Geometric Variations of Local Systems and Elliptic Surfaces

Charles Doran; Jordan Kostiuk

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arXiv · 1911.00402

Geometric Variations of Local Systems and Elliptic Surfaces

Charles Doran·Jordan Kostiuk

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Abstract

Geometric variations of local systems are families of variations of Hodge structure; they typically correspond to fibrations of K\"{a}hler manifolds for which each fibre itself is fibred by codimension one K\"{a}hler manifolds. In this article, we introduce the formalism of geometric variations of local systems and then specialize the theory to study families of elliptic surfaces. We interpret a construction of twisted elliptic surface families used by Besser-Livn\'{e} in terms of the middle convolution functor, and use explicit methods to calculate the variations of Hodge structure underlying the universal families of $M_N$-polarized K3 surfaces. Finally, we explain the connection between geometric variations of local systems and geometric isomonodromic deformations, which were originally considered by the first author in 1999.

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Charles Doran, Jordan Kostiuk. 2019-11-01. Geometric Variations of Local Systems and Elliptic Surfaces. https://arxiv.org/abs/1911.00402

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