Analytic torsion for log-Enriques surfaces and Borcherds product
Analytic torsion for log-Enriques surfaces and Borcherds product
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DOI：
10.1017/fms.2022.66
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发表时间：
2020-09
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通讯作者：
X. Dai;K. Yoshikawa
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文献类型：
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作者：
X. Dai;K. Yoshikawa
摘要引入了具有$\frac{1}{4}(1，1)$型循环商奇点的指数为2的对数Enrique曲面的一个全纯扭转不变量。这类具有k个奇点的对数-Enrique曲面的模空间是与签名为$(2，10-k)$的么模格相关的正交型模簇。我们证明了作为模簇的函数的不变量是由显式Borcherds积的Petersson范数给出的。我们注意到这个挠率不变量本质上是复数维$2$中的BCOV不变量。因此，与Calabi-Yau的情况不同，这种情况下的BCOV不变量不是一个二元不变量。
Abstract We introduce a holomorphic torsion invariant of log-Enriques surfaces of index two with cyclic quotient singularities of type $\frac {1}{4}(1,1)$ . The moduli space of such log-Enriques surfaces with k singular points is a modular variety of orthogonal type associated with a unimodular lattice of signature $(2,10-k)$ . We prove that the invariant, viewed as a function of the modular variety, is given by the Petersson norm of an explicit Borcherds product. We note that this torsion invariant is essentially the BCOV invariant in the complex dimension $2$ . As a consequence, the BCOV invariant in this case is not a birational invariant, unlike the Calabi-Yau case.