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Small symplectic $4$-manifolds via contact gluing and some applications

Weimin Chen

TL;DR

The paper develops a streamlined contact-gluing approach to build small symplectic 4-manifolds by composing convex fillings with concave caps, adapting Gay's technique to ${\\mathbb S}^1$-invariant Seifert data. It constructs explicit caps $V_0$ with concave boundaries and combines them with fillings $W_0$ to produce closed manifolds such as ${\\mathbb C}\\mathbb P^2\\#\\overline{\\mathbb C}\\mathbb P^2}$ and ${\\mathbb S}^2\\times\\mathbb S^2$, and to realize embedded singular Lagrangian ${\\mathbb R}\\mathbb P^2$ and Lagrangian Klein bottles. The work develops the theory of ${\\mathbb S}^1$-invariant contact structures, rational open books with periodic monodromy, and relative 2-handle attachments, enabling precise control of boundary Seifert data and canonical class signs. It then relates these constructions to algebraic geometry via rational unicuspidal curves, Milnor fillings, and Zariski-type questions, establishing bounds on self-intersections and producing infinite families of lens spaces embedded as hypersurfaces of contact type in symplectic surfaces. Overall, the paper provides a robust framework connecting contact geometry, symplectic caps, and complex-analytic singularities to generate and classify small, sometimes exotic, symplectic 4-manifolds and their fillings.

Abstract

In this paper, we introduce a streamlined procedure for constructing small symplectic $4$-manifolds via contact gluing, based on a technique invented by David Gay around 2000. We also give a few applications, ranging from embeddings of singular Lagrangian $RP^2$s to realizing an infinite family of lens spaces as a hypersurface of contact type in a symplectic Hirzebruch surface. Furthermore, our investigations on $S^1$-invariant contact structures also suggest an interesting and fairly strong upper bound for the self-intersection of a rational unicuspidal curve with one Puiseux pair $(p,q)$ in any algebraic surface (the bound depends only on the values of $(p,q)$).

Small symplectic $4$-manifolds via contact gluing and some applications

TL;DR

The paper develops a streamlined contact-gluing approach to build small symplectic 4-manifolds by composing convex fillings with concave caps, adapting Gay's technique to -invariant Seifert data. It constructs explicit caps with concave boundaries and combines them with fillings to produce closed manifolds such as and , and to realize embedded singular Lagrangian and Lagrangian Klein bottles. The work develops the theory of -invariant contact structures, rational open books with periodic monodromy, and relative 2-handle attachments, enabling precise control of boundary Seifert data and canonical class signs. It then relates these constructions to algebraic geometry via rational unicuspidal curves, Milnor fillings, and Zariski-type questions, establishing bounds on self-intersections and producing infinite families of lens spaces embedded as hypersurfaces of contact type in symplectic surfaces. Overall, the paper provides a robust framework connecting contact geometry, symplectic caps, and complex-analytic singularities to generate and classify small, sometimes exotic, symplectic 4-manifolds and their fillings.

Abstract

In this paper, we introduce a streamlined procedure for constructing small symplectic -manifolds via contact gluing, based on a technique invented by David Gay around 2000. We also give a few applications, ranging from embeddings of singular Lagrangian s to realizing an infinite family of lens spaces as a hypersurface of contact type in a symplectic Hirzebruch surface. Furthermore, our investigations on -invariant contact structures also suggest an interesting and fairly strong upper bound for the self-intersection of a rational unicuspidal curve with one Puiseux pair in any algebraic surface (the bound depends only on the values of ).

Paper Structure

This paper contains 9 sections, 49 theorems, 154 equations.

Table of Contents

  1. Introduction and the main results
  2. ${\mathbb S}^1$-invariant contact structures and transverse surgery
  3. Rational open books with periodic monodromy
  4. Relative handlebodies of symplectic $2$-handles on a Seifert manifold
  5. The sign of the canonical class
  6. Rational unicuspidal curves with one Puiseux pair
  7. Embeddings of singular Lagrangian ${\mathbb R}{\mathbb P}^2$s
  8. Symplectic fillings of the small Seifert space $M((13,6), (3,2), (2,-1))$
  9. Embeddings of other Stein surfaces and an infinite family of lens spaces

Key Result

Theorem 1.1

The $4$-manifold $V_0$ admits a symplectic structure with a concave contact boundary $(M_0,\xi_{inv})$, where $\xi_{inv}$ is a certain ${\mathbb S}^1$-invariant, tight contact structure on the small Seifert space $M_0=M((2,1),(2,-1), (2,-1))$. Furthermore, the following properties hold true: As a corollary, $W_0$ can be realized as a compact domain with convex contact boundary in both a symplecti

Theorems & Definitions (90)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Theorem 1.9
  • Corollary 1.10
  • Theorem 1.11
  • ...and 80 more