Lagrangian Spheres Research Articles

Lagrangian submanifolds in the homogeneous nearly Kähler $${\mathbb {S}}^3 \times {\mathbb {S}}^3$$ S 3 × S 3

In this paper, we investigate Lagrangian submanifolds in the homogeneous nearly Kahler $$\mathbb {S}^3 \times \mathbb {S}^3$$ . We introduce and make use of a triplet of angle functions to describe the geometry of a Lagrangian submanifold in $$\mathbb {S}^3 \times \mathbb {S}^3$$ . We construct a new example of a flat Lagrangian torus and give a complete classification of all the Lagrangian immersions of spaces of constant sectional curvature. As a corollary of our main result, we obtain that the radius of a round Lagrangian sphere in the homogeneous nearly Kahler $$\mathbb {S}^3 \times \mathbb {S}^3$$ can only be $$\frac{2}{\sqrt{3}}$$ or $$\frac{4}{\sqrt{3}}$$ .

Symplectic mapping class group relations generalizing the chain relation

In this paper, we examine mapping class group relations of some symplectic manifolds. For each [Formula: see text] and [Formula: see text], we show that the [Formula: see text]-dimensional Weinstein domain [Formula: see text], determined by the degree [Formula: see text] homogeneous polynomial [Formula: see text], has a Boothby–Wang type boundary and a right-handed fibered Dehn twist along the boundary that is symplectically isotopic to a product of right-handed Dehn twists along Lagrangian spheres. We also present explicit descriptions of the symplectomorphisms in the case [Formula: see text] recovering the classical chain relation for the torus with two boundary components.

Geodesics of Positive Lagrangians in Milnor Fibers

The space of positive Lagrangians in an almost Calabi-Yau manifold is an open set in the space of all Lagrangian submanifolds. A Hamiltonian isotopy class of positive Lagrangians admits a natural Riemannian metric $\Upsilon$, which gives rise to a notion of geodesics. We study geodesics of positive $O_n(\mathbb{R})$ invariant Lagrangian spheres in $n$-dimensional $A_m$ Milnor fibers. We show the existence and uniqueness of smooth solutions to the initial value problem and the boundary value problem. In particular, we obtain examples of smooth geodesics of positive Lagrangians in arbitrary dimension. As an application, we show that the Riemannian metric $\Upsilon$ induces a metric space structure on the space of positive $O_n(\mathbb{R})$ invariant Lagrangian spheres in the above mentioned Milnor fibers.

On the Fukaya category of a Fano hypersurface in projective space

This paper is about the Fukaya category of a Fano hypersurface $X \subset \mathbb{CP}^n$. Because these symplectic manifolds are monotone, both the analysis and the algebra involved in the definition of the Fukaya category simplify considerably. The first part of the paper is devoted to establishing the main structures of the Fukaya category in the monotone case: the closed-open string maps, weak proper Calabi-Yau structure, Abouzaid's split-generation criterion, and their analogues when weak bounding cochains are included. We then turn to computations of the Fukaya category of the hypersurface $X$: we construct a configuration of monotone Lagrangian spheres in $X$, and compute the associated disc potential. The result coincides with the Hori-Vafa superpotential for the mirror of $X$ (up to a constant shift in the Fano index $1$ case). As a consequence, we give a proof of Kontsevich's homological mirror symmetry conjecture for $X$. We also explain how to extract non-trivial information about Gromov-Witten invariants of $X$ from its Fukaya category.

Commuting symplectomorphisms and Dehn twists in divisors

Two commuting symplectomorphisms of a symplectic manifold give rise to actions on Floer cohomologies of each other. We prove the elliptic relation saying that the supertraces of these two actions are equal. In the case when a symplectomorphism $f$ commutes with a symplectic involution, the elliptic relation provides a lower bound on the dimension of $HF^*(f)$ in terms of the Lefschetz number of $f$ restricted to the fixed locus of the involution. We apply this bound to prove that Dehn twists around vanishing Lagrangian spheres inside most hypersurfaces in Grassmannians have infinite order in the symplectic mapping class group.

Lagrangian tori in four-dimensional Milnor fibres

The Milnor fibre of any isolated hypersurface singularity contains many exact Lagrangian spheres: the vanishing cycles associated to a Morsification of the singularity. Moreover, for simple singularities, it is known that the only possible exact Lagrangians are spheres. We construct exact Lagrangian tori in the Milnor fibres of all non-simple singularities of real dimension four. This gives examples of Milnor fibres whose Fukaya categories are not generated by vanishing cycles. Also, this allows progress towards mirror symmetry for unimodal singularities, which are one level of complexity up from the simple ones.

Smoothing singular constant scalar curvature Kähler surfaces and minimal Lagrangians

Given a complex surface X with singularities of class T and no nontrivial holomorphic vector field, endowed with a Kähler class Ω0, we consider smoothings (Mt,Ωt), where Ωt is a Kähler class on Mt degenerating to Ω0. Under an hypothesis of nondegeneracy of the smoothing at each singular point, we prove that if X admits a constant scalar curvature Kähler metric in Ω0, then Mt admits a constant scalar curvature Kähler metric in Ωt for small t.In addition, we construct small Lagrangian stationary spheres which represent Lagrangian vanishing cycles when t is small.

Lefschetz fibrations and exotic symplectic structures on cotangent bundles of spheres: Figure 1.

(J. Topol. 3 (2010) 157–180) In the proof of [4, Lemma 1.1], we appealed to an explicit isotopy of totally real spheres, constructed in [3, Section 5]. That construction works in the lowest dimension ( n = 2 ), but is wrong in general (one of the endpoints is not the desired sphere). Here, we explain a different approach, leading to a corrected version of [4, Lemma 1.1], which requires an additional assumption. Independently, while [4, Lemma 1.2] makes a statement about homotopy classes of almost complex structures, its proof only determines the isomorphism class of the tangent bundle as an abstract complex vector bundle, which is a priori a weaker statement. The argument here also fills that gap. The rest of the original paper is unaffected. Consider M = M m as in [4], in complex dimension n > 2 . The construction of E depends on a choice of Lagrangian sphere S = S δ m + 1 ⊂ M . By [1], the smooth isotopy class of S depends only on [ S ] ∈ H n ( M ) . Since S is Lagrangian, it comes with a canonical formal Legendrian structure (more precisely, a formal Legendrian structure for { 0 } × S ⊂ R × M , as defined in [5]). Given two homologous Lagrangian spheres, we can use a smooth isotopy between them to compare their canonical formal Legendrian structures. If these coincide, then the resulting manifolds E are diffeomorphic, compatibly with the homotopy classes of their almost complex structures. In general, the difference between two formal Legendrian structures for a given n-sphere is described by an element of π n + 1 ( V n , 2 n + 1 , U n ) , where V n , 2 n + 1 is the Stiefel manifold. That homotopy group was analyzed in [5, Lemmas A.5–A.7], with the following implications for our situation (cf. [5, Theorem A.4]). Crossing over one marked point. Lemma 1.1 (Corrected).Suppose that n = 2 . Then, any choice of δ m + 1 leads to a manifold E which is diffeomorphic to T * S n + 1 , and this diffeomorphism is compatible with the homotopy classes of almost complex structures. For higher even n , the same holds under the following additional assumption: ( *) δ m + 1 can be connected to a ‘standard path’ by an isotopy (rel endpoints ) which crosses over an even number of marked points in the plane (here, the ‘standard paths’ are the δ k , l from [4, Section 7]). It remains to consider the case n = 6 . Then, for any choice of δ m + 1 , the resulting E will be diffeomorphic to T * S 7 ≅ S 7 × R 7 (one shows this using the h-cobordism theorem, and the fact that any seven-dimensional vector bundle over S 7 is trivial). However, there are two possible homotopy classes of almost complex structures ( π 7 ( O 14 / U 7 ) ≅ π 7 ( O ∞ / U ∞ ) ≅ Z / 2 ), and it is not clear which one will arise if ( *) is dropped. In particular, we still do not know what element of (2) appears there. We thank Emmy Murphy for helpful suggestions.

On Lagrangian spheres in the flag variety F 3

On Lagrangian spheres in the flag variety F 3

Exotic spheres and the topology of symplectomorphism groups

We show that, for certain families $\phi_{\mathbf{s}}$ of diffeomorphisms of high-dimensional spheres, the commutator of the Dehn twist along the zero-section of $T^*S^n$ with the family of pullbacks $\phi^*_{\mathbf{s}}$ gives a noncontractible family of compactly-supported symplectomorphisms. In particular, we find examples: where the Dehn twist along a parametrised Lagrangian sphere depends up to Hamiltonian isotopy on its parametrisation; where the symplectomorphism group is not simply-connected, and where the symplectomorphism group does not have the homotopy-type of a finite CW-complex. We show that these phenomena persist for Dehn twists along the standard matching spheres of the $A_m$-Milnor fibre. The nontriviality is detected by considering the action of symplectomorphisms on the space of parametrised Lagrangian submanifolds. We find related examples of symplectic mapping classes for $T^*(S^n\times S^1)$ and of an exotic symplectic structure on $T^*(S^n\times S^1)$ standard at infinity.

Tight Lagrangian homology spheres in compact homogeneous Kähler manifolds

For any irreducible compact homogeneous Kähler manifold, we classify the compact tight Lagrangian submanifolds which have the ℤ2-homology of a sphere.

Pseudotoric structures and Lagrangian spheres in the flag variety F 3

Pseudotoric structures and Lagrangian spheres in the flag variety F 3

Homological mirror symmetry for Calabi–Yau hypersurfaces in projective space

We prove Homological Mirror Symmetry for a smooth d-dimensional Calabi-Yau hypersurface in projective space, for any d > 2 (for example, d = 3 is the quintic three-fold). The main techniques involved in the proof are: the construction of an immersed Lagrangian sphere in the `d-dimensional pair of pants'; the introduction of the `relative Fukaya category', and an understanding of its grading structure; a description of the behaviour of this category with respect to branched covers (via an `orbifold' Fukaya category); a Morse-Bott model for the relative Fukaya category that allows one to make explicit computations; and the introduction of certain graded categories of matrix factorizations mirror to the relative Fukaya category.

Exact Lagrangians in $$A_n$$ A n -surface singularities

In this paper we classify Lagrangian spheres in \(A_n\)-surface singularities up to Hamiltonian isotopy. Combining with a result of Ritter (Geom Funct Anal 20(3):779–816, 2010), this yields a complete classification of exact Lagrangians in \(A_n\)-surface singularities. Our main new tool is the application of a technique which we call ball-swappings and its relative version.

Disjoinable Lagrangian spheres and dilations

We consider open symplectic manifolds which admit dilations (in the sense previously introduced by Solomon and the author). We obtain restrictions on collections of Lagrangian submanifolds which are pairwise disjoint (or pairwise disjoinable by Hamiltonian isotopies) inside such manifolds. This includes the Milnor fibres of isolated hypersurface singularities which have been stabilized (by adding quadratic terms) sufficiently often.

Lagrangian Floer homology of a pair of real forms in Hermitian symmetric spaces of compact type

In this paper we calculate the Lagrangian Floer homology $HF(L_0, L_1 : {\mathbb Z}_2)$ of a pair of real forms $(L_0,L_1)$ in a monotone Hermitian symmetric space $M$ of compact type in the case where $L_0$ is not necessarily congruent to $L_1$. In particular, we have a generalization of the Arnold-Givental inequality in the case where $M$ is irreducible. As its application, we prove that the totally geodesic Lagrangian sphere in the complex hyperquadric is globally volume minimizing under Hamiltonian deformations.

Dehn twists and free subgroups of symplectic mapping class groups

Given two Lagrangian spheres in an exact symplectic manifold, we find conditions under which the Dehn twists about them generate a free non-abelian subgroup of the symplectic mapping class group. This extends a result of Ishida for Riemann surfaces. The proof generalises the categorical version of Seidel's long exact sequence to arbitrary powers of a fixed Dehn twist. We also show that the Milnor fibre of any isolated degenerate hypersurface singularity contains such pairs of spheres.

Exact Lagrangians in plumbings

Consider a Stein manifold M obtained by plumbing cotangent bundles of manifolds of dimension greater than or equal to 3 at points. We prove that the Fukaya category of closed exact Spin Lagrangians with vanishing Maslov class in M is generated by the compact cores of the plumbing. As applications, we classify exact Lagrangian spheres in A2-Milnor fibres of arbitrary dimension, derive constraints on exact Lagrangian fillings of Legendrian unknots in disk cotangent bundles, and prove that the categorical equivalence given by the spherical twist in a homology sphere is typically not realised by any compactly supported symplectomorphism.

Lagrangian spheres, symplectic surfaces and the symplectic mapping class group

Given a Lagrangian sphere in a symplectic 4-manifold $(M, \omega)$ with $b^+=1$, we find embedded symplectic surfaces intersecting it minimally. When the Kodaira dimension $\kappa$ of $(M, \omega)$ is $-\infty$, this minimal intersection property turns out to be very powerful for both the uniqueness and existence problems of Lagrangian spheres. On the uniqueness side, for a symplectic rational manifold and any class which is not characteristic and ternary, we show that homologous Lagrangian spheres are smoothly isotopic, and when the Euler number is less than 8, we generalize Hind and Evans' Hamiltonian uniqueness in the monotone case. On the existence side, when $\kappa=-\infty$, we give a characterization of classes represented by Lagrangian spheres, which enables us to describe the non-Torelli part of the symplectic mapping class group.

Lagrangian unknottedness in Stein surfaces

We show that the space of Lagrangian spheres inside the cotangent bundle of the 2-sphere is contractible. We then discuss the phenomenon of Lagrangian unknottedness in other Stein surfaces. There exist homotopic Lagrangian spheres which are not Hamiltonian isotopic, but we show that in a typical case all such spheres are still equivalent under a symplectomorphism.