Algebra and Geometry Seminar

USC, Kapraelian Hall, rm 414

The amoeba and coamoeba of a subvariety $Z \subset (\bC^\times)^n$ are the images under its projections to $\bR^n$ and $T^n$, respectively. In this talk we discuss joint work with Chris Kuo studying the coamoebae of Lagrangian submanifolds of $(\bC^\times)^n$, specifically how the combinatorics of their degenerations encodes the homological algebra of mirror coherent sheaves. Concretely, we associate to a free resolution $F^\bullet$ of a coherent sheaf on $(\bC^\times)^n$ a tropical Lagrangian coamoeba $T(F^\bullet)$, a certain simplicial complex in $T^n$. We show that the discrete information in $F^\bullet$ can be recovered from $T(F^\bullet)$ in common situations, and that in general there is a constructible sheaf supported on $T(F^\bullet)$ which is mirror to the coherent sheaf in the relevant sense. This sheaf can be interpreted as a singular Lagrangian brane supported on the stratified conormal bundle of $T(F^\bullet)$, and in some cases can be expressed as a degeneration of smooth Lagrangian branes. The resulting interplay between coherent sheaves on~$(\bC^\times)^n$ and simplicial complexes in $T^n$ provides a higher-dimensional generalization of the spectral theory of dimer models in~$T^2$, as well as a symplectic counterpart to the theory of brane brick models.