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Title:
When is a Stein structure merely symplectic?
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Abstract:
The study of Stein manifolds and their symplectic geometry has increasingly been dominated by the question of "rigid vs. flexible", e.g. subcritical Stein manifolds satisfy an h-principle, so their Stein homotopy type is determined by the homotopy class of the almost complex structure. I will show that in dimension 4, there is a much larger class of Stein domains that exist somewhere between rigid and flexible: while the h-principle does not hold in a strict sense, their Stein deformation type is completely determined by their symplectic deformation type. This result depends on some joint work with Sam Lisi and Jeremy Van Horn-Morris involving the relationship between Stein structures and Lefschetz fibrations, which can sometimes be realised as foliations by J-holomorphic curves.
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