Complex Form
المؤلف:
المرجع الالكتروني للمعلوماتيه
المصدر:
المرجع الالكتروني للمعلوماتيه
الجزء والصفحة:
...
27-11-2018
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Complex Form
The differential forms on 
decompose into forms of type 
, sometimes called 
-forms. For example, on 
, the exterior algebra decomposes into four types:
where 
, 
, and 
denotes the direct sum. In general, a 
-form is the sum of terms with 
s and 
s. A 
-form decomposes into a sum of 
-forms, where 
.
For example, the 2-forms on 
decompose as
The decomposition into forms of type 
is preserved by holomorphic functions. More precisely, when 
is holomorphic and 
is a 
-form on 
, then the pullback 
is a 
-form on 
.
Recall that the exterior algebra is generated by the one-forms, by wedge product and addition. Then the forms of type 
are generated by
 |
(5)
|
The subspace 
of the complex one-forms can be identified as the 
-eigenspace of the almost complex structure 
, which satisfies 
. Similarly, the 
-eigenspace is the subspace 
. In fact, the decomposition of 
determines the almost complex structure 
on 
.
More abstractly, the forms into type 
are a group representation of 
, where 
acts by multiplication by 
.
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