On the Uniqueness of Kähler-Einstein Polygons in Mutation-Equivalence Classes

Abstract

We study a subclass of Kähler-Einstein Fano polygons and how they behave under mutation. The polygons of interest are Kähler-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 Kähler-Einstein Fano triangles are mutation-equivalent, then they are isomorphic. Finally, we show that if a symmetric Fano polygon is mutation-equivalent to a Kähler-Einstein triangle, then they are isomorphic. Thus, each mutation-equivalence class has at most one Fano polygon which is either a Kähler-Einstein triangle or symmetric. A recent conjecture states that all Kähler-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 Kähler-Einstein property is not preserved by the barycentric transformation, and (b) $P$ is of strict type $B_2$. Finally, we find examples of Kähler-Einstein Fano polygons which are not minimal.

Figures (8)

• Figure 1: In red: the polygon $\mathop{\mathrm{conv}}\nolimits\left\{ \pm u_1, \pm u_2 \right\}$. In grey: the regions which can be ruled out from $h_2 P^*$ given that their apexes are not in $h_2 P^*$.
• Figure 2: In red: the polygon $\mathop{\mathrm{conv}}\nolimits\left\{ u,uG,uG^2 \right\}$. In grey: the regions which can be ruled out from $hP^*$ given that their apexes are not in $hP^*$.
• Figure 3: The six symmetric Fano polygons with $n$ primitive T-singularities and empty basket $\mathcal{B} = \varnothing$.
• Figure 4: An example to demonstrate Lemma \ref{['lemma:sym_det_fixed']}. Here, $E = \mathop{\mathrm{conv}}\nolimits\left\{ (2,-3), (1,0) \right\}$ and $H = \left(1304\right)$. The red points are the three choices for the second vertex $\bm{v}_2$ of $F$; the only valid choice for $\bm{v}_2$ is $(-1,4)$.
• Figure 5: An example with $n=2$ and $P$ is assumed to be $3$-symmetric. The points $\bm{v}_1$ and $\bm{v}_2$ are the only two vertices of $P$ below the $x$-axis. One of these must also be a vertex of the triangle $T$.

Theorems & Definitions (105)

• Theorem 1.1: minimality
• Theorem 1.2: bat_sym, on_KEs
• Conjecture 1.3: on_KEs
• Theorem 1.4
• Proposition 1.5
• Proposition 1.6: see Proposition \ref{['prop:KE_nonsym_from_3sym']}
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.4