Corollary 11.1.10`2C05FFD39C6D`

Corollary 11.1.10.2C05FFD39C6D If $n=2$, then for every smooth real function $u$ on $X$,

\[ \mathrm M_\omega (u)=\int _ X\omega ^2+2\int _ Xu\mathrm{dd}^{\mathrm{c}}\omega . \]

Consequently, $\mathrm M_\omega $ is constant on $\mathcal H_\omega $ if and only if $\mathrm{dd}^{\mathrm{c}}\omega =0$.

Proof. The formula is the case $n=2$ of (11.1.6), and it shows that $\mathrm{dd}^{\mathrm{c}}\omega =0$ implies constancy. Conversely, if the mass is constant, proposition 11.1.8 at $u=0$ gives $\mathrm{dd}^{\mathrm{c}}\omega =0$. $\square$

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BibTeX citation

@misc{pome-tag-2C05FFD39C6D,
  author = {{Pome contributors}},
  title = {Corollary 11.1.10},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/2C05FFD39C6D/}
}

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Corollary 11.1.102C05FFD39C6D

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Corollary 11.1.10`2C05FFD39C6D`