Twistor triangles in the period domain of complex tori

TL;DR

This paper explores the geometry of twistor triangles in the period domain of complex tori using representation theory and introduces invariants to classify these triangles up to symmetry.

Contribution

It introduces a new geometric framework for analyzing twistor triangles in the period domain via representation theory and defines invariants for their classification.

Findings

Defined a G-invariant pseudometric on the period domain.

Introduced pseudometric invariants to distinguish triangles.

Analyzed the structure of twistor triangles using algebraic methods.

Abstract

We study the geometry of the (generalized) twistor triangles $\triangle J_1J_2J_3$ in the period domain of compact complex tori of complex dimension $2n$ by the means of the representation theory of the algebras (of real dimension 8) generated by the complex structures $J_1,J_2,J_3$. Considering the period domain as the homogeneous space for $G=GL_{4n}(\mathbb{R})$, we introduce on it a $G$-invariant pseudometric and define pseudometric invariants, helping us to distinguish triangles from a reasonable class up to $G$-equivalence.