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

Complexified Morse functions 11
defined on the regular fiber? Naively, this seems plausible since M is obtained, roughly
speaking, by plumbing several disk cotangent bundles together (see §3.4), and one
would hope that the model complexifications on each of these disk cotangent bundles
can be made to agree on the overlaps, so that they patch together to yield a fibration
on M. (The actual construction, though, must combine the fibrations on each disk
cotangent bundle in a more sophisticated way, by combining the regular fibers of each
of the fibrations into one new regular fiber for the putative fibration on M.) In any case,
once this is known one would like to extend this to the slightly more sophisticated set
up of a bifibration on E. Roughly, this is a holomorphic map E −→ C 2 , with generic
singularities, encoding a family of Lefschetz fibrations on the fibers of a Lefschetz
fibration on E. (For the precise definition, see [S08A, §15e].) As pointed out to me
by Maydanskiy, one potential application of such a bifibration (together with work
in progress of Seidel) would be to construct exotic cotangent bundles along the same
lines as Maydanskiy’s recent work on exotic sphere cotangent bundles [M09]. More
tentatively, such bifibrations (and similar structures on the fibers of π2, etc.) may lead
to interesting matching relations among Lagrangian submanifolds in E, M, etc., in a
spirit similar to [S03B], [S08A]. (See section 1.3.2 below for more about matching
conditions which apply to Lagrangian submanifolds more general than spheres.)
1.3.2 Donaldson’s decomposition and generalized matching paths
Donaldson’s idea is as follows. First, one assumes that π maps L onto an embedded
path γ such that
f = γ −1 ◦ (π|L): L −→ [0, 1]
is a Morse function, either by constructing a suitable π for a given L (as achieved
in [AMP05]), or perhaps by deforming the given L and (E,π). Then, each critical
point of f is a critical point of π lying on L, and each unstable and stable manifold
of f is part of a Lefschetz thimble of π. The expectation is that L is isotopic to a
surgery-theoretic combination of all these Lefschetz thimbles. This is well-understood
when L is a sphere and γ runs between just two critical values: L is then the union
of two Lefschetz thimbles meeting at a common vanishing sphere and γ is called a
matching path, see [S08A, §16g], [S03B].
As we mentioned above, the proof of Theorem A involves constructing N ⊂ E by
doing successive surgery operations involving the Lefschetz thimbles, just as in Donaldson’s
proposed decomposition. More precisely, let us assume for convenience of