On Type IIA AdS$_3$ solutions and massive GK geometries

We give the necessary and sufficient conditions for warped AdS$_3$ (and Mink$_3$) solutions of Type II supergravities to preserve ${\cal N}=(2,0)$ supersymmetry, in terms of geometric conditions on their internal space M$_7$. Such solutions possess a canonical ten-dimensional Killing vector that can be either time-like or null. In this work we classify the null case in massive Type IIA supergravity which necessitates that M$_7$ decomposes as a circle fibration over a six-dimensional base with orthogonal SU(2)-structure containing a complex four-manifold. We narrow our focus to solutions for which M$_7$ becomes $\mathbb{T}^2$ fibred over a foliation of a K\"{a}hler manifold over an interval. We find a class of solutions which are the massive Type IIA version of GK geometries and present an extremal problem which computes the central charge of the solution using just topology. Finally, we present geometric conditions for AdS$_3$ solutions to preserve arbitrary extended chiral supersymmetry.