[Paper Review] The Kodaira dimension of contact 3-manifolds and geography of symplectic fillings

Tianjun Li, Cheuk Yu Mak|arXiv (Cornell University)|Oct 21, 2016

Geometric and Algebraic Topology26 references3 citations

TL;DR

This paper introduces the Kodaira dimension for contact 3-manifolds using symplectic caps, establishing a classification that governs the geography of symplectic fillings. It proves a universal lower bound of $2\chi + 3\sigma$ for all minimal strong symplectic fillings, extending Stipsicz's result for Stein fillings, and shows that exact self-cobordisms of fillable 3-manifolds are bounded below by 0, with non-trivial monoids possible even for uniruled and Calabi-Yau cases.

ABSTRACT

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.

Motivation & Objective

• To define and study the Kodaira dimension of contact 3-manifolds using symplectic caps, particularly uniruled and Calabi-Yau caps.
• To establish a comprehensive geography of symplectic fillings based on Kodaira dimension, generalizing Stipsicz's conjecture.
• To prove a universal lower bound on $2\chi + 3\sigma$ for all minimal strong symplectic fillings of any contact 3-manifold.
• To investigate the structure of exact self-cobordisms and their monoids, especially in relation to Heegaard Floer homology.

Proposed method

• Introduces the Kodaira dimension of a contact 3-manifold via the existence of uniruled or Calabi-Yau caps, defining $Kod = -\infty$, $0$, or $1$ accordingly.
• Uses the notion of maximal surfaces in symplectic 4-manifolds to bound exceptional curves and constrain filling invariants.
• Applies Donaldson hypersurfaces and polarized symplectic caps to construct a universal minimal symplectic cap for any minimal filling.
• Employs the Lefschetz duality and rational period condition to define Donaldson caps as duals to the cohomology class of the cap's symplectic form and contact form.
• Constructs explicit cobordisms via Weinstein handle attachments and cancellations to generate non-trivial elements in the self-cobordism monoid.
• Uses Heegaard Floer homology to distinguish cobordisms by the rank of induced maps, showing non-triviality of constructed cobordisms.

Experimental results

Research questions

• RQ1 Does the symplectic filling version of Stipsicz’s conjecture hold for all contact 3-manifolds with Kodaira dimension $-\infty$?
• RQ2 Can a universal minimal symplectic cap be constructed for any minimal strong filling of a contact 3-manifold?
• RQ3 Is the set of $2\chi + 3\sigma$ values for minimal strong fillings bounded below, and can this bound be explicitly computed?
• RQ4 Are there non-trivial Stein self-cobordisms of Stein fillable contact 3-manifolds that induce isomorphisms on Heegaard Floer homology?
• RQ5 Can the self-cobordism monoid of a contact 3-manifold be non-trivial even when its Stein fillings are unique up to symplectic deformation?

Key findings

• The set of $2\chi + 3\sigma$ values for minimal strong symplectic fillings of any contact 3-manifold is bounded from below, generalizing Stipsicz’s result for Stein fillings.
• For any contact 3-manifold, there exists a universal symplectic cap such that the union with any minimal filling yields a minimal closed symplectic 4-manifold.
• The lower bound on $2\chi + 3\sigma$ is explicitly calculable from a polarized symplectic cap, providing a topological obstruction applicable to all minimal fillings.
• There exist infinitely many Stein fillable uniruled contact 3-manifolds whose Stein self-cobordism monoids are non-trivial and have arbitrarily large finite cardinality.
• Non-trivial elements in the self-cobordism monoid can be constructed via sequences of $1$- and $2$-handle attachments and cancellations, with distinct cobordisms distinguished by relative homology groups.
• The induced map on Heegaard Floer homology of the constructed cobordisms has rank $2^l$, proving non-triviality and showing that such cobordisms do not induce isomorphisms.