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

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].

First, assume $\cap_{i}\{s_{i}=0\}\cup\cap_{i}\{t_{i}=0\}=\emptyset$. Hence the family of equations under consideration can be rewritten as:

$F$ being smooth.

Tsuji’s trick is as follows. By Kodaira’s lemma, there exists $E$ an effective Cartier divisor of $X$ such that $[\omega_{o}]=[\kappa_{\varepsilon}]+\varepsilon[E]$ where $[\kappa_{\varepsilon}]$ is ample, hence we may choose a representative $\kappa_{\varepsilon}$ which is a Kähler form for every $\varepsilon>0$ small enough. We may actually assume $E$ contains $B$ and use a family of $E$ such that $\cap Supp(E)=B$, by Nakamaye’s theorem on base loci [Na].

Actually, despite the notation, it will NOT be necessary to let $\varepsilon$ decrease to $0$ ⁶⁶ 6 This technical device could be useful to study finer regularity results and we will fix once for all such an $\varepsilon>0$.

Let $\sigma\in H^{0}(X,\mathcal{O}_{X}(E))$ be the canonical section vanishing on $E$ with the appropriate multiplicity. We can fix a smooth hermitian metric on this line bundle such that the Poincaré Lelong equation holds,

The function $\varphi_{t}:=\varphi_{t}-\varepsilon\log|\sigma|^{2}$ is smooth in $X\setminus E$ and is a classical solution to the PDE

where $(F_{\varepsilon,t})_{1\geq t>0}$ is uniformly bounded in the ${\mathcal{C}}^{\infty}(X)$-topology of functions and $\kappa_{t}=\kappa_{\varepsilon}+t\Omega$ is uniformly bounded in the ${\mathcal{C}}^{\infty}$-topology of Kähler forms on $X$.

We can use the result of the calculation in [Y], section 2. The important formula is (2.22) p. 351 and in a subsidiary fashion (2.21). In these formulae, at each point $p\in X-E$, an adequate system of normal coordinates for $\kappa_{t}$ is constructed and comparing the notations here and there, we substitute $n$ for $m$, $\kappa_{t}$ for $g_{i\bar{j}}$, $\kappa_{t}+dd^{c}\varphi_{t}$ for $g_{i\bar{j}}^{\prime}$, $\varphi_{t}$ for $\varphi$ and $F_{\varepsilon,t}$ for $F$. The operator $\Delta$ is the Laplace operator (with the analyst’s sign) of $\kappa_{t}$ and $\Delta^{\prime}$ the Laplace operator of $\kappa_{t}+dd^{c}\varphi_{t}$. Also $R_{i\bar{i}l\bar{l}}=R^{t}_{i\bar{i}l\bar{l}}$ is the holomorphic bissectional curvature of $\kappa_{t}$ expressed in the above system of normal coordinates.

Since $\kappa_{t}$ is uniformly bounded in the ${\mathcal{C}}^{2}$ topology of Kähler forms then certainly there is constant $C=C_{\varepsilon}$ independent of $t$ such that (2.21) holds and $C^{\prime}$ also independent of $t$ such that $C^{\prime}>\inf R^{t}_{i\bar{i}l\bar{l}}$.

After these substitutions are made, (2.22) p. 351 reads:

We can fix constants $C_{i}$ independent of $t$ such that

Thus setting $y=n+\Delta\varphi_{t}$ yields

Now by definition

For each $t>0$ the functions $\varphi_{t}$, $\varepsilon\Delta\log|\sigma|^{2}$ and $\Delta\varphi_{t}$ are bounded on $X$. Hence the positive function $e^{-C\varphi_{t}}(n+\Delta\varphi_{t})$ is continuous on $X$, vanishes on $E$ and is smooth on $X-E$. Its maximum is achieved at some point $p_{t}\not\in E$. It follows from the maximum principle that

Therefore $y\leq C_{8}$ with a constant independent of $t>0$. Now $e^{-C\varphi_{t}(p_{t})}=|\sigma(p_{t})|^{+C\varepsilon}e^{-C\varphi_{t}(p_{t})}$. Using the uniform ${\mathcal{C}}^{0}$ estimate for $\varphi_{t}$, we get $0\leq(n+\Delta_{t}\varphi_{t})\leq C_{9}e^{+C\varphi_{t}}$. Since $|\varphi_{t}|$ and $\varepsilon\Delta\log|\sigma|^{2}$ are uniformly bounded by a constant independent of $t>0$, we infer

This yields a $t$-independent ${\mathcal{C}}^{0}$- estimate of $dd^{c}\varphi_{t}$ on the compact subsets of $X-E$ ⁷⁷ 7 Note that $C=C_{\varepsilon}$ and that $\varepsilon C_{\varepsilon}$ might blow up as $\varepsilon$ goes to $0$. .

Standard arguments of the theory of complex Monge-Ampère equations give an interior estimate of $\varphi_{t}$ in ${\mathcal{C}}^{k,\alpha}_{loc}(X-E)$ for every $k\geq 2$, $\alpha\in]0,1[$ which is independent of $t>0$ (see for instance Theorem 5.1, p. 15 in [Bl2]). Hence the family $(\varphi_{t})_{t>0}$ is precompact in every ${\mathcal{C}}^{k,\alpha}_{loc}(X-E)$. Its cluster values are cluster values in ${\mathcal{C}}^{O}(X-E)$ hence they are all equal to $\varphi|_{X-E}$. This implies $\varphi\in{\mathcal{C}}^{k,\alpha}_{loc}(X-E)$, hence that $\varphi\in{\mathcal{C}}^{\infty}(X-E)$.

We study here the equation

The first few steps of the preceding argument can be repeated without changes. Next we apply formula (2.22) in [Y] as earlier, except that we set $F=F_{\varepsilon,t}+\log||s||^{(2k)}$, where $||s||^{(2k)}:=|s_{1}|^{2k}+\ldots+|s_{p}|^{2k}$. This yields

We recall the two preceding inequalities and observe two new ones that are available:

Setting as earlier $y=n+\Delta\varphi_{t}$, we get

After this point, the proof is entirely the same as before.