Thanks to Lemma 3.2 – here we use that $D^{-\varepsilon}\in L^{1}$ – and Theorem 2.1, for every $t\in[0,1]$ there is a unique continuous function $\varphi_{t}\in PSH(X,\omega_{0}+t\Omega)$ such that

$$(\omega_{o}+t\Omega+dd^{c}\varphi_{t})^{n}=C_{t}\frac{|s_{1}|^{2k}+\ldots+|s_{p}|^{2k}}{|t_{1}|^{2l}+\ldots+|t_{q}|^{2l}}e^{F}\Omega^{n}\text{ and }\sup_{X}\varphi_{t}=-1,$$

where $C_{t}>0$ is an adequate normalisation constant and $\|\varphi_{t}\|_{{\mathcal{C}}^{0}(X)}$ is uniformly bounded by a constant independant of $t\geq 0$.

We cannot use right away [Y], Theorem 8 p. 403, to ensure that $(\varphi_{t})$ be smooth outside $B$ for $t>0$, since our integral condition is stronger than his. However we can use [Y], Thm 3, p 365 to conclude that, in case $\cap_{i}\{t_{i}=0\}=\emptyset$, $(\varphi_{t})$ is smooth outside $B$ and $dd^{c}\varphi_{t}$ is a form whose coefficients are globally bounded on $X$, hence $\varphi_{t}\in{\mathcal{C}}^{1,1}(X)$ for $t>0$. Since this does not imply ellipticity if $\cap_{i}\{s_{i}=0\}\not=\emptyset$, this does not imply higher regularity on the whole of $X$.

The required uniformity in $t>0$ is not proved in [Y]. To deal with this case, we use a nice trick due to H.Tsuji [Ts].