We investigate symmetric and Kähler–Einstein Fano polytopes mainly focusing on the two dimensional singular case. In particular, we construct some examples of symmetric but not Kähler–Einstein singular Fano polygons which cannot exist in the smooth case. In fact, we completely classified all such cases. We also show that symmetric and Kähler–Einstein Fano polygon admits a non-trivial rotation and construct examples of a Fano polygon that isKähler–Einstein but not symmetric.
To understand more about Kähler–Einstein Fano polygons and symmetric Fano polygons, we introduce a new notion called the B-transformation of a Fano polygon.
In particular, if a Fano polygon is symmetric or Kähler–Einstein then the B-transformation of the Fano polygon is also a Fano polygon.
Then, we show that the property of being symmetric for Fano polygons is invariant under the B-transformation.
