Proposition 23.2.20`B365A2A75E69`

Proposition 23.2.20.B365A2A75E69 Let $u$ be a smooth real function. Then

\[ \mathrm{d}(\omega +\mathrm{dd}^{\mathrm{c}}u)-\lambda \wedge (\omega +\mathrm{dd}^{\mathrm{c}}u) =-\lambda \wedge \mathrm{dd}^{\mathrm{c}}u, \]

whereas

\[ \mathrm{d}(\omega +D_\lambda u) =\lambda \wedge (\omega +D_\lambda u). \]

Consequently, an ordinary potential preserves the LCK equation with the same Lee form precisely when $\lambda \wedge \mathrm{dd}^{\mathrm{c}}u=0$. A twisted potential always preserves that equation, and it gives an LCK metric whenever $\omega +D_\lambda u>0$.

Proof. The first identity follows from $\mathrm{d}\mathrm{dd}^{\mathrm{c}}u=0$ and $\mathrm{d}\omega =\lambda \wedge \omega $. The second follows from $\mathrm{d}_\lambda \omega =0$ and (23.2.19). $\square$

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BibTeX citation

@misc{pome-tag-B365A2A75E69,
  author = {{Pome contributors}},
  title = {Proposition 23.2.20},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/B365A2A75E69/}
}

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Proposition 23.2.20B365A2A75E69

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Proposition 23.2.20`B365A2A75E69`