论文标题
计算Riemann-Roch多项式和分类Hyper-Kähler四倍
Computing Riemann-Roch polynomials and classifying hyper-Kähler fourfolds
论文作者
论文摘要
我们证明，满足温和拓扑假设的Hyper-Kähler四倍是K3 $^{[2]} $变形类型。这尤其证明了O'Grady的猜想，表明K3 $^{[2]} $数值类型的Hyper-Kähler四倍是K3 $^{[2]} $变形类型的数值类型。我们的拓扑假设涉及两个积分度-2共同学类别的存在，满足了某些数值交叉条件。 证明中有两种主要成分。我们首先证明了该语句的拓扑版本，它表明我们的拓扑假设迫使Betti数字，Fujiki常数和Huybrechts-Riemann-Roch the Hyper-Kähler四倍的多项式与K3 $^{[2] $ hyper-hyper-hyper-hyper-kähllerfourdfolds相同。然后，本文的关键部分是要证明Hyper-KählerSyz的猜想是针对满足上述数值条件的分层类别的Hyper-Kähler四倍。
We prove that a hyper-Kähler fourfold satisfying a mild topological assumption is of K3$^{[2]}$ deformation type. This proves in particular a conjecture of O'Grady stating that hyper-Kähler fourfolds of K3$^{[2]}$ numerical type are of K3$^{[2]}$ deformation type. Our topological assumption concerns the existence of two integral degree-2 cohomology classes satisfying certain numerical intersection conditions. There are two main ingredients in the proof. We first prove a topological version of the statement, by showing that our topological assumption forces the Betti numbers, the Fujiki constant, and the Huybrechts-Riemann-Roch polynomial of the hyper-Kähler fourfold to be the same as those of K3$^{[2]}$ hyper-Kähler fourfolds. The key part of the article is then to prove the hyper-Kähler SYZ conjecture for hyper-Kähler fourfolds for divisor classes satisfying the numerical condition mentioned above.
