Kanda classified tight contact structures on the 3-torus. Which of them is Stein fillable? Is there any good reference?
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Eliashberg proved only one of them is Stein fillable (and in fact it's the only one that's strongly fillable). The paper is quite short.
Y. Eliashberg, Unique holomorphically fillable contact structure on the 3-torus, Internat. Math. Res. Notices (1996), no. 2, 77–82.
I also highly recommend Chris Wendl's 2014 series of blog posts surveying symplectic fillings. The fifth entry in the series specifically addresses the question you ask (assuming you know that all of the tight contact structures on $T^3$ are those given by $(T^3,\xi_k)$). I would be remiss if I didn't also mention Wendl's paper, which further proves that there is a unique minimal strong filling (up to the proper notion of equivalence), which is the Stein filling $D^*T^2 = T^2 \times D^2$ you know and love.
Wendl, Chris, Strongly fillable contact manifolds and $J$-holomorphic foliations. Duke Math. J. 151 (2010), no. 3, 337–384.
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$\begingroup$ These are all stated up to contactomorphism. Up to contact isotopy, there are infinitely distinct many tight contact structures that are all contactomorphic to the standard Stein fillable one. $\endgroup$ Commented Nov 1, 2021 at 19:34
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$\begingroup$ Because contact isotopy generates contactomorphisms, we have that up to contact isotopy, tight contact structures up to isotopy are in bijection with $MCG(T^3)/\mathrm{Stab}(\xi_0)$. On the one hand, $MCG(T^3) = GL_3(\mathbb{Z})$. On the other hand, given $\xi_0$, $T^3$ comes with a canonical generator $\gamma \in H_1(T^3)$ coming from representing the unique class nulhomologous with respect to the Stein filling, and $\mathrm{Stab}(\xi_0)$ consists precisely of those elements fixing this class, i.e. $\mathrm{Stab}(\xi_0) = \mathbb{Z}^2 \rtimes \mathrm{GL}_2(\mathbb{Z})$. $\endgroup$– KSackelCommented Nov 1, 2021 at 20:53
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$\begingroup$ To clarify, the quotient is therefore just in bijection with primitive elements in $\mathbb{Z}^3$ (given by the image of $\gamma$). $\endgroup$– KSackelCommented Nov 1, 2021 at 21:20