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

64 Joe Johns
First, set
and construct
M1 = M
π1 : E1 −→ D(c1),
as in §5.2 (but using the local model π 1 loc based on q1 = z 2 1 +z2 2 −z2 3 ) so that π1 has fiber
π −1
1 (c1 + 1) canonically isomorphic to M1 = M with vanishing sphere corresponding
to L1 ⊂ M.
Next, set
M2 = T L2
π/2 (M)
where T L2
π/2 is analogous to the twist operation we defined in §5.3. It is easy to see that
M2 = T L2
π/2 (M) has an exact Lagrangian sphere say L′ 2 ⊂ M2 which corresponds in a
straight-forward way to L2 ⊂ M1 = M. (The precise definition of L2 is analogous to
that of (L j
2 )′ ⊂ M2 for any j, in 5.3.) Define π2 : E2 −→ D(c2) as in §5.2 (but using
the local model π2 loc based on q2 = z2 1 − z22 − z23 ) so that π2 has fiber π −1
2 (c2 + 1)
canonically isomorphic to M2 and vanishing sphere corresponding to L ′ 2 ⊂ M2.
Now define
M0 = T L1
−π/2 (M).
By a construction analogous to that in §5.4, there is an exact Lagrangian sphere L ′ 0
corresponding to L0 ⊂ M. It is an “untwisted” version of L0. (The precise definition is
is the union of T \ N(α)
analogous to that of L ′ 4 ⊂ M2 in §5.4.) Roughly speaking, L ′ 0
and φL1 (Dr(ν ∗K−)), where N(α) is a tubular neighborhood of α, and Dr(ν ∗K−) is
“flat”, i.e. it is no longer “twisted”. Construct π0 : E0 −→ D(c0) as in §5.2 (but using
the local model π0 loc based on q0 = z2 1 +z22 +z23 ) so that π0 is a Lefschetz fibration with
fiber π −1
0 (c0 + 1) canonically isomorphic to M0 and vanishing sphere corresponding
to L ′ 0 ⊂ M2.
Finally let
M3 = M2.
There is an exact Lagrangian submanifold L ′ 3 ⊂ M3 = M2 which corresponds to
L3 ⊂ M. Again, it is the ”untwisted” version of L3 ⊂ M, analogous to L ′ 0 ⊂ M0
before. Construct π3 : E3 −→ D(c3) as in §5.4 (using the local model π3 loc based on
q3 = −z2 1 − z22 − z23 ) so that π3 is a Lefschetz fibration with fiber π −1
3 (c3 − 1) canonically
isomorphic to M3. (Note we have used c3 −1 here rather than c3+1, as in §5.4.)