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Cheuk Yu Mak

Publications and source records attributed to Cheuk Yu Mak.

23 records · Page 2Linked to original sources
arXiv2016

The Kodaira dimension of contact 3-manifolds and geography of symplectic fillings

We introduce the Kodaira dimension of contact 3-manifolds and establish some basic properties. In particular, contact 3-manifolds with distinct Kodaria dimensions behave differently when it comes to the geography of various kinds of fillings. On the other hand, we also prove that, given any contact 3-manifold, there is a lower bound of $2χ+3σ$ for all its minimal symplectic fillings. This is motivated by the bound of Stipsicz for Stein fillings. Finally, we discuss various aspects of exact self cobordisms of fillable 3-manifolds.

math.SG↗
arXiv2016

Calabi-Yau Caps, Uniruled Caps and Symplectic Fillings

We introduce symplectic Calabi-Yau caps to obtain new obstructions to exact fillings. In particular, it implies that any exact filling of the standard unit cotangent bundle of a hyperbolic surface has vanishing first Chern class and has the same integral homology and intersection form as its disk cotangent bundle. This gives evidence to a conjecture that all of its exact fillings are diffeomorphic to the disk cotangent bundle. As a result, we also obtain the first infinitely family of Stein fillable contact 3-manifolds with uniform bounds on the Betti numbers of its exact fillings but admitting minimal strong fillings of arbitrarily large $b_2$. Moreover, we introduce the notion of symplectic uniruled/adjunction caps and uniruled/adjunction contact structures to present a unified picture to the existing finiteness results on the topological invariants of exact/strong fillings of a contact 3-manifold. As a byproduct, we find new classes of contact 3-manifolds with the finiteness property and extend Wand's obstruction of planar contact 3-manifolds to uniruled/adjunction contact structures with complexity zero. structures with complexity zero.

math.SG↗
arXiv2015

Dehn twists exact sequences through Lagrangian cobordism

In this paper we introduce the following new ingredients: (1) rework on part of the Lagrangian surgery theory; (2) constructions of Lagrangian cobordisms on product symplectic manifolds; (3) extending Biran-Cornea Lagrangian cobordism theory to the immersed category. As a result, we manifest Seidel's exact sequences (both the Lagrangian version and the symplectomorphism version), as well as Wehrheim-Woodward's family Dehn twist sequence (including the codimension-1 case missing in the literature) as consequences of our surgery/cobordism constructions. Moreover, we obtain an expression of the autoequivalence of Fukaya category induced by Dehn twists along Lagrangian $\mathbb{RP}^n$, $\mathbb{CP}^n$ and $\mathbb{HP}^n$, which matches Huybrechts-Thomas's mirror prediction of the $\mathbb{CP}^n$ case modulo connecting maps. We also prove the split generation of any symplectomorphism by Dehn twists in $ADE$-type Milnor fibers.

math.SG↗
arXiv2014

Symplectic Divisorial Capping in Dimension 4

We investigate the notion of symplectic divisorial compactification for symplectic 4-manifolds with either convex or concave type boundary. This is motivated by the notion of compactifying divisors for open algebraic surfaces. We give a sufficient and necessary criterion, which is simple and also works in higher dimensions, to determine whether an arbitrarily small concave/convex neighborhood exist for an $ω$-orthogonal symplectic divisor (a symplectic plumbing). If deformation of symplectic form is allowed, we show that a symplectic divisor has either a concave or convex neighborhood whenever the symplectic form is exact on the boundary of its plumbing. As an application, we classify symplectic compactifying divisors having finite boundary fundamental group. We also obtain a finiteness result of fillings when the boundary can be capped by a symplectic divisor with finite boundary fundamental group.

math.SG↗