The monodromy groups of Dolgachev's CY moduli spaces are Zariski dense☆
The monodromy groups of Dolgachev's CY moduli spaces are Zariski dense☆
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DOI：
10.1016/j.aim.2014.12.017
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发表时间：
2014-07
影响因子：
1.7
通讯作者：
Mao Sheng;Jinxing Xu;K. Zuo
中科院分区：
文献类型：
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作者：
Mao Sheng;Jinxing Xu;K. Zuo
设Mn，2 n+ 2是CY流形的粗模空间，它是由Pn沿沿着2 n+ 2超平面分支的双覆盖在一般位置上的一个收缩分解产生的。证明了当n≥ 3时，Mn，2n + 2的好族的单值群在相应的辛群或正交群中是Zerkki稠密的.特别地，当n≥ 3时，周期映射不给出粗模空间作为Shimura簇的任何部分紧化的一致化。这就推翻了多尔加乔夫的一个猜想。因此，基本群的粗模空间的m有序点在Pn被证明是大的，一旦它不是一个点。对于由Pn沿沿着m个超平面在一般位置分支的循环覆盖所产生的CY流形的模空间，得到了类似的Zerkiki-密度结果.对B的几何实现问题进行了分类。给出了A型有界对称域的Gross.
Abstract Let M n, 2 n+ 2 be the coarse moduli space of CY manifolds arising from a crepant resolution of double covers of P n branched along 2 n+ 2 hyperplanes in general position. We show that the monodromy group of a good family for M n, 2 n+ 2 is Zariski dense in the corresponding symplectic or orthogonal group if n≥ 3. In particular, the period map does not give a uniformization of any partial compactification of the coarse moduli space as a Shimura variety whenever n≥ 3. This disproves a conjecture of Dolgachev. As a consequence, the fundamental group of the coarse moduli space of m ordered points in P n is shown to be large once it is not a point. Similar Zariski-density result is obtained for moduli spaces of CY manifolds arising from cyclic covers of P n branched along m hyperplanes in general position. A classification towards the geometric realization problem of B. Gross for type A bounded symmetric domains is given.