Twistor lines in the period domain of complex tori

TL;DR

The paper explores the structure of the period domain of complex tori, demonstrating that twistor lines connect different complex structures and analyzing their geometric properties and implications for Hodge theory.

Contribution

It establishes the connectivity of the period domain via twistor lines and provides criteria for twistor path connectivity in Hodge loci.

Findings

Any two complex tori's periods can be connected by a chain of twistor lines.

Twistor lines are holomorphic submanifolds of the period domain.

The paper characterizes the degree of twistor lines in the Plücker embedding.

Abstract

As in the case of irreducible holomorphic symplectic manifolds, the period domain $Compl$ of compact complex tori of even dimension $2n$ contains twistor lines. These are special $2$-spheres parametrizing complex tori whose complex structures arise from a given quaternionic structure. In analogy with the case of irreducible holomorphic symplectic manifolds, we show that the periods of any two complex tori can be joined by a {\em generic} chain of twistor lines. We also prove a criterion of twistor path connectivity of loci in $Compl$ where a fixed second cohomology class stays of Hodge type (1,1). Furthermore, we show that twistor lines are holomorphic submanifolds of $Compl$, of degree $2n$ in the Pl\"ucker embedding of $Compl$.