Symmetric and Kähler–Einstein Fano polygons

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We investigate singular symmetric and Kähler–Einstein Fano polytopes. More precisely, we show that every symmetric Fano polytope is Kähler–Einstein generalizing the work by Batyrev and Selivanova, and study the automorphism groups of symmetric and Kähler–Einstein Fano polygons in detail. In particular, every finite subgroup of G L 2 ( Z ) GL_2(\mathbb {Z}) is an automorphism group of a Kähler–Einstein Fano polygon.