Download L6: Almost complex structures To study general symplectic

L6: Almost complex structures
To study general symplectic manifolds, rather
than Kähler manifolds, it is helpful to extract
the “homotopy-theoretic essence” of having a
complex structure. An almost complex structure is a bundle automorphism J : T M → T M
with J 2 = −I. If M is a complex manifold,
with complex co-ordinates zj = xj + iyj , then
there is a distinguished J defined by
∂
∂
J
=
;
∂x
∂y
µ
¶
J
Ã
∂
∂y
!
∂
=− .
∂x
The tangent spaces to M are naturally complex
vector spaces, which carry multiplication by
i. More generally, for almost complex (M, J),
J extends complex-linearly to T M ⊗ C , and
splits this space into ±i-eigenspaces T M ⊗ C =
∼ T M , and
T 1,0(M ) ⊕ T 0,1(M ). So T 1,0(M ) =
R
in the complex case can also be viewed as
∂ i, with ∂ = ∂ − i ∂ . If J
T 1,0(M ) = C h ∂z
∂zj
∂xj
∂yj
j
comes from a system of complex co-ordinates,
we say it is integrable.
Definition: An a.c.s. J on (M, ω) is compatible
with a symplectic form ω if
ω(Jv, Ju) = ω(u, v);
ω(v, Jv) > 0 ∀v 6= 0
The symmetric bilinear form g(u, v) = ω(u, Jv)
defines a Riemannian metric in this case.
Proposition: (M, ω) admits a compatible J.
Proof: First we show this is true for a symplectic vector space V . Choose some metric g on
V , and define A by ω(u, v) = g(Au, v). Then
A∗ = −A so AA∗ is symmetric positive definite:
g(AA∗v, v) = g(A∗v, A∗v) > 0 ∀v 6= 0
−1 is diagoThus AA∗ = B.diag(λ1, . . . , λ2n
).B
√
nalisable, with λi > 0, and AA∗ exists and
√
√
equals B.diag( λ1, . . . , λ2n).B −1. Set:
√
J = ( AA∗)−1A ⇒ JJ ∗ = I and J ∗ = −J
Thus J 2 = −I is an a.c.s. and
ω(Ju, Jv) = g(AJu, Jv) = g(JAu,
Jv) = g(Au, v)
√
ω(u, Ju) = g(−JAu, u) = g( AA∗u, u) > 0
so J is compatible. Use this construction on
each TxM , since it’s canonical it works globally.
The proof actually shows that the space of
compatible J is the same as the space of metrics, which is convex, hence contractible. Thus
introducing a compatible J involves “essentially no choice”.
Definition: a symplectic vector bundle is a vector bundle π : E → B such that each fibre Eb
has a linear symplectic form ωb, these form a
smooth global section Ω of Λ2E ∗, and locally
in B things are trivial:
∼ (U × R 2n, ω )
(π −1(U ), Ω) =
0
Corollary: such an E canonically admits the
structure of a complex vector bundle. Hence,
it has a first Chern class. This is a “characteristic class”; we assign to each complex vector
bundle E → B an element c1(E) ∈ H 2(B; Z)
such that c1(f ∗E) = f ∗c1(E) is natural under
continuous maps and pullback of vector bundles. Note that choices of a.c.s J on E are all
homotopic, so c1(E) does not change as it lives
in the integral cohomology which is discrete. If
E = T M we write c1(M ) for c1(E).
There are many definitions of c1(E):
(i) The determinant line bundle Λr E → B is
pulled back by a classifying map φE : B →
BU (1) = C P∞, i.e. Λr E = φ∗E Ltaut, and we
] is generated by
know H ∗(C P∞; Z) = Z[cuniv
1
an element of degree 2; set c1(E) = φ∗E (cuniv
).
1
(ii) Take a generic section s : B → Λr E which
is transverse to the zero-section, which implies
its zero-set Z(s) is a smooth submanifold of B
of real codimension 2. Set c1(E) = PD[Z(s)].
(iii) Pick a connexion dA on E, given by a matrix of one-forms (θij ) w.r.t. a basis sj ∈ Γ(E)
of local sections of E:
dA : Γ(E) → Γ(E ⊗ T ∗B); sj 7→
X
i
si ⊗ θij
The curvature FA = dA ◦ dA, locally dθ + θ ∧ θ,
is a matrix-valued 2-form in Ω2(End(E)), so
T r(FA) ∈ Ω2(B) is a 2-form. The Bianchi
identity says this is closed, so defines a cohomology class, and c1(E) = [T r(FA)/2iπ]. One
can check this class is independent of choice
of connexion dA.
We noted c1(E) ∈ H 2(M ; Z) is an integral class,
so does not change as we vary JE continuously.
Here are some other properties:
(i) c1(E ∗) = −c1(E);
(ii) c1(E ⊕E 0) = c1(E)⊕c1(E 0); given any short
exact sequence 0 → E → E → E 0 → 0, the same
formula holds;
(iii) c1(E ⊗ F ) = rk(E).c1(F ) + rk(F ).c1(E); in
particular for a line bundle c1(L⊗k ) = kc1(L).
(iv) If M is compact, there is an isomorphism
{C −vector bundles/M } c1
−→ H 2(M ; Z)
Isomorphism
If M is complex, this can fail holomorphically.
Examples:
(i) c1(Σg ) = 2 − 2g. This is Gauss-Bonnet: on
a surface, both c1 and the Euler characteristic
count (signed) zeroes of a generic vector field.
∼
(ii) c1(Ltaut → C Pn) = −[H] ∈ H 2(C Pn) =
ZhHi, where [H] = PD[C Pn−1] is the hyperplane class; c1(C Pn) = n + 1, by considering
poles of a holoc n-form (dz0∧· · ·∧dzn)/(z0 . . . zn).
Adjunction formula: if C ⊂ (X 4, J) is an almost
complex curve in an almost complex surface,
i.e. if J preserves T C ⊂ T X, then
2g(C) − 2 = −c1(X) · [C] + [C]2
So the genus of an almost complex curve is
determined by its homology class.
Proof: We have a SES of complex v.bundles
0 → T C → T X|C → νC/X → 0
Now deduce c1(T X|C ) = c1(T C)⊕c1(νC/X ) and
evaluate the terms. A small (smooth, generic)
displacement of C in X gives a section of νC/X
with exactly [C]2 zeroes, to sign. ¥
A four-manifold X has a (non-degenerate, symmetric) intersection form on H 2(X; R ). Let b+
and b− denote the number of positive and negative eigenvalues, so the signature σ(X) is the
difference b+ −b−, whilst b+ +b− = b2(X). Now
give X an a.c.s J, defining c1 = c1(T X, J).
Signature theorem (Hirzebruch):
c2
1 = 2e(X) + 3σ(X) is a topological invariant.
Despite this, there are lots of possible c1’s. Say
M 4 is even if every class a ∈ H 2(M ) has even
square. Algebraic topology shows
(i) if M is even and H 2(M ; Z) has no 2-torsion,
any h s.t. h2 = 2e + 3σ and h ≡ 0 ∈ H 2(M ; Z2)
is c1 of some almost complex structure, and
(ii) for any a.c.s. c2
1 ≡ σ (8).
Corollary: if X 4 admits an a.c.s. then the selfconnect sum X]X does not, X]X]X does, etc.
Proof: The condition c2
1 ≡ σ(8) & the signature theorem show 1 − b1 + b+ must be even.
But b1(X]X) = 2b1(X), b+(X]X) = 2b+(X)
etc, so for X]X we have 1 − b1 + b+ is odd.
S 4 was not symplectic since H 2(S 4) = 0. Now
we see C P2]C P2 is not symplectic since it has
no a.c.s. The proof that C P3]C P3]C P3 is not
symplectic takes most of gauge theory and a
few hundred pages. In dimension 2n > 4, there
is no known example of a triple (X, [ω], J) with
[ω] ∈ H 2(X) s.t. [ω]n 6= 0 and J an a.c.s. and
X known to admit no symplectic structure.