ScalingStacks

Lemma 2.7 . [0402]

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Lemma 2.7.

By construction

{∂β1∂μ1=2∂∂η(α1+α3)=2∂v11∂η,∂β1∂μ2=2​∂v12∂η,∂β2∂μ1=2∂v21∂η,∂β2∂μ2=2​∂v22∂η∂β0∂μ1=−2∂∂η(v11+v21),∂β0∂μ2=−2​∂∂η​(v12+v22).\begin{cases}\frac{\partial\beta_{1}}{\partial\mu_{1}}=2\frac{\partial}{\partial\eta}(\alpha_{1}+\alpha_{3})=2\frac{\partial v^{11}}{\partial\eta},\quad&\frac{\partial\beta_{1}}{\partial\mu_{2}}=2\frac{\partial v^{12}}{\partial\eta},\\ \frac{\partial\beta_{2}}{\partial\mu_{1}}=2\frac{\partial v^{21}}{\partial\eta},\quad&\frac{\partial\beta_{2}}{\partial\mu_{2}}=2\frac{\partial v^{22}}{\partial\eta}\\ \frac{\partial\beta_{0}}{\partial\mu_{1}}=-2\frac{\partial}{\partial\eta}(v^{11}+v^{21}),\quad&\frac{\partial\beta_{0}}{\partial\mu_{2}}=-2\frac{\partial}{\partial\eta}(v^{12}+v^{22}).\end{cases}\quad

Morever,

∂β1∂η¯=−12​∂w∂μ1,∂β2∂η¯=−12​∂w∂μ2,∂β0∂η¯=12​(∂w∂μ1+∂w∂μ2).\frac{\partial\beta_{1}}{\partial\bar{\eta}}=-\frac{1}{2}\frac{\partial w}{\partial\mu_{1}},\quad\frac{\partial\beta_{2}}{\partial\bar{\eta}}=-\frac{1}{2}\frac{\partial w}{\partial\mu_{2}},\quad\frac{\partial\beta_{0}}{\partial\bar{\eta}}=\frac{1}{2}(\frac{\partial w}{\partial\mu_{1}}+\frac{\partial w}{\partial\mu_{2}}).

Therefore the type (1,0) forms

(2.15) ζ1′=ζ1+β1​d​η,ζ2′=ζ2+β2​d​η,ζ0′=−ζ1−ζ2+β0​d​η\zeta_{1}^{\prime}=\zeta_{1}+\beta_{1}d\eta,\quad\zeta_{2}^{\prime}=\zeta_{2}+\beta_{2}d\eta,\quad\zeta_{0}^{\prime}=-\zeta_{1}-\zeta_{2}+\beta_{0}d\eta

are closed, namely they are holomorphic differentials.

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