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Lemma 3.8 . [042K]

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Lemma 3.8.

Let 0≤θ1∞,θ2∞≤2​π0\leq\theta_{1}^{\infty},\theta_{2}^{\infty}\leq 2\pi be two real numbers to be determined. The holomorphic 1-forms

{ζ~1′=ζ~1+(β~1+β1​(0,0,1)​π​−1−−1​θ1∞)​d​η,ζ~2′=ζ~2+(β~2+β2​(0,0,1)​π​−1−−1​θ2∞)​d​η,ζ~0′=ζ~0+(β~0+β0​(0,0,1)​π​−1+−1​θ1∞+−1​θ2∞)​d​η\begin{cases}\tilde{\zeta}_{1}^{\prime}=\tilde{\zeta}_{1}+(\tilde{\beta}_{1}+\beta_{1}(0,0,1)\pi\sqrt{-1}-\sqrt{-1}\theta_{1}^{\infty})d\eta,\\ \tilde{\zeta}_{2}^{\prime}=\tilde{\zeta}_{2}+(\tilde{\beta}_{2}+\beta_{2}(0,0,1)\pi\sqrt{-1}-\sqrt{-1}\theta_{2}^{\infty})d\eta,\\ \tilde{\zeta}_{0}^{\prime}=\tilde{\zeta}_{0}+(\tilde{\beta}_{0}+\beta_{0}(0,0,1)\pi\sqrt{-1}+\sqrt{-1}\theta_{1}^{\infty}+\sqrt{-1}\theta_{2}^{\infty})d\eta\end{cases}

are closed, namely they are holomorphic differentials.

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