具有猜想最小体积的逻辑规范对

Pub Date : 2024-08-19 DOI:10.1007/s00229-024-01588-6
Louis Esser, Burt Totaro
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引用次数: 0

摘要

我们构建了对数规范对(X, B),其中 B 是一个非零还原除数,并且 \(K_X+B\) 充裕,具有已知最小的体积。我们猜想我们的例子在每个维度上都具有最小的体积。这个猜想在维度 2 中是真的,由 Liu 和 Shokurov 提出。这些例子都是非准光滑的加权投影超曲面。我们还提出了一个相关极值问题的例子。Esser 构建了一个 klt Calabi-Yau 变体,猜想它在每个维度上都有最小的 mld(例如,维度 2 中的 mld 为 1/13,维度 3 中的 mld 为 1/311)。然而,这个例子只在最多 18 维的情况下被完全证明。现在我们将证明埃塞尔的例子在所有维度上所需的性质(特别是确定其 mld)。
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Log canonical pairs with conjecturally minimal volume

We construct log canonical pairs (XB) with B a nonzero reduced divisor and \(K_X+B\) ample that have the smallest known volume. We conjecture that our examples have the smallest volume in each dimension. The conjecture is true in dimension 2, by Liu and Shokurov. The examples are weighted projective hypersurfaces that are not quasi-smooth. We also develop an example for a related extremal problem. Esser constructed a klt Calabi–Yau variety which conjecturally has the smallest mld in each dimension (for example, mld 1/13 in dimension 2 and 1/311 in dimension 3). However, the example was only worked out completely in dimensions at most 18. We now prove the desired properties of Esser’s example in all dimensions (in particular, determining its mld).

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