(Good test function) Given the divisor $E_{0}$, we can choose a $C^{2}$ test function $v$ on $X_{t}$ such that the following hold uniformly for small $t\neq 0$:

• •

  $v$ is zero for $h\geq\delta$.

• •

  Globally $0\leq v\leq-\log|t|$.

• •

  For any divisor $E_{j}$ intersecting $E_{0}$, there is a subset of $E_{j}^{0}$ with measure at least $C_{2}$ on which $\sqrt{-1}\partial\bar{\partial}v\wedge\omega_{\mathcal{X}}|_{X_{t}}^{n-1}\geq C_{3}\omega_{\mathcal{X}}|_{X_{t}}^{n}$.

• •

  For $C_{4}|t|^{2}\leq h\leq\delta$, the form $\sqrt{-1}\partial\bar{\partial}v\wedge\omega_{\mathcal{X}}|_{X_{t}}^{n-1}\geq 0$.

• •

  For $h\leq C_{4}|t|^{2}$, the form $\sqrt{-1}\partial\bar{\partial}v\wedge\omega_{\mathcal{X}}|_{X_{t}}^{n-1}\geq-C_{5}\omega_{\mathcal{X}}|_{X_{t}}^{n}$.