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Relative methods in symplectic topology.

Dorfmeister, Josef Gerhard (2009)

Relative methods in symplectic topology.

Thesis or Dissertation

In the first part we define relative Ruan invariants that count embedded connected symplectic submanifolds which contact a fixed symplectic hypersurface $V$ in a symplectic 4-manifold $(X,\omega)$ at prescribed points with prescribed contact orders (in addition to insertions on $X\backslash V$) for generic $V$. We obtain invariants of the deformation class of $(X,V,\omega)$. Two large issues must be tackled to define such invariants: (1) Curves lying in the divisor $V$ and (2) genericity results for almost complex structures constrained to make $V$ symplectic. Moreover, these invariants are refined to take into account rim tori decompositions. In the latter part of the paper, we extend the definition to disconnected submanifolds and construct relative Gromov-Taubes invariants.
In the second part we introduce the notion of the relative symplectic cone
$\mathcal C_M^V$. As an application, we determine the {\it symplectic cone $\mathcal C_M$} of certain $T^2$-fibrations. In
particular, for some elliptic surfaces we verify the conjecture in
\cite{TJL1}: If $M$ underlies a minimal K\"ahler surface with
$p_g>0$, the symplectic cone $\mathcal C_M$ is equal to $\mathcal
P^{c_1(M)}\cup \mathcal P^{-c_1(M)}$, where $\mathcal
P^{\alpha}=\{e\in H^2(M;\mathbb R)|e\cdot e>0 \hbox{ and } e\cdot
\alpha>0\}$ for nonzero $\alpha \in H^2(M;\mathbb R)$ and $\mathcal
P^{0}=\{e\in H^2(M;\mathbb R)|e\cdot e>0 \}$.

University of Minnesota Ph.D. dissertation. July 2009. Major: Mathematics. Advisor: Tian-Jun Li. 1 computer file (PDF); v, 110 pages.

Dorfmeister, Josef Gerhard.
(2009).
Relative methods in symplectic topology..
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