<s>
In	O
mathematics	O
,	O
plurisubharmonic	B-Algorithm
functions	I-Algorithm
(	O
sometimes	O
abbreviated	O
as	O
psh	O
,	O
plsh	B-Algorithm
,	O
or	O
plush	B-Algorithm
functions	I-Algorithm
)	O
form	O
an	O
important	O
class	O
of	O
functions	O
used	O
in	O
complex	O
analysis	O
.	O
</s>
<s>
On	O
a	O
Kähler	O
manifold	O
,	O
plurisubharmonic	B-Algorithm
functions	I-Algorithm
form	O
a	O
subset	O
of	O
the	O
subharmonic	O
functions	O
.	O
</s>
<s>
However	O
,	O
unlike	O
subharmonic	O
functions	O
(	O
which	O
are	O
defined	O
on	O
a	O
Riemannian	B-Architecture
manifold	I-Architecture
)	O
plurisubharmonic	B-Algorithm
functions	I-Algorithm
can	O
be	O
defined	O
in	O
full	O
generality	O
on	O
complex	B-Algorithm
analytic	I-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
In	O
full	O
generality	O
,	O
the	O
notion	O
can	O
be	O
defined	O
on	O
an	O
arbitrary	O
complex	O
manifold	O
or	O
even	O
a	O
complex	B-Algorithm
analytic	I-Algorithm
space	I-Algorithm
as	O
follows	O
.	O
</s>
<s>
Equivalently	O
,	O
a	O
-function	O
f	O
is	O
plurisubharmonic	B-Algorithm
if	O
and	O
only	O
if	O
is	O
a	O
positive	O
(	O
1	O
,	O
1	O
)	O
-form	O
.	O
</s>
<s>
Relation	O
to	O
Kähler	O
manifold	O
:	O
On	O
n-dimensional	O
complex	O
Euclidean	O
space	O
,	O
is	O
plurisubharmonic	B-Algorithm
.	O
</s>
<s>
for	O
some	O
Kähler	O
form	O
,	O
then	O
is	O
plurisubharmonic	B-Algorithm
,	O
which	O
is	O
called	O
Kähler	O
potential	O
.	O
</s>
<s>
Relation	O
to	O
Dirac	O
Delta	O
:	O
On	O
1-dimensional	O
complex	O
Euclidean	O
space	O
,	O
is	O
plurisubharmonic	B-Algorithm
.	O
</s>
<s>
If	O
is	O
an	O
analytic	O
function	O
on	O
an	O
open	O
set	O
,	O
then	O
is	O
plurisubharmonic	B-Algorithm
on	O
that	O
open	O
set	O
.	O
</s>
<s>
The	O
set	O
of	O
plurisubharmonic	B-Algorithm
functions	I-Algorithm
has	O
the	O
following	O
properties	O
like	O
a	O
convex	O
cone	O
:	O
</s>
<s>
if	O
is	O
a	O
plurisubharmonic	B-Algorithm
function	I-Algorithm
and	O
a	O
positive	O
real	O
number	O
,	O
then	O
the	O
function	O
is	O
plurisubharmonic	B-Algorithm
,	O
</s>
<s>
if	O
and	O
are	O
plurisubharmonic	B-Algorithm
functions	I-Algorithm
,	O
then	O
the	O
sum	O
is	O
a	O
plurisubharmonic	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
a	O
function	O
is	O
plurisubharmonic	B-Algorithm
if	O
and	O
only	O
if	O
it	O
is	O
plurisubharmonic	B-Algorithm
in	O
a	O
neighborhood	O
of	O
each	O
point	O
.	O
</s>
<s>
If	O
is	O
plurisubharmonic	B-Algorithm
and	O
a	O
monotonically	O
increasing	O
,	O
convex	O
function	O
then	O
is	O
plurisubharmonic	B-Algorithm
.	O
</s>
<s>
If	O
and	O
are	O
plurisubharmonic	B-Algorithm
functions	I-Algorithm
,	O
then	O
the	O
function	O
is	O
plurisubharmonic	B-Algorithm
.	O
</s>
<s>
then	O
is	O
plurisubharmonic	B-Algorithm
.	O
</s>
<s>
Every	O
continuous	O
plurisubharmonic	B-Algorithm
function	I-Algorithm
can	O
be	O
obtained	O
as	O
the	O
limit	O
of	O
a	O
monotonically	O
decreasing	O
sequence	O
of	O
smooth	O
plurisubharmonic	B-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
(	O
see	O
limit	B-Algorithm
superior	I-Algorithm
and	I-Algorithm
limit	I-Algorithm
inferior	I-Algorithm
for	O
the	O
definition	O
of	O
lim	B-Algorithm
sup	I-Algorithm
)	O
.	O
</s>
<s>
Plurisubharmonic	B-Algorithm
functions	I-Algorithm
are	O
subharmonic	O
,	O
for	O
any	O
Kähler	O
metric	O
.	O
</s>
<s>
Therefore	O
,	O
plurisubharmonic	B-Algorithm
functions	I-Algorithm
satisfy	O
the	O
maximum	O
principle	O
,	O
i.e.	O
</s>
<s>
In	O
complex	O
analysis	O
,	O
plurisubharmonic	B-Algorithm
functions	I-Algorithm
are	O
used	O
to	O
describe	O
pseudoconvex	B-Algorithm
domains	I-Algorithm
,	O
domains	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
and	O
Stein	O
manifolds	O
.	O
</s>
<s>
The	O
main	O
geometric	O
application	O
of	O
the	O
theory	O
of	O
plurisubharmonic	B-Algorithm
functions	I-Algorithm
is	O
the	O
famous	O
theorem	O
proven	O
by	O
Kiyoshi	O
Oka	O
in	O
1942	O
.	O
</s>
<s>
admitting	O
a	O
smooth	O
,	O
exhaustive	O
,	O
strongly	O
plurisubharmonic	B-Algorithm
function	I-Algorithm
.	O
</s>
