We define an simple invariant of an embedded nullhomologous Lagrangian torus and use this invariant to show that many symplectic 4-manifolds have infinitely many pairwise symplectically inequivalent nullhomologous Lagrangian tori. We further show that for a large class of examples that lambda(T) is actually a C-infinity invariant. In addition, this invariant is used to show that many symplectic 4-manifolds have nontrivial homology classes which are represented by infinitely many pairwise inequivalent Lagrangian tori, a result first proved by S Vidussi for the homotopy K3-surface obtained from knot surgery using the trefoil knot in [Lagrangian surfaces in a fixed homology class: existence of knotted Lagrangian tori, J. Diff. Geom. (to appear)].
In recent papers, summarized in survey [1], we construct a number of examples of non standard lagrangian tori on compact toric varieties and as well on certain non toric varieties which admit pseudotoric structures. Using this pseudotoric technique we explain how non standard lagrangian tori of Chekanov type can be constructed and what is the topological difference between standard Liouville tori and the non standard ones. However we have not discussed the natural question about the periods of the constructed twist tori; in particular the monotonicity problem for the monotonic case was not stu…
We prove the Hamiltonian unknottedness of real Lagrangian tori in the monotone $S^2\times S^2$, namely any real Lagrangian torus in $S^2\times S^2$ is Hamiltonian isotopic to the Clifford torus $\mathbb{T}_{\text{Clif}}$. The proof is based on a neck-stretching argument, Gromov's foliation theorem, and the Cieliebak-Schwingenheuer criterion.
Fix a symplectic K3 surface X homologically mirror to an algebraic K3 surface Y by an equivalence taking a graded Lagrangian torus L in X to the skyscraper sheaf of a point y of Y. We show there are Lagrangian tori with vanishing Maslov class in X whose class in the Grothendieck group of the Fukaya category is not generated by Lagrangian spheres. This is mirror to a statement about the `Beauville--Voisin subring' in the Chow groups of Y, and fits into a conjectural relationship between Lagrangian cobordism and rational equivalence of algebraic cycles.
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 cohomology groups of these tori are nontrivial. The proof uses the idea of toric degeneration to analyze the full potential functions with bulk deformations of these tori.
We prove some results about linking of Lagrangian tori in the symplectic vector space $(\mathbb{R}^4, ω)$. We show that certain enumerative counts of holomophic disks give useful information about linking. This enables us to prove, for example, that any two Clifford tori are unlinked in a strong sense. We extend work of Dimitroglou Rizell and Evans on linking of monotone Lagrangian tori to a class of non-monotone tori in $\mathbb{R}^4$ and also strengthen their conclusions in the monotone case in $\mathbb{R}^4$.
We derive constraints on Lagrangian embeddings in completions of certain stable symplectic fillings with semisimple symplectic cohomologies. Manifolds with these properties can be constructed by generalizing the boundary connected sum operation to our setting, and are related to certain birational surgeries like blow-downs and flips. As a consequence, there are many non-toric (non-compact) monotone symplectic manifolds whose wrapped Fukaya categories are proper.
We prove that every smoothly immersed 2-torus of $\mathbb{R}^4$ can be approximated, in the C0-sense, by immersed polyhedral Lagrangian tori. In the case of a smoothly immersed (resp. embedded) Lagrangian torus of $\mathbb{R}^4$, the surface can be approximated in the C1-sense by immersed (resp. embedded) polyhedral Lagrangian tori. Similar statements are proved for isotropic 2-tori of $\mathbb{R}^{2n}$.
We investigate which orbits of an $n$-dimensional torus action on a $2n$-dimensional toric Kähler manifold $M$ are minimal. In other words, we study minimal submanifolds appearing as the fibres of the moment map on a toric Kähler manifold. Amongst other questions we investigate and give partial answers to the following: (1) How many such minimal Lagrangian tori exist? (2) Can their stability, as critical points of the area functional, be characterised just from the ambient geometry? (3) Given a toric symplectic manifold, for which sets of orbits $S$, is there a compatible toric Kähler metric w…
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 energy of nearby Lagrangians. Applying this obstruction to toric fibres, we obtain that the central fibre of many (and probably all) toric monotone symplectic manifolds is real only if the corresponding moment polytope is centrally symmetric. Furthermore, we embed the Chekanov torus in all toric monotone symplectic manifolds and show that it is exotic and not re…
We prove a general form of the wall-crossing formula which relates the disk potentials of monotone Lagrangian submanifolds with their Floer-theoretic behavior away from a Donaldson divisor. We define geometric operations called mutations of Lagrangian tori in del Pezzo surfaces and in toric Fano varieties of higher dimension, and study the corresponding wall-crossing formulas that compute the disk potential of a mutated torus from that of the original one. In the case of del Pezzo surfaces, this justifies the connection between Vianna's tori and the theory of mutations of Landau-Ginzburg seeds…
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 use to "read off" a lot of interesting geometry of the total space. Topics covered include: action-angle coordinates, symplectic reduction, toric manifolds, visible and tropical Lagrangians, almost toric systems, Milnor fibres of cyclic quotient singularities, mutation of polygons, non-toric blow-up, an almost toric view on Lisca's classification of fillings…
We classify weakly exact, rational Lagrangian tori in $T^* \mathbb{T}^2- 0_{\mathbb{T}^2}$ up to Hamiltonian isotopy. This result is related to the classification theory of closed $1$-forms on $\mathbb{T}^n$ and also has applications to symplectic topology. As a first corollary, we strengthen a result due independently to Eliashberg-Polterovich and to Giroux describing Lagrangian tori in $T^* \mathbb{T}^2-0_{\mathbb{T}^2}$ which are homologous to the zero section. As a second corollary, we exhibit pairs of disjoint totally real tori $K_1, K_2 \subset T^*\mathbb{T}^2$, each of which is isotopic…
