Abstract
We construct almost toric fibrations (ATFs) on all del Pezzo surfaces, endowed with a monotone symplectic form. Except for {mathbb {C}}P^2# overline{{mathbb {C}}P^2}, {mathbb {C}}P^2# 2overline{{mathbb {C}}P^2}, we are able to get almost toric base diagrams (ATBDs) of triangular shape and prove the existence of infinitely many symplectomorphism (in particular Hamiltonian isotopy) classes of monotone Lagrangian tori in {mathbb {C}}P^2# koverline{{mathbb {C}}P^2} for k=0,3,4,5,6,7,8. We name these tori Theta ^{n_1,n_2,n_3}_{p,q,r}. Using the work of Karpov-Nogin, we are able to classify all ATBDs of triangular shape. We are able to prove that {mathbb {C}}P^2# overline{{mathbb {C}}P^2} also has infinitely many monotone Lagrangian tori up to symplectomorphism and we conjecture that the same holds for {mathbb {C}}P^2# 2overline{{mathbb {C}}P^2}. Finally, the Lagrangian tori Theta ^{n_1,n_2,n_3}_{p,q,r} subset X can be seen as monotone fibres of ATFs, such that, over its edge lies a fixed anticanonical symplectic torus Sigma . We argue that Theta ^{n_1,n_2,n_3}_{p,q,r} give rise to infinitely many exact Lagrangian tori in X setminus Sigma , even after attaching the positive end of a symplectization to partial (X setminus Sigma ).
Highlights
We say that two Lagrangians submanifolds of a symplectic manifold X belong to the same symplectomorphism class if there is a symplectomorphism of X sending one Lagrangian to the other
Even though we do not have a toric structure in CP2#kCP2, 3 ≤ k ≤ 8 endowed with a monotone symplectic form, in this paper we show that we can construct almost toric fibrations (ATFs) in all monotone del Pezzo surfaces, i.e., CP1 × CP1 and CP2#kCP2, 0 ≤ k ≤ 8
To prove Theorem 1.1(a) we show that for CP1 × CP1 and CP2#kCP2, k = 0, 3, 4, 5, 6, 7, 8, we can build the almost toric base diagrams of triangular shape described in Figs. 1, 2, 3 and 4
Summary
We say that two Lagrangians submanifolds of a symplectic manifold X belong to the same symplectomorphism class if there is a symplectomorphism of X sending one Lagrangian to the other. In view of that we conjecture: Conjecture 1.4 The boundary Maslov-2 convex hull of a monotone Lagrangian fibre of an ATF described by an ATBD (here we assume that cuts are always taken as eigenrays, which are fixed by the monodromy—see [31, Definition 4.11]) is determined by the limit orbifold (Definition 2.14). 6 we relate our work with [30], by pointing out that the complement of the symplectic torus in the anti-canonical class is obtained from attaching (Weinstein handles along the boundary of) Lagrangian disks to the (co-disk bundle of the) monotone fibre of each ATBD These tori are exact in the complement of. CP2#3CP2, and for some family of tori in the monotone (CP1)2m
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
