ScalingStacks

Fix once and for all an $\varepsilon>0$ small enough so that Kodaira’s lemma holds. First of all notice that the classes $\alpha_{t}-\varepsilon E=\kappa_{\varepsilon}^{t}$ are all Kähler when $t$ is close to $1$. Choose a Kähler form $\chi_{\varepsilon}\in\kappa_{\varepsilon}$, let $\sigma\in H^{0}(X,\mathcal{O}_{X}(E))$ be the canonical section, and fix a Hermitian metric $|\cdot|$ on $E$ such that the following Poicaré-Lelong equation holds

(4.4)

$$\omega-\varepsilon[E]=\chi_{\varepsilon}-\varepsilon\sqrt{-1}\partial\overline{\partial}\log|\sigma|,$$

where $[E]$ denotes the current of integration on $E$. Then we have

$$\beta_{t}-\varepsilon[E]=\chi_{\varepsilon}+(\beta_{t}-\omega)-\varepsilon\sqrt{-1}\partial\overline{\partial}\log|\sigma|,$$

and $\chi_{\varepsilon}^{t}=\chi_{\varepsilon}+(\beta_{t}-\omega)$ is Kähler for $t$ close to $1$. There are smooth functions $\varphi_{t}$ solutions of

(4.5)

$$\omega_{t}^{n}=(\beta_{t}+\sqrt{-1}\partial\overline{\partial}\varphi_{t})^{n}=\alpha_{t}^{n}\Omega,$$

where the positive constants $\alpha_{t}^{n}$ approach $\alpha^{n}$ as $t$ goes to $1$, and $\sup_{X}\varphi_{t}=0$. We apply Theorem 2.1 and Proposition 3.1 of [EGZ] again, and get uniform estimates $\|\varphi_{t}\|_{C^{0}}\leq C_{0}$ independent of $t$. Outside $E$ we have

$$\beta_{t}=\chi_{\varepsilon}^{t}-\varepsilon\sqrt{-1}\partial\overline{\partial}\log|\sigma|,$$

so that the functions $\psi_{t}=\varphi_{t}-\varepsilon\log|\sigma|$ solve

(4.6)

$$(\chi_{\varepsilon}^{t}+\sqrt{-1}\partial\overline{\partial}\psi_{t})^{n}=\alpha_{t}^{n}\Omega=e^{F_{\varepsilon}^{t}}(\chi_{\varepsilon}^{t})^{n}$$

there, for some appropriate smooth functions $F_{\varepsilon}^{t}$, defined on the whole of $X$. As $t$ approaches $1$, the Kähler forms $\chi_{\varepsilon}^{t}$ are uniformly bounded in the smooth topology (with eigenvalues bounded away from $0$ uniformly), and so are the functions $F_{\varepsilon}^{t}$. Yau’s second order estimates [Y2] for the Monge-Ampère equation (4.6) give

(4.7)

$$\triangle^{\prime}_{t}(e^{-A\psi_{t}}(n+\triangle_{t}\psi_{t}))\geq e^{-A\psi_{t}}\left(-C_{1}-C_{2}(n+\triangle_{t}\psi_{t})+(n+\triangle_{t}\psi_{t})^{\frac{n}{n-1}}\right),$$

where $A,C_{1}$ and $C_{2}$ are uniform positive constants, $\triangle_{t}$ is the Laplacian of $\chi_{\varepsilon}^{t}$ and $\triangle^{\prime}_{t}$ is the Laplacian of $\chi_{\varepsilon}^{t}+\sqrt{-1}\partial\overline{\partial}\psi_{t}$. Now notice that on $X\backslash E$ we have

$$e^{-A\psi_{t}}(n+\triangle_{t}\psi_{t})=|\sigma|^{A\varepsilon}e^{-A\varphi_{t}}(n+\triangle_{t}\varphi_{t}-\varepsilon\triangle_{t}\log|\sigma|),$$

and

$$\left|\triangle_{t}\log|\sigma|\right|\leq C,$$

for some uniform constant $C$. Hence the function $e^{-A\psi_{t}}(n+\triangle_{t}\psi_{t})$ goes to zero when we approach $E$, and so its maximum will be attained. The maximum principle applied to (4.7) then gives

$$n+\triangle_{t}\psi_{t}\leq Ce^{A(\psi_{t}-\inf_{X\backslash E}\psi_{t})},$$

on the whole of $X\backslash E$. But noticing that $\inf_{X\backslash E}\psi_{t}\geq\inf_{X}\varphi_{t}-C$ for a uniform constant $C$, and recalling that $|\varphi_{t}|\leq C_{0}$, we get

$$n+\triangle_{t}\varphi_{t}\leq C+n+\triangle_{t}\psi_{t}\leq C(1+|\sigma|^{-A\varepsilon}).$$

This gives uniform interior $C^{2}$ estimates of $\varphi_{t}$ and $\psi_{t}$ on compact sets of $X\backslash E$. Then the Harnack estimate of Evans-Krylov gives uniform $C^{2,\gamma}$ estimates, for some $0<\gamma<1$, and a standard bootstrapping argument gives uniform $C^{k,\gamma}$ estimates for all $k\geq 2$, on compact sets of $X\backslash E$, independent of $t<1$. Thus the family $(\varphi_{t})$ is precompact $C^{k,\gamma^{\prime}}(X\backslash E)$ for any $0<\gamma^{\prime}<\gamma$, and any limit point $\psi$ belongs to $PSH(X\backslash E,\omega)$, it satisfies

$$(\omega+\sqrt{-1}\partial\overline{\partial}\psi)^{n}=\alpha^{n}\Omega$$

on $X\backslash E$, and is bounded near $E$. Hence $\psi$ extends to a bounded function in $PSH(X,\omega)$ and the above Monge-Ampère equation holds on $X$ because the Borel measure $(\omega+\sqrt{-1}\partial\overline{\partial}\psi)^{n}$ doesn’t charge the analytic set $E$. Then by the uniqueness part of Theorem 2.1 of [EGZ], we must have $\psi=\varphi$. This implies that $\varphi_{t}\to\varphi$ in $C^{\infty}$ on compact sets of $X\backslash E$, and that $\varphi$ is smooth there.