and \(L \subset M\) a Lagrangian submanifold. Let \(\omega : \pi_2 (M, L) \to {\mathbb R}\) and \(\mu : \pi_2 (M, L) \to {\mathbb Z}\) be respectively the area and Maslov homomorphisms (see [\textit{V. I. Arnol'd}, Funkts. Anal. Prilozh. 1, No. 1, 1--14 (1967; Zbl 0175.20303)] for the construction of the Maslov homomorphism). \(L\) is monotone when \(\omega\) is a positive multiple of \(\mu\). \N\NThe authors take up the question as to how many distinct monotone Lagrangian tori exist in the complex projective space. Let \(\pi : {\mathbb C}^n \setminus \{ 0 \} \to {\mathbb C}P^{n-1}\) be the projection. Then \(\pi \Big( \prod_{j=1}^n S^1 \big( r_j \big) \Big)\) with \(r_j = r > 0\) (the Clifford torus in \({\mathbb C}P^{n-1}\)) is a monotone Lagrangian torus and monotone Lagrangian tori not Hamiltonian isotopic to a Clifford torus are exotic. Let \((a, b, c) \in {\mathbb N}^3\) such that \(a^2 + b^2 + c^2 = 3 a b c\) (a Markov triple) and let\N\[\N{\mathbb C}P \big( a^2 , b^2 , c^2 \big) = {\mathbb C}P^2 \big/ I (a^2 ) \times I(b^2 ) \times I (c^2 )\N\]\Nbe the weighted projective space [\(I(n) = \{ z \in {\mathbb C} : z^n =1 \}\) and the action is diagonal]. Let \(T_{(a, b, c)} \subset {\mathbb C}P^2\) be the Vianna torus associated to the Markov triple \((a, b, c)\), that is, the central fiber of the almost toric fibration of \({\mathbb C}P^2\) obtained from the rational blow-down of \({\mathbb C}P \big( a^2 , b^2 , c^2 \big)\), see [\textit{R. Ferreira de Velloso Vianna}, J. Topol. 9, No. 2, 535--551 (2016; Zbl 1350.53102)]. The main contribution of this paper is to lift the Vianna tori to monotone Lagrangian tori in \({\mathbb C}P^n\), \(n \geq 3\). Let\N\[\N\mu_n : {\mathbb C}P^n \to {\mathbb R}^{n-2} , \;\;\; \mu_n \big( [z] \big) = |z|^{-2} \big( \big| z_3 \big|^2 , \ \cdots , \big| z_n \big|^2 \big) ,\N\]\N(the moment map of the standard Hamiltonian \({\mathbb T}^{n-2}\) action on the last \(n-2\) homogeneous coordinates). Let \(\{ e_i : 1 \leq i \leq n - 2 \} \subset {\mathbb R}^{n-2}\) be the canonical linear basis. \({\mathbb T}^{n-2}\) acts freely on\N\[\NF_n = \mu_n^{-1} \Big( \Big\{ \frac{1}{n+1} \sum_{i=1}^{n-2} e_i \Big\} \Big)\N\]\Nand \(F_n \big/ {\mathbb T}^{n-2} \simeq {\mathbb C}P^2\) (a symplectomorphism). Let \(q : F_n \to {\mathbb C}P^2\) be the projection. The lifted Vianna torus in the paper under review is \(q^{-1} \big( T_{(a, b, c)} \big)\). As a corollary to their construction the authors find infinitely many distinct exotic monotone Lagrangian tori in \({\mathbb C}P^n\) for every \(n \geq 2\).