Bott vanishing for algebraic surfaces
Bott vanishing for algebraic surfaces
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DOI：
10.1090/tran/8045
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发表时间：
2018-12
影响因子：
1.3
通讯作者：
B. Totaro
中科院分区：
文献类型：
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作者：
B. Totaro
Bott证明了射影空间上的层上同调的一个强消去定理。它适用于多年生品种，但不适用于大多数其他品种。我们证明了五次del Pezzo曲面，也称为零亏格的五点稳定曲线的模空间M_(0，5)^bar的Bott为零。这是第一个非Toric Fano变种，回答了Achinger，Witaszek和Zdanowicz提出的问题。在另一个方向上，我们证明了对于许多K3曲面，包括20次或至少24次的非常一般的K3曲面，Bott是零的。这是建立在Beauville和Mukai关于K3曲面的模空间的工作的基础上的。准确地确定哪些K3曲面满足Bott消失将是一件有趣的事情。
Bott proved a strong vanishing theorem for sheaf cohomology on projective space. It holds for toric varieties, but not for most other varieties. We prove Bott vanishing for the quintic del Pezzo surface, also known as the moduli space M_{0,5}^bar of 5-pointed stable curves of genus zero. This is the first non-toric Fano variety for which Bott vanishing has been shown, answering a question by Achinger, Witaszek, and Zdanowicz. In another direction, we prove Bott vanishing for many K3 surfaces, including very general K3 surfaces of degree 20 or at least 24. This builds on Beauville and Mukai's work on moduli spaces of K3 surfaces. It would be interesting to determine exactly which K3 surfaces satisfy Bott vanishing.