The Cone Conjecture for Some Calabi-Yau Varieties
The Cone Conjecture for Some Calabi-Yau Varieties
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一些 Calabi-Yau 品种的锥猜想
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发表时间：
2017
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通讯作者：
Bjørn Skauli
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作者：
Bjørn Skauli
在这篇论文中，我们介绍了最小模型程序和Morison-Kawamata锥猜想。然后，我们给出了一些关于反正则超曲面的已知结果，这些结果与Kollár在[Bor91，附录]中证明的结果相一致，将反正则超曲面的nef锥与环境簇的nef锥联系起来。我们将详细介绍这一点的证明，包括定义和必要的背景材料，以遵循证明。我们对几个背景结果给出了我们自己的证明，包括用环面几何做的两个证明。然后举例说明了Kolár结果的可能推广是如何失败的。这里的大多数例子都是新的，除了在[HLW02]中找到的一个例子。对于这个例子，我们给出了我们自己的新描述。然后，我们继续证明关于Enrique Calabi-Yau上的nef锥的Morison-Kawamata锥猜想，以及更一般地关于作为某些有限商数出现的Calabi-Yau变种的猜想。这在[OS01]中已经得到了证明，我们使用这篇论文作为背景材料，但我们对锥形猜想的证明是基于不同的想法，尽管两者最终都依赖于Torelli定理。然后，我们研究了射影空间乘积中的某些Calabi-Yau完全交，特别证明了活动锥的Morison-Kawamata锥猜想。这推广了[CO15]中的结构和结果。这里的结果和证明是新的，尽管许多想法是受到[CO15]和[Ogu14]的启发。最后，我们证明了反正则超曲面在一点和两点爆破时关于动锥的Morison-Kawamata锥猜想，并计算了这些Calabi-Yau簇的双态自同构群。据我们所知，这最后一节的结果是新的。
In this thesis we give an introduction to the Minimal Model Program and the Morrison-Kawamata cone conjecture. We then present some known results on anticanonical hypersurfaces building up to the result proven by Kollár in [Bor91, Appendix] relating the nef cone of an anticanonical hypersurface to the nef cone of the ambient variety. We will present the proof of this in detail including the definitions and necessary background material to follow the proof. We give our own proofs for several of the background results including two proofs done by toric geometry. Then we give examples illustrating how possible extension of Kollár’s result fail. Most of the examples here are new, except for one example found in [HLW02]. For this example we give our own new description. We then move on to proving the Morrison-Kawamata cone conjecture for the nef cone on the Enriques Calabi-Yau, and more generally on Calabi-Yau varieties arising as certain finite étale quotients. This is already proven in [OS01], and we use this paper for background material, but our proof of the cone conjecture is based on a different idea, although both ultimately rely on the Torelli theorem. We then study certain Calabi-Yau complete intersections in products of projective spaces and in particular prove the Morrison-Kawamata cone conjecture for the movable cone. This generalizes the constructions and results in [CO15]. Here the results and proofs are new, although many ideas are inspired by [CO15] and [Ogu14]. Finally we prove the Morrison-Kawamata cone conjecture for the movable cone for an anticanonical hypersurface in the blowup of P in one and two points, and compute the birational automorphism group for these Calabi-Yau varieties. To our knowledge the results of this final section are new.