Linear complex structure

In mathematics, a complex structure on a real vector space ${\displaystyle V}$ is an automorphism of ${\displaystyle V}$ that squares to the minus identity, ${\displaystyle -Id_{V}}$. Such a structure on ${\displaystyle V}$ allows one to define multiplication by complex scalars in a canonical fashion so as to regard ${\displaystyle V}$ as a complex vector space.

Every complex vector space can be equipped with a compatible complex structure in a canonical way; however, there is in general no canonical complex structure. Complex structures have applications in representation theory as well as in complex geometry where they play an essential role in the definition of almost complex manifolds, by contrast to complex manifolds. The term "complex structure" often refers to this structure on manifolds; when it refers instead to a structure on vector spaces, it may be called a linear complex structure.