On the Uniqueness of K\"ahler-Einstein Polygons in Mutation-Equivalence Classes

We study a subclass of K\"ahler-Einstein Fano polygons and how they behave under mutation. The polygons of interest are K\"ahler-Einstein Fano triangles and symmetric Fano polygons. In particular, we find an explicit bound for the number of these polygons in an arbitrary mutation-equivalence class. An important mutation-invariant of a Fano polygon is its singularity content. We extend the notion of singularity content and prove that it is still a mutation-invariant. We use this to show that if two symmetric Fano polygons are mutation-equivalent, then they are isomorphic. We further show that if two K\"ahler-Einstein Fano triangles are mutation-equivalent, then they are isomorphic. Finally, we show that if a symmetric Fano polygon is mutation-equivalent to a K\"ahler-Einstein triangle, then they are isomorphic. Thus, each mutation-equivalence class has at most one Fano polygon which is either a K\"ahler-Einstein triangle or symmetric. A recent conjecture states that all K\"ahler-Einstein Fano polygons are either triangles or are symmetric. We provide a counterexample $P$ to this conjecture and discuss several of its properties. For instance, we compute iterated barycentric transformations of $P$ and find that (a) the K\"ahler-Einstein property is not preserved by the barycentric transformation, and (b) $P$ is of strict type $B_2$. Finally, we find examples of K\"ahler-Einstein Fano polygons which are not minimal.