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arXiv · 1810.10058

Constructing a CM Mumford fourfold from Shioda's fourfold

Yuwei Zhu
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Abstract

Shioda proved that the Jacobian $A_S$ of the curve $y^2 = x^9 -1$ is a 4-dimensional CM abelian variety with codimension 2 Hodge cycles not generated by divisors. It was noted by Shioda that this behavior resembles the abelian varieties constructed by Mumford. We prove that Shioda's fourfold $A_S$ cannot be realized as a special case of Mumford's construction. However, by modifying its Hodge structure, we construct a basis for computing the period matrix of a CM Mumford fourfold with multiplication by $\sqrt{-3}$.

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Yuwei Zhu. 2018-10-23. Constructing a CM Mumford fourfold from Shioda's fourfold. https://arxiv.org/abs/1810.10058

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