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Holomorphic forms, the
β
Λ
\bar{\partial}
β
Λ
-equation, and duality on a reduced complex space
Authors
H\ue5kan Samuelsson Kalm
Publication date
1 January 2015
Publisher
Abstract
We study two natural notions of holomorphic forms on a reducedpure
n
n
n
-dimensional complex space
X
X
X
: sections of the sheaves
Ξ©
X
β
\Omega_X^{\bullet}
Ξ©
X
β
β
of germs ofholomorphic forms on
X
r
e
g
X_{reg}
X
re
g
β
that have a holomorphic extension to some ambient complex manifold, and sections of the sheaves
Ο
X
β
\omega_X^{\bullet}
Ο
X
β
β
introduced by Barlet. We show that
Ξ©
X
p
\Omega_X^p
Ξ©
X
p
β
and
Ο
X
n
β
p
\omega_X^{n-p}
Ο
X
n
β
p
β
are Serre dual to each otherin a certain sense. We also provide explicit, intrinsic and semi-global Koppelman formulas for the
β
Λ
\bar{\partial}
β
Λ
-equation on
X
X
X
and introduce fine sheaves
A
X
p
,
q
\mathscr{A}_X^{p,q}
A
X
p
,
q
β
and
B
X
p
,
q
\mathscr{B}_X^{p,q}
B
X
p
,
q
β
of
(
p
,
q
)
(p,q)
(
p
,
q
)
-currents on
X
X
X
, that are smooth on
X
r
e
g
X_{reg}
X
re
g
β
,such that
(
A
X
p
,
β
,
β
Λ
)
(\mathscr{A}_X^{p,\bullet},\bar{\partial})
(
A
X
p
,
β
β
,
β
Λ
)
is a resolution of
\Om_X^p
and, if
Ξ©
X
n
β
p
\Omega_X^{n-p}
Ξ©
X
n
β
p
β
is Cohen-Macaulay,
(
B
X
p
,
β
,
β
Λ
)
(\mathscr{B}_X^{p,\bullet},\bar{\partial})
(
B
X
p
,
β
β
,
β
Λ
)
is a resolution of
Ο
X
p
\omega_X^{p}
Ο
X
p
β
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Last time updated on 07/05/2019