+
�
18.966
–
Homework
3
–
due
Thursday
April
19,
2007.
1.
Let (
M, ω
) be a symplectic manifold,
J
a compatible almostcomplex
structure,
and
g
the corresponding Riemannian
metric. Show that twodimensional almostcomplex
sub
′
manifolds of
M
are absolutely
volume minimizing in
their homology
class, i.e.: let
C
,
C
be twodimensional
compact
closed
oriented
submanifolds
of
M
, representing the same
ho
mology
class [
C
] = [
C
′
]
∈
H
2
(
M,
Z
). Assume that
J
(
TC
) =
TC
(and
the orientation of
C
agrees with
that induced
by
J
).
Then
vol
g
(
C
)
≤
vol
g
(
C
′
).
Hint:
compare
ω

C
′
and
the area form induced
by
g
.
2.
We will
admit
the fact
that the cohomology
ring of
CP
n
(the set of complex lines
through 0 in
C
n
+1
) is
H
∗
(
CP
n
,
Z
) =
Z
[
h
]
/h
n
+1
, where
h
∈
H
2
(
CP
n
,
Z
) is
Poincar´
e dual
to
the homology
class represented
by
a linear
CP
n
−
1
⊂
CP
n
.
The tautological
line bundle
L
→
CP
n
is
the subbundle of the trivial bundle
C
n
+1
×
CP
n
whose fiber at
a point
of
CP
n
is the corresponding line in
C
n
+1
. The homogeneous
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 Spring '10
 DenisAuroux
 Geometry, Algebraic Topology, Manifold, Chern, Vector bundle, CPN, Cn+1

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