On the Period Map for Polarized Hyperkähler Fourfolds
On the Period Map for Polarized Hyperkähler Fourfolds
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偏振超克勒四倍周期图
DOI：
10.1093/imrn/rnx333
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发表时间：
2018
影响因子：
1
通讯作者：
Macrì, Emanuele
中科院分区：
文献类型：
--
作者：
Debarre, Olivier;Macrì, Emanuele
我们研究了K3曲面的Hilbert正方形的变形的光滑射影Hyperkähler四重曲面，它们具有固定的阶可除性极化。它们被一个拟射影不可约20维模空间参数化，Verbitksy的Torelli定理暗示它们的周期映射是开嵌入的。我们的主要结果是周期映射的像的补集是我们所描述的显式Heegner因子的有限并。我们还证明了给定周期空间中无限多Heegner因子的一般点对应于四重同构于K3曲面的Hilbert平方或双EPW(Eisenbud-Popescu-Walter)六重。在两个附录中，我们确定了Picard数为1或2的各种射影超kähler四重环的双正则自同构群或双正则自同构群。
We study smooth projective hyperkähler fourfolds that are deformations of Hilbert squares of K3 surfaces and are equipped with a polarization of fixed degree and divisibility. They are parametrized by a quasi-projective irreducible 20-dimensional moduli space and Verbitksy’s Torelli theorem implies that their period map is an open embedding. Our main result is that the complement of the image of the period map is a finite union of explicit Heegner divisors that we describe. We also prove that infinitely many Heegner divisors in a given period space have the property that their general points correspond to fourfolds which are isomorphic to Hilbert squares of a K3 surfaces, or to double EPW (Eisenbud–Popescu–Walter) sextics. In two appendices, we determine the groups of biregular or birational automorphisms of various projective hyperkähler fourfolds with Picard number 1 or 2.