发表机构
Northwestern University（西北大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明仿射对数Calabi--Yau簇的特殊骨架不依赖紧化选择且在自同构下不变，刻画了有限生成赋值为特殊的充要条件，并通过对Markov三次曲面补集的分析给出作者与Xu猜想的一个反例。
AI 中文摘要
设$U$是一个仿射对数Calabi--Yau簇。尽管特殊赋值和有限生成赋值是通过$U$的对数CY-Fano紧化来定义的，我们证明了在复形中对应对应的骨架不依赖于该紧化的选择，因此它们在$\mathrm{Aut}(U)$下不变。我们还证明了有限生成赋值是特殊的当且仅当它相对于由$U$上的正则函数诱导的自然偏序是极大的。接着我们考虑由$\mathbb{A}^3$中Markov三次曲面的补集得到的仿射对数Calabi--Yau三维簇。我们描述了三个Vieta对合在特殊骨架上的作用，并将其与双曲平面上的$(\infty,\infty,\infty)$-三角形反射群联系起来。我们还证明了，对于对偶复形的每个有限三角剖分，特殊骨架在某个单纯形上不是局部闭的。通过锥构造，这为作者和Xu的一个猜想提供了反例。
英文摘要
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.
Comments53 pages, 4 figures. Comments are welcome
