We discuss joint work with Chris Kuo and earlier work with David Treumann and Eric Zaslow on which it builds. This earlier work gave a combinatorial description of how homological mirror symmetry acts on one-dimensional coherent sheaves on a toric surface. Namely, it takes them to Lagrangians which are perturbations of the conormals to a suitable bipartite graph in $T^2$. The map on moduli is implemented by the Kasteleyn operator of the graph, hence is governed by the enumeration of perfect matchings. In current work we show that such a description holds more generally for coherent sheaves of codimension one on a toric variety of dimension $n \geq 2$, now involving bipartite graphs in $T^n$.
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