Two results on the Hodge structure of complex tori

TL;DR

This paper investigates the Hodge structures in the cohomology of complex tori, establishing their relation to abelian varieties and extending the Kuga-Satake construction to non-polarized Hodge structures of K3 type.

Contribution

It proves that polarizable Hodge structures in complex tori cohomology originate from abelian varieties and extends the Kuga-Satake construction to non-polarized K3 type structures.

Findings

Polarizable Hodge structures in tori cohomology come from abelian varieties.

A universality result for the Kuga-Satake construction for non-polarized K3 type structures.

Extension of Hodge structure analysis beyond polarized cases.

Abstract

We prove two results regarding Hodge structures appearing in the cohomology of complex tori. First, we prove that if a polarizable Hodge structure appears in the cohomology of a complex torus $T$, it appears in the cohomology of an abelian variety isomorphic to a subquotient of $T$. Second, we prove a universality result for the Kuga-Satake construction applied to Hodge structures of $K3$ type that might not be polarized.