Holomorphic curves in Stein domains and the tau-invariant

The scope of the paper is threefold. First, we build on recent work by Hayden to compute Hedden’s tau-invariant ${\tau }_{\xi }(L)$ in the case when $\xi$ 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 ${\tau }_{\xi }$ to the Grigsby-Ruberman-Strle topological tau-invariant ${\tau }_{\mathfrak{s}}$, associated to the ${\text{Spin}}^c$-structure $\mathfrak{s}={\mathfrak{s}}_{\xi }$ of the contact structure $\xi$, 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 ${\tau }_{\mathfrak{s}}$-invariants of certain links in lens spaces, and estimate their PL slice genus.