A complex structure on a real vector space 
is a linear
transformation 
satisfying 
,
where 
is the identity map.
For example, the standard complex structure on the plane is the linear transformation 
given by
 |
(1)
|
which corresponds to a counterclockwise rotation by 
. For 
, this map satisfies
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
Every complex structure on a two-dimensional real vector space has the displayed coordinate form in a suitable vector basis.
On a connected two-dimensional real manifold without boundary, a complex structure is an equivalence
class of atlases whose coordinate
charts take values in open sets of 
and whose transition functions
are holomorphic. Two such atlases
are equivalent when their union is again such an atlas.
The resulting manifold is a Riemann
surface (Milne 2021, online version pp. 91-92).
