Intersections of complex structures

TL;DR

This paper investigates the geometric configurations of planes in even-dimensional real vector spaces stabilized by pairs of complex structures, providing explicit counts and bounds for these special planes.

Contribution

It offers a complete description of planes stabilized by orthogonal complex structures and establishes a lower bound for generic pairs, advancing understanding of complex structure interactions.

Findings

Finite number of such planes for generic pairs

Explicit count of stabilized planes for orthogonal structures

Lower bound for stabilized planes in generic cases

Abstract

We study the sets of planes in an even dimensional real vector space $V$ which are simultaneously stabilised by a pair of complex structures on $V$. We completely describe these sets of planes for pairs of orthogonal complex structures. Generically, the number of such planes is finite. We compute this number for orthogonal complex structures and prove that it gives a lower bound for the number of planes simultaneously stabilised by a generic pair of complex structures on $V$.