ScalingStacks

In this setting we look at the Kähler forms $\omega_{t}=\omega_{0}+t\omega_{X}$ for $0<t\leq 1$, which are cohomologous to the Ricci-flat metrics $\tilde{\omega}_{t}$. We then define a smooth function $E$ by

$$\mathrm{Ric}(\omega_{X})=\sqrt{-1}\partial\overline{\partial}E,\quad\int_{X}e^{E}\omega_{X}^{n}=\int_{X}\omega_{1}^{n},$$

which is possible thanks to the $\partial\overline{\partial}$-lemma. Then the equation $\mathrm{Ric}(\tilde{\omega}_{t})=0$ is equivalent to

$$\tilde{\omega}_{t}^{n}=a_{t}e^{E}\omega_{X}^{n},$$

where

$$a_{t}=\frac{\int_{X}\omega_{t}^{n}}{\int_{X}\omega_{1}^{n}}.$$

Using the $\partial\overline{\partial}$-lemma again, we can find smooth functions $\varphi_{t}$ for $0<t\leq 1$ so that $\tilde{\omega}_{t}=\omega_{t}+\sqrt{-1}\partial\overline{\partial}\varphi_{t}$, $\sup_{X}\varphi_{t}=0$ and we have

(2.5)

$$(\omega_{t}+\sqrt{-1}\partial\overline{\partial}\varphi_{t})^{n}=a_{t}e^{E}\omega_{X}^{n}.$$

Notice that as $t$ approaches zero, the constants $a_{t}$ behave like

(2.6)

$$\binom{n}{m}\frac{\int_{X}\omega_{0}^{m}\wedge\omega_{X}^{n-m}}{\int_{X}\omega_{1}^{n}}t^{n-m}+O(t^{n-m+1}).$$

We can then write (2.5) as

(2.7)

$$(\omega_{t}+\sqrt{-1}\partial\overline{\partial}\varphi_{t})^{n}=c_{t}t^{n-m}e^{E}\omega_{X}^{n},$$

where the constant $c_{t}$ is bounded away from zero and infinity as $t$ goes to zero. Equation (2.7) has been studied for example in [KT] where a uniform $L^{\infty}$ bound on $\varphi_{t}$ was conjectured. When $m=1$ such a bound can be easily proved using the Moser iteration method (see [ST1]). The bound in the general case was then proved independently by Demailly and Pali [DP] and by Eyssidieux, Guedj and Zeriahi [EGZ2]: