ScalingStacks

Remark 3.7 . [04A6]

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Remark 3.7.

Using the Floer degree formula (63), the asymptotic behaviour of Ω⁡(v1,…​vn)\Omega(v_{1},\ldots v_{n}) at the corners can be extracted from the above proof: at the C​F0​(L,L′)CF^{0}(L,L^{\prime}) corner point pp

Ω⁡(v1,…​vn)=ap​z(θL′−θL)​(p)/π​(1+O⁡(z)),ap≠0,arg⁡ap=θL​(p)modπ​ℤ.\Omega(v_{1},\ldots v_{n})=a_{p}z^{(\theta_{L^{\prime}}-\theta_{L})(p)/\pi}(1+O(z)),\quad a_{p}\neq 0,\quad\arg a_{p}=\theta_{L}(p)\mod\pi\mathbb{Z}.

At the degree one self intersections pl∈C​F1​(L+,L−)p_{l}\in CF^{1}(L_{+},L_{-}) on LL or L′L^{\prime},

Ω⁡(v1,…​vn)=al​z(θL−−θL+)​(pl)/π​(1+O⁡(z)),al≠0,arg⁡al=θL+​(pl)modπ​ℤ.\Omega(v_{1},\ldots v_{n})=a_{l}z^{(\theta_{L_{-}}-\theta_{L_{+}})(p_{l})/\pi}(1+O(z)),\quad a_{l}\neq 0,\quad\arg a_{l}=\theta_{L_{+}}(p_{l})\mod\pi\mathbb{Z}.

At the degree nn output qq,

Ω⁡(v1,…​vn)=aq​z(θL−θL′)​(q)/π​(1+O⁡(z)),aq≠0,arg⁡aq=θL′​(q)modπ​ℤ.\Omega(v_{1},\ldots v_{n})=a_{q}z^{(\theta_{L}-\theta_{L^{\prime}})(q)/\pi}(1+O(z)),\quad a_{q}\neq 0,\quad\arg a_{q}=\theta_{L^{\prime}}(q)\mod\pi\mathbb{Z}.

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