Fillable contact structures on torus bundles over circles (Q2759001)

The author provides an explicit construction of a (strongly) symplectic filling for the \(T^2\)-bundle over \(S^1\) with parabolic monodromy \(\left(\begin{smallmatrix} 1 & n\\ 0 & 1\end{smallmatrix}\right)\), where \(n\) is a natural number. In the language of Thurston geometries, such manifolds arise as orbit spaces of the Lie group Nil\(^3\).NEWLINENEWLINE For \(n=1\) the filling is diffeomorphic to the neighbourhood of a singular fibre of type I\(_1\) in an elliptic fibration, and the symplectic structure is chosen so that the elliptic fibration becomes a singular Lagrangian fibration with singular fibre corresponding to a so-called `focus-focus' singularity.NEWLINENEWLINEFor \(n>1\) one must carry out a similar construction on an \(n\)-fold covering space.NEWLINENEWLINE The reviewer would like to point out that even more explicit examples are provided by the Brieskorn varieties \(V^3(a_0,a_1,a_2)\), where \(1/a_0 + 1/a_1 + 1/a_2 = 1\). These are known to be nilmanifolds, with a fillable contact structure defined analogously to the standard contact structure on \(S^3\). However these examples will not cover the more general construction for solvmanifolds promised by the author for a future paper.