Fourfolds of Weil type and the spinor map

TL;DR

This paper introduces the recent advances by Markman and O'Grady on abelian fourfolds of Weil type, emphasizing the spinor map and its role in understanding their structure and relation to Kuga Satake varieties.

Contribution

It provides a straightforward introduction to complex results on Weil type fourfolds, focusing on the spinor map and representation theory without triality.

Findings

Explicit descriptions of abelian fourfolds of Weil type with trivial discriminant.

Connection of these fourfolds to Kuga Satake varieties.

Use of the spinor map to analyze Hodge structures.

Abstract

Recent papers by Markman and O'Grady give, besides their main results on the Hodge conjecture and on hyperkaehler varieties, surprising and explicit descriptions of families of abelian fourfolds of Weil type with trivial discriminant. They also provide a new perspective on the well-known fact that these abelian varieties are Kuga Satake varieties for certain weight two Hodge structures of rank six. In this paper we give a pedestrian introduction to these results. The spinor map, which is defined using a half-spin representation of SO(8), is used intensively. For simplicity, we use basic representation theory and we avoid the use of triality.