ScalingStacks

Proposition 6.23 . [04KX]

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Proposition 6.23.

Let f:X→Bf:X\rightarrow B be a fibration satisfying Assumption 6.22 and let Fb¯=f−1​(b¯)F_{\bar{b}}=f^{-1}(\bar{b}) be a smooth fibre. There is a basis γ=(γ1,γ2,γ3)\gamma=(\gamma_{1},\gamma_{2},\gamma_{3}) of H1​(Fb¯,ℤ)H_{1}(F_{\bar{b}},\mathbb{Z}) and coordinates (b1,b2,b3)(b_{1},b_{2},b_{3}) on BB with respect to which the periods of f±:X±→B±f^{\pm}:X^{\pm}\rightarrow B^{\pm} can be written

λ1±=2​π​d​b1,λ2±=d​H±+λ0,λ3±=d​b3,\begin{array}[]{l}\lambda_{1}^{\pm}=2\pi db_{1},\\ \lambda_{2}^{\pm}=dH^{\pm}+\lambda_{0},\\ \lambda_{3}^{\pm}=db_{3},\end{array}

where λ0=arg⁡(b1+i​b2)​d​b1+log⁡|b1+i​b2|​d​b2\lambda_{0}=\arg(b_{1}+ib_{2})db_{1}+\log|b_{1}+ib_{2}|db_{2} and H±∈C∞​(B±)H^{\pm}\in C^{\infty}(B^{\pm}). Moreover, there is a fibre preserving symplectomorphism

Θ±:T∗​B±/ΛH±→(X#)±\Theta^{\pm}:T^{\ast}B^{\pm}/\penalty\Lambda_{H^{\pm}}\rightarrow(X^{\#})^{\pm} (63)

where ΛH±\Lambda_{H^{\pm}} is the integral lattice generated by λ1±,λ2±,λ3±\lambda_{1}^{\pm},\lambda_{2}^{\pm},\lambda_{3}^{\pm}.

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