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Corollary 15.19.12A1536B3C6F39

Corollary 15.19.12.A1536B3C6F39 For the preceding geometric data and every smooth positive probability volume form $\mu $ on $X$, the solutions of (15.19.2) have the bounds and subsequential convergence of theorem 15.19.7, with $Z=E$. Moreover, for each exceptional curve,

\[ \int _{E_ j}\omega _\varepsilon =\varepsilon \int _{E_ j}\eta \longrightarrow 0. \]

Proof. Apply the theorem using lemma 15.19.10. Since $\pi $ contracts $E_ j$, the restriction of $\theta $ to it is zero. Stokes on the compact smooth curve gives $\int _{E_ j}\mathrm{dd}^{\mathrm{c}}u_\varepsilon =0$, proving the area identity. $\square$

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