Hodge cycles and quadratic relations between holomorphic periods on CM abelian varieties

TL;DR

This paper demonstrates that for CM abelian varieties, all non-trivial Hodge relations among holomorphic periods can be generated by explicit monomial quadratic relations, with constructed auxiliary varieties aiding this process.

Contribution

It introduces a method to generate all Hodge relations from monomial quadratic ones for CM abelian varieties using auxiliary constructions.

Findings

All non-trivial Hodge relations are generated by explicit monomial quadratic relations.

Construction of a CM abelian variety B that splits over the Galois closure of A's CM field.

The approach emphasizes the importance of monomial quadratic relations in understanding periods.

Abstract

In this paper, we prove the following result advocating the importance of monomial quadratic relations between holomorphic CM periods. For any simple CM abelian variety $A$, we can construct a CM abelian variety $B$ such that all non-trivial Hodge relations between the holomorphic periods of the product $A\times B$ are generated by monomial quadratic ones which are also explicit. Moreover, $B$ splits over the Galois closure of the CM field associated with $A$.