PomeExample 14.18.3.8AC1A3B0F2AF If $n=2$ and $\gamma _0$ is Gauduchon, then $\Gamma _ u=\gamma _0+\mathrm{dd}^{\mathrm{c}}u=\eta _ u$, and $\eta _ u$ is also Gauduchon. The mass identity of theorem 11.1.7 gives $\int _ X\eta _ u^2=\int _ X\gamma _0^2$. Therefore
\[ b_\gamma (F)=\log \int _ X\gamma _0^2- \log \int _ X\mathrm{e}^ F\gamma ^2, \qquad \mathrm{d}\mu _ u^\gamma =\frac{\mathrm{e}^ F\gamma ^2}{\int _ X\mathrm{e}^ F\gamma ^2}. \]
The adjoint equation of proposition 14.12.3 gives the same measure, because $\mathrm{dd}^{\mathrm{c}}\eta _ u=0$ and hence $\rho _ u$ is constant. Along $F+tG$, the second derivative of $b_\gamma $ is minus the variance of $G$ with respect to this probability measure.
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