New constructions and computations in rigid and flexible symplectic geometry and applications to several complex variables
- This thesis is divided into three parts. In the first part, we give a complete characterization of those disk bundles over surfaces which embed as rationally convex strictly pseudoconvex domains in $\C^2$. We recall some classical obstructions and prove some deeper ones related to symplectic and contact topology. We explain the close connection to Lagrangian surfaces with isolated singularities and develop techniques for constructing such surfaces. Our proof also gives a complete characterization of Lagrangian surfaces with open Whitney umbrellas, answering a question first posed by Givental in 1986. In the second part, we introduce a class of Weinstein manifolds which are sublevel sets of flexible Weinstein manifolds but are not themselves flexible. These manifolds, called subflexible, exhibit rather subtle behavior with respect to both pseudoholomorphic curve invariants and symplectic flexibility. We construct a large class of examples and prove that every flexible Weinstein manifold can be Weinstein homotoped to have a nonflexible sublevel set. This resolves some recent open questions in symplectic flexibility. In the third part, we establish an infinitesimal version of fragility for squared Dehn twists around even dimensional Lagrangian spheres. The precise formulation involves twisting the Fukaya category by a closed two-form or bulk deforming it by a half-dimensional cycle. As our main application, we compute the twisted and bulk deformed symplectic cohomology of the subflexible Weinstein manifolds constructed in the second part.
|Type of resource
|electronic; electronic resource; remote
|1 online resource.
|Siegel, Kyler Bryce
|Stanford University, Department of Mathematics.
|Eliashberg, Y, 1946-
|Eliashberg, Y, 1946-
|Statement of responsibility
|Kyler Bryce Siegel.
|Submitted to the Department of Mathematics.
|Thesis (Ph.D.)--Stanford University, 2016.
- © 2016 by Kyler Bryce Siegel
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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