Example 10.18.3`0C57A6C5BFEC`

Example 10.18.3.0C57A6C5BFEC The weak closure $\mathscr C_\omega $ of the smooth strictly positive normalized Gauduchon forms can be smaller than $\mathscr G_\omega $, even on a Kähler surface. Let $X$ be the blowup of $\mathbb P^2$ at one point and $C$ its exceptional curve. Fix a Kähler form $\omega $ and a smooth closed representative $\beta _ C$ of the class of $[C]$. Then

\[ T_ C=\frac{[C]}{\int _ C\omega }\in \mathscr G_\omega , \qquad \int _ X\beta _ C\wedge T_ C=\frac{-1}{\int _ C\omega }<0, \]

whereas every smooth strictly positive $T\in \mathscr G_\omega $ satisfies $\int _ X\beta _ C\wedge T=\int _ C T>0$. In particular $T_ C\notin \mathscr C_\omega $.

Proof. The integration current $[C]$ is positive and closed, so its normalization belongs to $\mathscr G_\omega $. Its pairing with the smooth form $\beta _ C$ is the self-intersection $C^2=-1$, computed in example 12.6.3. For smooth $T$, the Poincaré–Lelong formula and $\mathrm{dd}^{\mathrm{c}}T=0$ give the second identity. Since the pairing with the fixed smooth form $\beta _ C$ is continuous under weak convergence of currents, it is nonnegative on the weak closure of the smooth tests, and this closure does not contain $T_ C$. $\square$

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@misc{pome-tag-0C57A6C5BFEC,
  author = {{Pome contributors}},
  title = {Example 10.18.3},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/0C57A6C5BFEC/}
}

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Example 10.18.30C57A6C5BFEC

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Example 10.18.3`0C57A6C5BFEC`