Constructing a CM Mumford fourfold from Shioda's fourfold

2018/10/23 by Yuwei Zhu, Zhu, Yuwei
Mathematics · #14C30 #14K22 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1810.10058

openalex publication_date 2018/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Shioda proved that the Jacobian A_S of the curve y² = x⁹ -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 √(-3).

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