Related papers: On an exotic Lagrangian torus in $\mathbb{C}P^2$

We construct a family of Lagrangian tori $\Theta^n_s$ $\subset$ $(\mathbb{C}P^1)^n$, $s \in (0,1)$, where $\Theta^n_{1/2} = \Theta^n$, is the monotone twist Lagrangian torus described by Chekanov-Schlenk. We show that for $n = 2m$ and $s…

This is a note on Beauville's problem (solved by Greb, Lehn and Rollenske in the non-algebraic case and by Hwang and Weiss in general) whether a lagrangian torus on an irreducible holomorphic symplectic manifold is a fiber of a lagrangian…

We consider various constructions of monotone Lagrangian submanifolds of $C P^n, S^2\times S^2$, and quadric hypersurfaces of $C P^n$. In $S^2\times S^2$ and $C P^2$ we show that several different known constructions of exotic monotone tori…

We show that the Lagrangian torus in the cotangent bundles of the 2-sphere obtained by applying the geodesic flow to the unit circle in a fibre is not displaceable by computing its Lagrangian Floer homology. The computation is based on a…

We show that the union of a meridian and a longitude of the symplectic 2-torus is superheavy in the sense of Entov-Polterovich. By a result of Entov-Polterovich, this implies that the product of this union and the Clifford torus of…

Can a given Lagrangian submanifold be realized as the fixed point set of an anti-symplectic involution? If so, it is called \emph{real}. We give an obstruction for a closed Lagrangian submanifold to be real in terms of the displacement…

We show that in many examples the non-displaceability of Lagrangian submanifolds by Hamiltonian isotopy can be proved via Lagrangian Floer cohomology with non-unitary line bundle. The examples include all monotone Lagrangian torus fibers in…

For a Lagrangian torus A in a simply-connected projective symplectic manifold M, we prove that M has a hypersurface disjoint from a deformation of A. This implies that a Lagrangian torus in a compact hyperk\"ahler manifold is a fiber of an…

This note is a proof of the fact that a lagrangian torus on a hyperkaehler fourfold is always a fiber of an almost holomorphic lagrangian fibration.

We define an equivariant Lagrangian Floer theory for Lagrangian torus fibers in a compact symplectic toric manifold equipped with a subtorus action. We show that the set of all Lagrangian torus fibers with weak bounding cochain data whose…

We show that, up to Lagrangian isotopy, there is a unique Lagrangian torus inside each of the following uniruled symplectic four-manifolds: the symplectic vector space $\mathbb{R}^4$, the projective plane $\mathbb{C}P^2$, and the monotone…

(Non-)displaceability of fibers of integrable systems has been an important problem in symplectic geometry. In this paper, for a large class of classical Liouville integrable systems containing the Lagrangian top, the Kovalevskaya top and…

We prove that a monotone Lagrangian torus in $S^2\times S^2$ which suitably sits in a symplectic fibration with two sections in its complement is Hamiltonian isotopic to the Clifford torus.

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…

Related to each degeneration from CP^2 to CP(a^2,b^2,c^2), for (a,b,c) a Markov triple - positive integers satisfying a^2 + b^2 + c^2 = 3abc - there is a monotone Lagrangian torus, which we call T(a^2,b^2,c^2). We employ techniques from…

We construct almost toric fibrations (ATFs) on all del Pezzo surfaces, endowed with a monotone symplectic form. Except for $\mathbb{C}P^2 \# 1 \overline{\mathbb{C}P^2}$ and $\mathbb{C}P^2 \# 2 \overline{\mathbb{C}P^2}$ , we are able to get…

We prove the existence of a one-parameter family of nondisplaceable Lagrangian tori near a linear chain of Lagrangian 2-spheres in a symplectic 4-manifold. When the symplectic structure is rational we prove that the deformed Floer…

We prove that symplectic quasi-states and quasi-morphisms on a symplectic manifold descend under symplectic reduction on a superheavy level set of a Hamiltonian torus action. Using a construction due to Abreu and Macarini, in each dimension…

For a broad class of symplectic manifolds of dimension at least six, we find the following new phenomenon: there exist local exotic Lagrangian tori. More specifically, let $X$ be a geometrically bounded symplectic manifold of dimension at…

This is a book aimed at graduate students and researchers in symplectic geometry, based on a course I taught in 2019. The primary message is that the base of a Lagrangian torus fibration inherits an integral affine structure, which you can…