# Relative methods in symplectic topology.

2009-07

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Relative methods in symplectic topology.

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2009-07

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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 $\mathcal
P^{0}=\{e\in H^2(M;\mathbb R)|e\cdot e>0 \}$.

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

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Dorfmeister, Josef Gerhard. (2009). Relative methods in symplectic topology.. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/53403.

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