Lagrangian Spheres Research Articles

Volcanic Skies: coupling explosive eruptions with atmospheric simulation to create consistent skyscapes

AbstractExplosive volcanic eruptions rank among the most terrifying natural phenomena, and are thus frequently depicted in films, games, and other media, usually with a bespoke once‐off solution. In this paper, we introduce the first general‐purpose model for bi‐directional interaction between the atmosphere and a volcano plume. In line with recent interactive volcano models, we approximate the plume dynamics with Lagrangian disks and spheres and the atmosphere with sparse layers of 2D Eulerian grids, enabling us to focus on the transfer of physical quantities such as temperature, ash, moisture, and wind velocity between these sub‐models. We subsequently generate volumetric animations by noise‐based procedural upsampling keyed to aspects of advection, convection, moisture, and ash content to generate a fully‐realized volcanic skyscape. Our model captures most of the visually salient features emerging from volcano‐sky interaction, such as windswept plumes, enmeshed cap, bell and skirt clouds, shockwave effects, ash rain, and sheathes of lightning visible in the dark.

Isolated hypersurface singularities, spectral invariants, and quantum cohomology

Abstract We study the relation between isolated hypersurface singularities (e.g., ADE) and the quantum cohomology ring by using spectral invariants, which are symplectic measurements coming from Floer theory. We prove, under the assumption that the quantum cohomology ring is semi-simple, that (1) if the smooth Fano variety degenerates to a Fano variety with an isolated hypersurface singularity, then the singularity has to be an A m {A_{m}} -singularity, (2) if the symplectic manifold contains an A m {A_{m}} -configuration of Lagrangian spheres, then there are consequences for the Hofer geometry, and that (3) the Dehn twist reduces spectral invariants.

Hofer geometry via toric degeneration

The main theme of this paper is to use toric degeneration to study Hofer geometry by producing distinct homogeneous quasimorphisms on the group of Hamiltonian diffeomorphisms. We focus on the (complex n-dimensional) quadric hypersurface and the del Pezzo surfaces, and study two classes of distinguished Lagrangian submanifolds that appear naturally in a toric degeneration, namely the Lagrangian torus which is the monotone fiber of a Lagrangian torus fibration, and the Lagrangian spheres that appear as vanishing cycles. For the quadrics, we prove that the group of Hamiltonian diffeomorphisms admits two distinct homogeneous quasimorphisms and derive some superheaviness results. Along the way, we show that the toric degeneration is compatible with the Biran decomposition. This implies that for n=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n=2$$\\end{document}, the Lagrangian fiber torus (Gelfand–Zeitlin torus) is Hamiltonian isotopic to the Chekanov torus, which answers a question of Y. Kim. We prove analogous results for the del Pezzo surfaces. We also discuss applications to C0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^0$$\\end{document} symplectic topology.

Exact Calabi–Yau categories and odd-dimensional Lagrangian spheres

An exact Calabi–Yau structure, originally introduced by Keller, is a special kind of smooth Calabi–Yau structure in the sense of Kontsevich–Vlassopoulos (2021). For a Weinstein manifold M , the existence of an exact Calabi–Yau structure on the wrapped Fukaya category \mathcal{W}(M) imposes strong restrictions on its symplectic topology. Under the cyclic open-closed map constructed by Ganatra (2019), an exact Calabi–Yau structure on \mathcal{W}(M) induces a class \tilde{b} in the degree one equivariant symplectic cohomology \mathrm{SH}_{S^1}^{1}(M) . Any Weinstein manifold admitting a quasi-dilation in the sense of Seidel–Solomon [Geom. Funct. Anal. 22 (2012), 443–477] has an exact Calabi–Yau structure on \mathcal{W}(M) . We prove that there are many Weinstein manifolds whose wrapped Fukaya categories are exact Calabi–Yau despite the fact that there is no quasi-dilation in \mathrm{SH}^{1}(M) ; a typical example is given by the affine hypersurface \{x^{3}+y^{3}+z^{3}+w^{3}=1\}\subset\mathbb{C}^{4} . As an application, we prove the homological essentiality of Lagrangian spheres in many odd-dimensional smooth affine varieties with exact Calabi–Yau wrapped Fukaya categories.

Refined disk potentials for immersed Lagrangian surfaces

We define a refined Gromov–Witten disk potential of monotone immersed Lagrangian surfaces in a symplectic 4‑manifold that are self-transverse as an element in a capped version of the Chekanov–Eliashberg dg‑algebra of the singularity links of the double points (a collection of Legendrian Hopf links). We give a surgery formula that expresses the potential after smoothing a double point. We study refined potentials of monotone immersed Lagrangian spheres in the complex projective plane and find monotone spheres that cannot be displaced from complex lines and conics by symplectomorphisms. We also derive general restrictions on sphere potentials using Legendrian lifts to the contact 5‑sphere.

Caustics of Lagrangian homotopy spheres with stably trivial Gauss map

For each positive integer $n$, we give a geometric description of the stably trivial elements of the group $\pi_n U_n/O_n$. In particular, we show that all such elements admit representatives whose tangencies with respect to a fixed Lagrangian plane consist only of folds. By the h-principle for the simplification of caustics, this has the following consequence: if a Lagrangian distribution is stably trivial from the viewpoint of a Lagrangian homotopy sphere, then by an ambient Hamiltonian isotopy one may deform the Lagrangian homotopy sphere so that its tangencies with respect to the Lagrangian distribution are only of fold type. Thus the stable triviality of the Lagrangian distribution, which is a necessary condition for the simplification of caustics to be possible, is also sufficient. We give applications of this result to the arborealization program and to the study of nearby Lagrangian homotopy spheres.

Anti‐symplectic involutions for Lagrangian spheres in a symplectic quadric surface

We show that the space of anti-symplectic involutions of a monotone $S^2\times S^2$ whose fixed points set is a Lagrangian sphere is connected. This follows from a stronger result, namely that any two anti-symplectic involutions in that space are Hamiltonian isotopic.

On the invariance of Welschinger invariants

We collect in this note some observations about original Welschinger invariants of real symplectic fourfolds. None of their proofs is difficult, nevertheless these remarks do not seem to have been made before. Our main result is that when $X$ is a real rational algebraic surface, Welschinger invariants only depend on the number of real interpolated points, and some homological data associated to $X$. This strengthened the invariance statement initially proved by Welschinger. This main result follows easily from a formula relating Welschinger invariants of two real symplectic manifolds differing by a surgery along a real Lagrangian sphere. In its turn, once one believes that such formula may hold, its proof is a mild adaptation of the proof of analogous formulas previously obtained by the author on the one hand, and by Itenberg, Kharlamov and Shustin on the other hand. We apply the two aforementioned results to complete the computation of Welschinger invariants of real rational algebraic surfaces, and to obtain vanishing, sign, and sharpness results for these invariants that generalize previously known statements. We also discuss some hypothetical relations of our work with tropical refined invariants defined by Block-G\ottsche and G\ottsche-Schroeter.

Squared Dehn twists and deformed symplectic invariants

We establish an infinitesimal version of fragility for squared Dehn twists around even dimensional Lagrangian spheres. The precise formulation involves twisting the Fukaya category by a closed two-form or bulk deforming it by a half-dimensional cycle. As our main application, we compute the twisted and bulk deformed symplectic cohomology of the subflexible Weinstein manifolds constructed in \cite{murphysiegel}.

A Note on the Evolution of the Whitney Sphere Along Mean Curvature Flow

We study the evolution of the Whitney sphere along the Lagrangian mean curvature flow. We show that equivariant Lagrangian spheres in {{mathbb {C}}^n} satisfying mild geometric assumptions collapse to a point in finite time and the tangent flows converge to a Lagrangian plane with multiplicity two.

Rational equivalence and Lagrangian tori on K3 surfaces

Fix a symplectic K3 surface X homologically mirror to an algebraic K3 surface Y by an equivalence taking a graded Lagrangian torus L \subset X to the skyscraper sheaf of a point y \in 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.

Lagrangian cobordisms, Lefschetz Fibrations and Quantum Invariants

We study Lagrangian cobordisms with the tools provided by Lagrangian quantum homology. In particular, we develop the theory for the setting of Lagrangian cobordisms or Lagrangians with cylindrical ends in a Lefschetz fibration, and put the different versions of the quantum homology groups into relation by a long exact sequence. We prove various practical relations of maps in this long exact sequence and we extract invariants that generalize the notion of discriminants to Lagrangian cobordisms in Lefschetz fibrations. We prove results on the relation of the discriminants of the ends of a cobordism and the cobordism itself. We also give examples arising from Lagrangian spheres and relate the discriminant to open Gromov Witten invariants. We show that for some configurations of Lagrangian spheres the discriminant always vanishes. We study a set of examples that arise from Lefschetz pencils of complex quadric hypersurfaces of the complex projective space. These quadrics are symplectic manifolds endowed with real structures and their real part are the Lagrangians of interest. Using the results established in this thesis, we compute the discriminants of all these Lagrangians by reducing the calculation to the previously established case of a real Lagrangian sphere in the quadric.

Tropically constructed Lagrangians in mirror quintic threefolds

AbstractWe use tropical curves and toric degeneration techniques to construct closed embedded Lagrangian rational homology spheres in a lot of Calabi-Yau threefolds. The homology spheres are mirror dual to the holomorphic curves contributing to the Gromov-Witten (GW) invariants. In view of Joyce’s conjecture, these Lagrangians are expected to have special Lagrangian representatives and hence solve a special Lagrangian enumerative problem in Calabi-Yau threefolds.We apply this construction to the tropical curves obtained from the 2,875 lines on the quintic Calabi-Yau threefold. Each admissible tropical curve gives a Lagrangian rational homology sphere in the corresponding mirror quintic threefold and the Joyce’s weight of each of these Lagrangians equals the multiplicity of the corresponding tropical curve.As applications, we show that disjoint curves give pairwise homologous but non-Hamiltonian isotopic Lagrangians and we check in an example that$>300$mutually disjoint curves (and hence Lagrangians) arise. Dehn twists along these Lagrangians generate an abelian subgroup of the symplectic mapping class group with that rank.

The classification of Lagrangians nearby the Whitney immersion

The Whitney immersion is a Lagrangian sphere inside the four-dimensional symplectic vector space which has a single transverse double point of Whitney self-intersection number $+1.$ This Lagrangian also arises as the Weinstein skeleton of the complement of a binodal cubic curve inside the projective plane, and the latter Weinstein manifold is thus the `standard' neighbourhood of Lagrangian immersions of this type. We classify the Lagrangians inside such a neighbourhood which are homologically essential, and which either are embedded or immersed with a single double point; they are shown to be Hamiltonian isotopic to either product tori, Chekanov tori, or rescalings of the Whitney immersion.

A Lagrangian sphere which is not a vanishing cycle

We give examples of Calabi–Yau threefolds containing Lagrangian spheres which are not vanishing cycles of nodal degenerations, answering a question of Donaldson in the negative.

Lagrangian mean curvature flow of Whitney spheres

It is shown that an equivariant Lagrangian sphere with a positivity condition on its Ricci curvature develops a type-II singularity under the Lagrangian mean curvature flow that rescales to the product of a grim reaper with a flat Lagrangian subspace. In particular this result applies to the Whitney spheres.

Viterbo’s transfer morphism for symplectomorphisms

We construct an analogue of Viterbo’s transfer morphism for Floer homology of an automorphism of a Liouville domain. As an application we prove that the Dehn–Seidel twist along any Lagrangian sphere in a Liouville domain of dimension [Formula: see text] has infinite order in the symplectic mapping class group.

Nearby Lagrangian fibers and Whitney sphere links

Let $n>3$, and let $L$ be a Lagrangian embedding of $\mathbb{R}^{n}$ into the cotangent bundle $T^{\ast }\mathbb{R}^{n}$ of $\mathbb{R}^{n}$ that agrees with the cotangent fiber $T_{x}^{\ast }\mathbb{R}^{n}$ over a point $x\neq 0$ outside a compact set. Assume that $L$ is disjoint from the cotangent fiber at the origin. The projection of $L$ to the base extends to a map of the $n$-sphere $S^{n}$ into $\mathbb{R}^{n}\setminus \{0\}$. We show that this map is homotopically trivial, answering a question of Eliashberg. We give a number of generalizations of this result, including homotopical constraints on embedded Lagrangian disks in the complement of another Lagrangian submanifold, and on two-component links of immersed Lagrangian spheres with one double point in $T^{\ast }\mathbb{R}^{n}$, under suitable dimension and Maslov index hypotheses. The proofs combine techniques from Ekholm and Smith [Exact Lagrangian immersions with a single double point, J. Amer. Math. Soc. 29 (2016), 1–59] and Ekholm and Smith [Exact Lagrangian immersions with one double point revisited, Math. Ann. 358 (2014), 195–240] with symplectic field theory.

On homological mirror symmetry of toric Calabi–Yau threefolds

We use Lagrangian torus fibrations on the mirror $X$ of a toric Calabi-Yau threefold $\check X$ to construct Lagrangian sections and various Lagrangian spheres on $X$. We then propose an explicit correspondence between the sections and line bundles on $\check X$ and between spheres and sheaves supported on the toric divisors of $\check X$. We conjecture that these correspondences induce an embedding of the relevant derived Fukaya category of $X$ inside the derived category of coherent sheaves on $\check X$.

Surgery of real symplectic fourfolds and Welschinger invariants

A surgery of a real symplectic manifold $X_{\mathbb R}$ along a real Lagrangian sphere $S$ is a modification of the symplectic and real structure on $X_{\mathbb R}$ in a neigborhood of $S$. Genus 0 Welschinger invariants of two real symplectic $4$-manifolds differing by such a surgery have been related in a previous work in collaboration with N. Puignau. In the present paper, we explore some particular situations where these general formulas greatly simplify. As an application, we complete the computation of genus 0 Welschinger invariants of all del~Pezzo surfaces, and of all $\mathbb R$-minimal real conic bundles. As a by-product, we establish the existence of some new relative Welschinger invariants. We also generalize our results to the enumeration of curves of higher genus, and give relations between hypothetical invariants defined in the same vein as a previous work by Shustin.