Special valuations and automorphisms of affine log Calabi--Yau varieties

Source abstract

Let $U$ be an affine log Calabi--Yau variety. Although special and finitely generated valuations are defined using a log CY-Fano compactification of $U$, we prove that the corresponding skeleta in the dual complex are independent of this choice, and hence invariant under $\mathrm{Aut}(U)$. We also show that a finitely generated valuation is special if and only if it is maximal with respect to a natural partial order induced by regular functions on $U$. We then consider the affine log Calabi--Yau threefold obtained as the complement of the Markov cubic surface in $\mathbb{A}^3$. We describe the action of the three Vieta involutions on the special skeleton and relate it to the $(\infty,\infty,\infty)$-triangle reflection group on the hyperbolic plane. We also show that, for every finite triangulation of the dual complex, the special skeleton fails to be locally closed on some simplex. Via the cone construction, this provides a counterexample to a conjecture of the author and Xu.