The Period Lattice for Enriques
The Period Lattice for Enriques
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恩里克斯的周期格
DOI：
10.1007/978-3-662-03073-8
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
S. Allcock
中科院分区：
文献类型：
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作者：
S. Allcock
众所周知，复Enrique曲面的同构类与Zariski-开子集(D？H)=？关于O(2；10)的厄米特对称空间D的商。这里H是D和？中的全测地因子。是一个特定的算术群。在这一结果的通常表述中5]，？被描述为签名(2；10)的某个整格N的等距群。此晶格相当复杂，需要复杂的技术才能使用。这个注记的目的是用简单得多的格I 2；10来代替N，即签名(2；10)的唯一奇么模格。这使得关于N的几个论点可以戏剧性地简化，用基本事实取代复杂的分析。例如，在此设置中，很容易看到H=？D=？是不可约的，而且D=？的Satake紧化的边界分支也很容易计数。用I2；10代替N也可以证明(D？h)=？具有可收缩的万能盖子。这些结果中的最后一个是新的，完整的证明出现在1]中；这里我们只给出主要思想。这篇论文的基础是格子理论技巧，这对从事格子工作的人来说是众所周知的；然而，它在这种背景下的应用似乎以前没有发表过。我们回顾了5]中的一些符号和事实。U表示具有内积矩阵？0 1 1 0的二维格子。如果A是格，则A(N)表示A的一个复本，A的所有内积都乘以n.一个Enrique曲面S有基本群Z=2，它的泛覆盖~S是K3曲面.覆盖变换{作用于L=H2(S；Z)=E8(？1)2U3，其并格M在L中是本原的，与E8(？2)U(2)同构.结果是N=M？与E_8(？2)U(2)U同构，则存在全纯2-形式！在~S上，对乘法常数唯一，且满足{(！)=？！这意味着作为H2(S；C)=L C的一个元素，它位于NC中。满足等式h！j！i=0和不等式h！j！i>；0，其中hj！i表示…
It is well-known that the isomorphism classes of complex Enriques surfaces are in 1-1 correspondence with a Zariski-open subset (D ? H)=? of the quotient of the Hermitian symmetric space D for O(2; 10). Here H is a totally geodesic divisor in D and ? is a certain arithmetic group. In the usual formulation of this result 5], ? is described as the isometry group of a certain integral lattice N of signature (2; 10). This lattice is quite complicated, and requires sophisticated techniques to work with. The purpose of this note is to replace N by the much simpler lattice I 2;10 , the unique odd unimodular lattice of signature (2; 10). This allows for dramatic simpliications in several arguments concerning N, replacing intricate analysis by elementary facts. For example, in this setting it is easy to see that H=? D=? is irreducible, and also easy to enumerate the boundary components in the Satake compactiication of D=?. Using I 2;10 in place of N also allows one to show that (D?H)=? has contractible universal cover. The last of these results is new, and the full proof appears in 1]; here we only give the main idea. The basis of this paper is a lattice-theoretic trick which is well-known to those who work with lattices; however, its applications in this setting do not appear to have been published before. We review some notation and facts from 5]. U denotes the two-dimensional lattice with inner product matrix ? 0 1 1 0. If A is a lattice then A(n) denotes a copy of A with all inner products multiplied by n. An Enriques surface S has fundamental group Z=2 and its universal cover ~ S is a K3 surface. The covering transformation { acts on L = H 2 (~ S; Z) = E 8 (?1) 2 U 3 , and its xed lattice M is primitive in L and isomorphic to E 8 (?2) U(2). It turns out that N = M ? is isomorphic to E 8 (?2)U(2)U. There is a holomorphic 2-form ! on ~ S, unique up to a multiplicative constant, and it satisses {(!) = ?!. This implies that as an element of H 2 (S; C) = L C , it lies in N C. Furthermore, ! satisses the equality h!j!i = 0 and the inequality h!j !i > 0, where h j i denotes …