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Mathematics > Algebraic Geometry

arXiv:1812.06882 (math)
[Submitted on 17 Dec 2018 (v1), last revised 15 Apr 2019 (this version, v2)]

Title:K3 surfaces associated to Abelian Fourfolds of Mumford's Type

Authors:Yuwei Zhu
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Abstract:Mumford constructed a family of abelian fourfolds with special stucture not characterized by endomorphism ring. Galluzzi showed that the weight 2 Hodge structure of such a variety decomposes into Hodge substructures via the action of Mumford-Tate group, one of which is of K3 type with Hodge number (1,7,1). We will compute the intersection form of such a Hodge structure and provide a canonical integral Hodge structure on this subspace. Furthermore, we shall show two applications of this invariant. Firstly, using the above formula we shall show that every K3 surface obtained in this way will admit an elliptic fibration. Comparing them with the list by Shimada, we will show that the Mordell-Weil group of all elliptic fibrations have either 2-torsion or trivial torsion group, and the conditions of the occurance of 2-torsions can be specified. Secondly, we shall use it to determine the quaternion that gives a special CM abelian fourfold that is derived equivalent to the Shioda fourfold.
Comments: Update April 3, 2019: minor changes
Subjects: Algebraic Geometry (math.AG)
Cite as: arXiv:1812.06882 [math.AG]
  (or arXiv:1812.06882v2 [math.AG] for this version)
  https://doi.org/10.48550/arXiv.1812.06882

Submission history

From: Yuwei Zhu [view email]
[v1] Mon, 17 Dec 2018 16:41:25 UTC (17 KB)
[v2] Mon, 15 Apr 2019 14:58:43 UTC (17 KB)
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