Holomorphic curves in Stein domains and the tau-invariant

TL;DR

The paper extends Hedden's $\tau$-invariant to Stein-fillable contact 3-manifolds by relating $2\tau_\xi(T) - |T|$ to the rational self-linking of transverse links bounding holomorphic curves in Stein fillings, providing a new relative Thom conjecture in this setting. It clarifies and interrelates multiple Heegaard Floer $\tau$-invariants ($\tau_\mathfrak{s}$, $\tau_\xi$, $\tau_\theta$) via Alexander filtrations and cobordism maps, and uses these to derive obstructions to holomorphic fillability, as well as sharp slice-Bennequin inequalities. The authors then apply lattice cohomology to compute $\tau_\mathfrak{s}$-invariants for certain links in lens spaces and to bound the piecewise-linear slice genus, producing concrete examples (notably in $L(4,1)$ and $L(9,2)$) that obstruct rational and PL concordance to links in $S^3$. Overall, the work furnishes both new theoretical links between contact/holomorphic geometry and Floer theory and practical invariants that obstruct holomorphic fillings and constrain slice genera in rational homology 4-balls.

Abstract

The scope of the paper is threefold. First, we build on recent work by Hayden to compute Hedden's tau-invariant $τ_ξ(L)$ in the case when $ξ$ is a Stein fillable contact structure on a rational homology sphere, and $L$ is a transverse link arising as the boundary of a pseudo-holomorphic curve. This leads to a new proof of the relative Thom conjecture for Stein domains. Secondly, we compare the invariant $τ_ξ$ to the Grigsby-Ruberman-Strle topological tau-invariant $τ_{\mathfrak s}$, associated to the $\text{Spin}^c$-structure $\mathfrak s=\mathfrak s_ξ$ of the contact structure $ξ$, to obtain topological obstructions for a link type to admit a holomorphically fillable transverse representative. Finally, we use our main result together with methods from lattice cohomology to compute the $τ_{\mathfrak s}$-invariants of certain links in lens spaces, and estimate their PL slice genus.

Figures (10)

• Figure 1: [1]Left: the $2d$-component link $L_{2d}$ in $L(4,1)$; right: the $d$-component torus link $T_{d,d}$ in $S^3$.
• Figure 2: [1]The link $M_{3d}$ in $L(9,2)$. The $-1$ denotes a negative full-twist. There are $d$ blue, $d$ purple and $d$ green components for a total of $3d$ unframed components.
• Figure 3: [1]Attaching positively twisting bands in a Bennequin surface. The picture is taken from Hayden.
• Figure 4: [1]The surface on the right-hand side is obtained by removing a negative elliptic point.
• Figure 5: [1]The characteristic foliation after removing the ribbon singularity.

Theorems & Definitions (55)

• Remark 1.2
• Theorem 1.3
• Theorem 1.4: Relative Thom conjecture for Stein fillings
• Proposition 1.5
• Corollary 1.6
• Theorem 1.7
• Corollary 1.8
• Corollary 1.9
• Theorem 1.10
• Proposition 1.11