Download My talk on Almost Complex Structures

My talk on Almost Complex Structures
Apurv Nakade
1. Integrability of almost complex structures
Definition 1.1. The Nijenhuis tensor on an almost complex manifold (M, J) is defined by
N (X, Y ) := [JX, JY ] − J[JX, Y ] − J[X, JY ] − [X, Y ]
Proposition 1.2. N is a tensor.
Proof: Enough to check N (X, f Y ) − f N (X, Y ) = 0 for any smooth function f . Use the property that
[X, f Y ] − f [X, Y ] = Y (f )X
Theorem 1.3 (Newlander-Nirenberg). J is integrable if and only if N vanishes identically.
I do not know how to prove this.
Define the space of (0, 1) vector fields to be the vector fields of the form X − iJX.
Lemma 1.4. N vanishes identically if and only if the space of (0, 1) vector fields is closed under Lie brackets.
Proof:
[X − iJX, Y − iJY ] = [X, Y ] − [JX, JY ] + i([JX, Y ] + [X, JY ])
This is of the form Z − iJZ if and only if
J([X, Y ] − [JX, JY ]) = [JX, Y ] + [X, JY ]
which is equivalent to saying that N ∼
= 0.
Theorem 1.5. Any almost complex structure on a Riemann surface is integrable.
Proof: It is easy to check that N (X, JX) = 0. Now the tangent space at each point is spanned by X and
JX for a Riemann surface.
2. Spaces of Almost complex structures
Fix a symplectic vector space (V, ω). We denote the standard structures by (R2n , ω0 , J0 ).
We say J is tamed by ω if ω(X, JX) > 0. Further if ω(JX, JY ) = ω(X, Y ) then J is said to be compatible
with ω.
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APURV NAKADE
2.1. Polar decomposition: Given a metric g on (V, ω) we can construct a compatible almost complex
structure J as follows:
Let ω(X, Y ) = g(X, AY ) and let A† denote the dual of A with respect to g, then by the anti-symmetry of
ω we have A† = −A and so −AA† is symmetric positive definite and hence has a well defined square root,
define
p
J := −A2 A−1
To see that J is tame note that
p
−A2 A−1 X)
p
= g(X, −A2 X)
ω(X, JX) = ω(X,
which is bigger than 0 because
To see that J is compatible
√
−A2 is positive definite.
p
p
ω(JX, JY ) = ω( −A2 A−1 X, −A2 A−1 Y )
p
p
= g( −A2 A−1 X, −A2 Y )
= g(−A2 A−1 X, Y )
= g(−A1 X, Y )
= ω(X, Y )
We are interested in the space of all tame and compatible almost complex structures.
Definition 2.1. Denote by Jc (V, ω) (Jt (V, ω)) the space of all compatible (tame) almost complex structures.
Proposition 2.2. Jc (V, ω) is contractible.
Proof: Assume for simplicity that we have a standard structure Jc (R2n , ω0 ). It suffices to construct a
canonical path between J ∈ Jc (R2n , ω0 ) and J0 . We use the polar decomposition. Let g be the metric
corresponding to J. Then gt = g.t + g0 (1 − t) is continuous family of metrics. We can use the polar
decomposition to create a continuous family of almost complex structures Jt connecting J0 to J.
With a little more effort one can show that the space Jc (V, ω) is in fact isomorphic to Sp(2n)/U (n) which
is affine as U (n) is the largest compact subgroup inside Sp(2n).
Proposition 2.3. Jt (V, ω) is contractible.
Proof: By the previous proposition it suffices to show that Jc (V, ω) is in fact a retract of Jt (V, ω). We again
use the polar decomposition to construct a map Jt (V, ω) → Jc (V, ω). From a tame almost complex structure
one can create a metric by defining
g(X, Y ) = (ω(JX, Y ) + ω(X, JY ))/2
from g we can create a compatible almost complex structure using Polar decomposition.
It is also possible to identify these spaces explicitly. Let J0 denote the standard almost complex structure
(multiplication by i).
Proposition 2.4. The map J 7→ (1 + J0−1 J)−1 (1 − J0−1 J) maps Jt (V, ω) diffeomorphically onto
{S : ||S|| ≤ 1 and SJ0 + J0 S = 0}
and Jc (V, ω) diffeomorphically onto
{S : ||S|| ≤ 1 and SJ0 + J0 S = 0 and S T = S}
By Darboux’ theorem, given a symplectic manifold we have natural fiber bundles with fibres Jt and Jc over
them. Because the fibers are contractible we can find global sections to these bundles and hence we get,
MY TALK ON ALMOST COMPLEX STRUCTURES
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Theorem 2.5. On every symplectic manifold (M, ω) there exists a compatible almost complex structure.
Further such an almost complex structure is unique up to homotopy.
3. J-holomorphic curves
Now onwards assume that our symplectic manifold is endowed with an almost complex structure (M, ω, J).
Definition 3.1. A J-homolomorphic curve is map
f : (Σ, j) → (M, J)
where Σ is a Riemann surface with almost complex structure j and we require the compatibility condition
df ◦ j = J ◦ df
We are interested in the local structure of J-holomorphic curves. So we restrict our attention to maps of the
form
f : (D, i) → (Cn , J)
0
where D is the unit disk in R and i is the standard complex structure (multiplication by
1
2
−1
).
0
In this case df ◦ i = J ◦ df is equivalent to
∂f
∂f
= J(f )
∂y
∂x
(3.1)
by comparing the first column. The second condition is equivalent to J 2 = −1.
Now we define the following operators:
∂
∂
∂
=(
− i )/2
∂z
∂x
∂y
∂
∂
∂
=(
+ i )/2
∂ z̄
∂x
∂y
Using these new operators the equation 3.1 we get
∂f
∂f
∂f
∂f
i(
−
) = J(f )(
+
)
∂z
∂ z̄
∂z
∂ z̄
∂f
∂f
=⇒
+ (1 + iJ(f ))−1 (1 − iJ(f ))
=0
∂ z̄
∂z
Theorem 3.2 (Poincare ∂¯ lemma). For any smooth function f ∈ C ∞ (D) define
Z Z
f (τ )
P f (z) =
dτ dτ̄
τ
−z
D
then we have
∂P f
f (z) =
∂ z̄
Using the Poincare ∂¯ lemma the above equation can be rewritten as
∂
∂f
(f + P (1 + iJ(f ))−1 (1 − iJ(f )) ) = 0
∂ z̄
∂z
∂f
−1
Substituting g(z) = f + P (1 + iJ(f )) (1 − iJ(f )) ∂z this reduces to the familiar equation
∂g
=0
∂ z̄
Theorem 3.3. Critical points of J-holomorphic curves are isolated.
Theorem 3.4. ∀p ∈ M and ∀v ∈ Tp M sufficiently small ∃ a J-holomorphic curve such that 0 to p and
df0 (1) = v.