The Picard-Lefschetz theory of complexified Morse functions 1 ...

58 Joe Johns
We, on the other hand, identify
f −1
2 ([c2 − 1/2, c2 + 1/2]) ∩ (k 2 ) −1 ([4(r/2) 2 , 4r 2 ]) with D [r/2,r](ν ∗ K+) ∼ = S 1 × D 2 [r/2,r]
in just one step using Φ 2 . To compare our approach to Milnor’s we express Φ 2
in two steps as follows. Recall that lemma 4.1 says Φ 2 = ρ0 ◦ Φ 2 , where Φ2
is radial symplectic flow to π −1
2 (0) \ {0}. This implies that Φ2 can be expressed
in two steps as symplectic flow to π −1
2 (−1/2), followed by Φ2 −1/2
. Lemma 8.3
below shows that symplectic flow along the real part and gradient flow agree up to
reparameterization, so that implies that the first step of Milnor’s approach agrees with
our first step (up to isotopy). As for comparing Φ −1/2 and η, recall that for each
x ∈ f −1
2 (c2 − 1/2) ∩ (k 2 ) −1 ([4(r/2) 2 , 4r 2 ]), we have
for some
Then since
and
it follows that Φ 2 −1/3
Φ2(x) = (u, 0) + i(0,λv)
(u,λv) ∈ S 1 × D 2 [r/2,r]
η(u,θv) = (cosh θu, sinh θv)
k2((cosh θu, 0) + (0, sinh θv)) = sinh 2 θ + 2 sinh θ
(Above, ϕ is the inverse of this map.)
and η differ only by a radial diffeomorphism
θ ↦→ λ = sinh 2 θ + 2 sinh θ.
Here is the precise statement and proof of the claim in the last remark concerning
symplectic transport on the real part.
Lemma 8.3 Let π : E −→ C be a symplectic Lefschetz fibration, where E is equipped
with symplectic structure ω, such that the regular fibers of π are symplectic, and J
is an almost complex structure on E compatible with ω such that π∗(Jv) = iJπ∗(v).
Suppose E has an anti-symplectic, anti-complex involution
that is,
ι: E −→ E,
ι 2 = id
ι ∗ ω = −ω