Orbifold Aspects of Certain Occult Period Maps

TL;DR

This paper investigates the automorphism groups of Hodge structures in certain algebraic varieties, confirming conjectures about their actions on cyclic covers, with implications for understanding orbifold structures and period maps.

Contribution

It characterizes automorphism groups of Hodge structures for cubic threefolds, fourfolds, and related varieties, confirming conjectures by Kudla and Rapoport.

Findings

Automorphism groups of Hodge structures are explicitly characterized.

Actions of automorphism groups on cyclic covers are determined.

Conjectures by Kudla and Rapoport are confirmed for specific varieties.

Abstract

We first characterize the automorphism groups of Hodge structures of cubic threefolds and cubic fourfolds. Then we determine for some complex projective manifolds of small dimension (cubic surfaces, cubic threefolds, and non-hyperelliptic curves of genus 3 or 4), the action of their automorphism groups on Hodge structures of associated cyclic covers, and thus confirm conjectures made by Kudla and Rapoport.