Let
be a complex manifold of complex dimension
. If
is a
-smooth real submanifold of real codimension
in
, let us denote by
the tangential complex space at
. Such a manifold
can be locally represented in the form:
, where all
are real
-functions in an open subset
of X. The submanifold
is called
if the number
is independent of the point
. A submanifold
is called CR generic if
for every
.
Let us consider
to be an oriented
-smooth
generic submanifold of real codimension
in an
-dimensional complex manifold
, and let us denote by
the ideal sheaf in the Grassmann algebra
of germs of complex valued
-forms on
, that are locally generated by functions (which vanish on
), and by their anti-holomorphic differentials. One also has on
the Dolbeault complexes for the sheaves of germs of smooth forms:
where
is the sheaf of germs of complex valued
-forms of bidegree
, for
. Let us also set
.
As
, for each
we now have the categorical sequence
of subcomplexes of the complex
written as :
Therefore, we also have the quotient complexes
defined by the exact sequences of fine sheaves complexes:
With the induced differentials denoted by
we can now write the quotient complex-which is called the tangential Cauchy-Riemann complex of
-smooth forms- as follows:
Remarks: There are two distinct ways of defining the tangential Cauchy-Riemann complex:
For further, full details the reader is referred to the recent textbook by Burgess (2000) on this subject.
The cohomology groups of
on
, for
being an open subset
of
, are then appropriately denoted here as
.
As of this snapshot date, this entry was owned by bci1.