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Cheuk Yu Mak

Publications and source records attributed to Cheuk Yu Mak.

At least 19 recordsLinked to original sources
arXiv2025

Orbifold Floer spectral invariants, symmetric product links and Weyl laws

We explain a strategy, based on spectral invariants on symmetric product orbifolds, for proving the smooth closing lemma for Hamiltonian diffeomorphisms of a symplectic manifold when the orbifold quantum cohomologies of its symmetric products possess suitable idempotents. We relate the existence of such idempotents to the manifold containing a sequence of Lagrangian links, whose number of components tends to infinity, satisfying a number of properties. Orbifold Floer cohomology for global quotient orbifolds is used axiomatically, and is constructed in a companion paper. We illustrate this strategy by giving a new proof of the smooth closing lemma for area-preserving diffeomorphisms of the 2-sphere. The construction of suitable Lagrangian links in higher dimensions remains an intriguing open problem.

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arXiv2025

$C^0$-rigidity of the Hamiltonian diffeomorphism group of symplectic rational surfaces

We investigate the $C^0$-topology of the group of symplectic diffeomorphisms of positive symplectic rational surfaces. For all but a few exceptions, we prove that the group of Hamiltonian diffeomorphisms forms a connected component in the $C^0$-topology. This provides the first nontrivial case in which the group of Hamiltonian diffeomorphisms is known to be $C^0$-closed inside the group of symplectic diffeomorphisms. The key to our approach is to build a bridge between techniques from symplectic mapping class groups and problems in $C^0$-symplectic topology. Via a careful adaptation of tools from $J$-holomorphic foliation and inflation, we establish the necessary $C^0$-distance estimates. We hope that this serves as an example of how these two subfields can interact fruitfully, and also propose several questions arising from this interplay.

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arXiv2025

Orbifold Hamiltonian Floer theory for global quotients

We construct bulk-deformed orbifold Hamiltonian Floer theory for a global quotient orbifold, that is the quotient of a smooth closed symplectic manifold by a finite group acting faithfully via symplectomorphisms. The moduli spaces define an `ordered marked Flow category', which we equip with a coherent presentation via derived orbifolds. The global charts for orbifold Floer cylinders are built from moduli spaces of holomorphic curves in a quotient of projective space by a free action of the given finite group.

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arXiv2024

Symplectic annular Khovanov homology and fixed point localizations

We introduce a new version of symplectic annular Khovanov homology and establish spectral sequences from (i) the symplectic annular Khovanov homology of a knot to the link Floer homology of the lift of the annular axis in the double branched cover; (ii) the symplectic Khovanov homology of a two-periodic knot to the symplectic annular Khovanov homology of its quotient; and (iii) the symplectic Khovanov homology of a strongly invertible knot to the cone of the axis-moving map between the symplectic annular Khovanov homology of the two resolutions of its quotient.

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arXiv2024

Fukaya categories of hyperplane arrangements

To a simple polarized hyperplane arrangement (not necessarily cyclic) $\mathbb{V}$, one can associate a stopped Liouville manifold (equivalently, a Liouville sector) $\left(M(\mathbb{V}),\xi\right)$, where $M(\mathbb{V})$ is the complement of finitely many hyperplanes in $\mathbb{C}^d$, obtained as the complexifications of the real hyperplanes in $\mathbb{V}$. The Liouville structure on $M(\mathbb{V})$ comes from a very affine embedding, and the stop $\xi$ is determined by the polarization. In this article, we study the symplectic topology of $\left(M(\mathbb{V}),\xi\right)$. In particular, we prove that their partially wrapped Fukaya categories are generated by Lagrangian submanifolds associated to the bounded and feasible chambers of $\mathbb{V}$. A computation of the Fukaya $A_\infty$-algebra of these Lagrangians then enables us to identity these wrapped Fukaya categories with the $\mathbb{G}_m^d$-equivariant hypertoric convolution algebras $\widetilde{B}(\mathbb{V})$ associated to $\mathbb{V}$. This confirms a conjecture of Lauda-Licata-Manion (arXiv:2009.03981) and provides evidence for the general conjecture of Lekili-Segal (arXiv:2304.10969) on the equivariant Fukaya categories of symplectic manifolds with Hamiltonian torus actions.

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arXiv2023

Hameomorphism Groups of Positive Genus Surfaces

In their previous works arXiv:2105.11026, arXiv:2206.10749, Cristofaro-Gardiner, Humili\`ere, Mak, Seyfaddini and Smith defined links spectral invariants on connected compact surfaces and used them to show various results on the algebraic structure of the group of area-preserving homeomorphisms of surfaces, particularly in cases where the surfaces have genus zero. We show that on surfaces with higher genus, for a certain class of links, the invariants will satisfy a local quasimorphism property. Subsequently, we generalize their results to surfaces of any genus. This extension includes the non-simplicity of (i) the group of hameomorphisms of a closed surface, and (ii) the kernel of the Calabi homomorphism inside the group of hameomorphisms of a surface with non-empty boundary. Moreover, we prove that the Calabi homomorphism extends (non-canonically) to the $C^0$-closure of the set of Hamiltonian diffeomorphisms of any surface. The local quasimorphism property is a consequence of a quantitative K\"unneth formula for a connected sum in Heegaard Floer homology, inspired by results of Ozsv\'{a}th and Szab\'{o}.

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arXiv2023

Coulomb branch algebras via symplectic cohomology

Let $(\bar{M}, \omega)$ be a compact symplectic manifold with convex boundary and $c_1(T\bar{M})=0$. Suppose that $(\bar{M}, \omega)$ is equipped with a convex Hamiltonian $G$-action for some connected, compact Lie group $G$. We construct an action of the pure Coulomb branch of $G$ on the $G$-equivariant symplectic cohomology of $\bar{M}.$ Building on work of Teleman, we use this construction to characterize the Coulomb branches of Braverman-Finkelberg-Nakajima in terms of equivariant symplectic cohomology.

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arXiv2023

A characterization of heaviness in terms of relative symplectic cohomology

For a compact subset $K$ of a closed symplectic manifold $(M, \omega)$, we prove that $K$ is heavy if and only if its relative symplectic cohomology over the Novikov field is non-zero. As an application we show that if two compact sets are not heavy and Poisson commuting, then their union is also not heavy. A discussion on superheaviness together with some partial results are also included.

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arXiv2022

Subleading asymptotics of link spectral invariants and homeomorphism groups of surfaces

This paper continues the study of link spectral invariants on compact surfaces, introduced in our previous work and shown to satisfy a Weyl law in which they asymptotically recover the Calabi invariant. Here we study their subleading asymptotics on surfaces of genus zero. We show the subleading asymptotics are bounded for smooth time-dependent Hamiltonians, and recover the Ruelle invariant for autonomous disc maps with finitely many critical values. We deduce that the Calabi homomorphism admits infinitely many extensions to the group of compactly supported area-preserving homeomorphisms, and that the kernel of the Calabi homomorphism on the group of hameomorphisms is not simple.

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arXiv2022

Affine nil-Hecke algebras and Quantum cohomology

Let $G$ be a compact, connected Lie group and $T \subset G$ a maximal torus. Let $(M,\omega)$ be a monotone closed symplectic manifold equipped with a Hamiltonian action of $G$. We construct a module action of the affine nil-Hecke algebra $\hat{H}_*^{S^1 \times T}(LG/T)$ on the $S^1 \times T$-equivariant quantum cohomology of $M$, $QH^*_{S^1 \times T}(M).$ Our construction generalizes the theory of shift operators for Hamiltonian torus actions [OP,LJ]. We show that, as in the abelian case, this action behaves well with respect to the quantum connection. As an application of our construction, we show that when $G$ is semi-simple, the $G$-equivariant quantum cohomology $QH_G^*(M)$ defines a canonical holomorphic Lagrangian subvariety $\mathbb{L}_G(M) \hookrightarrow BFM(G_{\mathbb{C}}^{\vee})$ in the BFM-space of the Langlands dual group, confirming an expectation of Teleman from [T1].

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arXiv2021

Quantitative Heegaard Floer cohomology and the Calabi invariant

We define a new family of spectral invariants associated to certain Lagrangian links in compact and connected surfaces of any genus. We show that our invariants recover the Calabi invariant of Hamiltonians in their limit. As applications, we resolve several open questions from topological surface dynamics and continuous symplectic topology: we show that the group of Hamiltonian homeomorphisms of any compact surface with (possibly empty) boundary is not simple; we extend the Calabi homomorphism to the group of Hameomorphisms constructed by Oh-M\"uller; and, we construct an infinite dimensional family of quasimorphisms on the group of area and orientation preserving homeomorphisms of the two-sphere. Our invariants are inspired by recent work of Polterovich and Shelukhin defining and applying spectral invariants for certain classes of links in the two-sphere.

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arXiv2020

Circular spherical divisors and their contact topology

This paper investigates the symplectic and contact topology associated to circular spherical divisors. We classify, up to toric equivalence, all concave circular spherical divisors $ D $ that can be embedded symplectically into a closed symplectic 4-manifold and show they are all realized as symplectic log Calabi-Yau pairs if their complements are minimal. We then determine the Stein fillability and rational homology type of all minimal symplectic fillings for the boundary torus bundles of such $D$. When $ D $ is anticanonical and convex, we give explicit betti number bounds for Stein fillings of its boundary contact torus bundle.

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arXiv2019

Non-displaceable Lagrangian links in four-manifolds

Let $\omega$ denote an area form on $S^2$. Consider the closed symplectic 4-manifold $M=(S^2\times S^2, A\omega \oplus a \omega)$ with $0<a<A$. We show that there are families of displaceable Lagrangian tori $L_{0,x},\, L_{1,x} \subset M$, for $x \in [0,1]$, such that the two-component link $L_{0,x} \cup L_{1,x}$ is non-displaceable for each $x$.

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arXiv2019

Fukaya-Seidel categories of Hilbert schemes and parabolic category $\mathcal{O}$

We realise Stroppel's extended arc algebra in the Fukaya-Seidel category of a natural Lefschetz fibration on the generic fiber of the adjoint quotient map on a type $A$ nilpotent slice with two Jordan blocks, and hence obtain a symplectic interpretation of certain parabolic two-block versions of Bernstein-Gelfan'd-Gelfan'd category $\mathcal{O}$. As an application, we give a new geometric construction of the spectral sequence from annular to ordinary Khovanov homology. The heart of the paper is the development of a cylindrical model to compute Fukaya categories of (affine open subsets of) Hilbert schemes of quasi-projective surfaces, which may be of independent interest.

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arXiv2019

Tropically constructed Lagrangians in mirror quintic threefolds

We use tropical curves and toric degeneration techniques to construct closed embedded Lagrangian rational homology spheres in a lot of Calabi-Yau threefolds. We apply this construction to the tropical curves obtained from the 2875 lines on the quintic Calabi-Yau threefold. Each admissible tropical curve gives a Lagrangian rational homology sphere in the corresponding mirror quintic threefold and disjoint curves give pairwise homologous but non-Hamiltonian isotopic Lagrangians. We check in an example that $>300$ mutually disjoint curves (and hence Lagrangians) arise. We show that the weight of each of these Lagrangians equals to the multiplicity of the corresponding tropical curve.

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arXiv2018

Spherical twists and Lagrangian spherical manifolds

We study Dehn twists along Lagrangian submanifolds that are finite quotients of spheres. We decribe the induced auto-equivalences to the derived Fukaya category and explain its relation to twists along spherical functors.

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