By assumption, we can write $T=\omega|_{V}+i\partial\overline{\partial}\varphi$ for some $L^{1}$ usc function $\varphi$ on $V$ which has analytic singularities on $V$. Our goal is to extend $\varphi$ to a function $\Phi$ on $X$ with $\omega+i\partial\overline{\partial}\Phi$ a Kähler current.

Thanks to [6, Lemma 2.1], there exists a function $\psi:X\rightarrow[-\infty,\infty)$ which is smooth on $X\backslash V$, with analytic singularities along $V$, and with $i\partial\overline{\partial}\psi\geqslant-A\omega$ as currents on $X$, for some large $A$. Then, for $\delta$ sufficiently small, we have that $R=\omega+\delta i\partial\overline{\partial}\psi$ is a Kähler current on $X$ with analytic singularities along $V$. Fix one such $\delta$ and define $F=\delta\psi$.

Choose $\varepsilon>0$ small enough so that

holds as currents on $V$. We can cover $V$ by finitely many charts $\{W_{j}\}_{1\leqslant j\leqslant N}$ such that on each $W_{j}$ there are local coordinates $(z_{1},\dots,z_{n})$ and so that $V\cap W_{j}=\{z_{1}=\dots=z_{n-k}=0\}$, where $k=\dim V$. Write $z=(z_{1},\dots,z_{n-k})$ and $z^{\prime}=(z_{n-k+1},\dots,z_{n})$ and define a function $\varphi_{j}$ on $W_{j}$ (with analytic singularities) by

where $A>0$ is a constant. If we shrink the $W_{j}$’s slightly, still preserving the property that $V\subset\cup_{j}W_{j}$, we can choose $A$ sufficiently large so that

holds on $W_{j}$ for all $j$. It will also be useful to fix slightly smaller open sets $W_{j}^{\prime}\Subset U_{j}\Subset W_{j}$ such that $\cup_{j}W_{j}^{\prime}$ still covers $V$. Note that since $\varphi$ is smooth at the generic point of $V$, by construction all functions $\varphi_{j}$ are also smooth in a neighborhood of the generic point of $V\cap W_{j}$.

We wish to glue the functions $\varphi_{j}$ together to produce a Kähler current defined in a neighborhood of $V$ in $X$. This would be straightforward if the functions $\varphi_{j}$ were continuous, thanks to a procedure of Richberg [13], but in our case the functions $\varphi_{j}|_{V}$ have poles along

We can still use the technique of Richberg to produce a Kähler current in an open neighborhood in $X$ which does not contain the polar set $P$. An argument using resolution of singularities will then allow us to get a Kähler current in a whole neighborhood of $V$.

The first step is to consider two open sets $W_{1},W_{2}$ in the covering with $W_{1}^{\prime}\cap W_{2}^{\prime}\cap V$ nonempty, and fix a compact set $K\subset V$ with $(W_{1}^{\prime}\cup W_{2}^{\prime})\cap V\subset K\subset(U_{1}\cup U_{2})\cap V$. Let $M_{1}=K\cap\partial U_{2},M_{2}=K\cap\partial U_{1},$ so that $M_{1}$ and $M_{2}$ are disjoint compact subsets of $V$. This setup is depicted in figure 1.

Pick $\theta_{1}$ a smooth nonnegative cutoff function which is identically $1$ in a neighborhood of $M_{2}$ in $X$ and $\theta_{2}$ a smooth nonnegative cutoff function which is identically $1$ in a neighborhood of $M_{1}$ in $X$ so that the supports of $\theta_{1}$ and $\theta_{2}$ are disjoint. Then, if we choose $\eta>0$ small, the functions

$j=1,2$ have analytic singularities and satisfy $\omega+i\partial\overline{\partial}\tilde{\varphi}_{j}\geqslant\varepsilon\omega$ on $W_{j}$. Furthermore they are smooth in a neighborhood of the generic point of $V\cap W_{j}$. On $W_{1}\cap W_{2}$ we then define

which satisfies $\omega+i\partial\overline{\partial}\tilde{\varphi}_{0}\geqslant\varepsilon\omega$ and equals $\varphi$ on $V\cap W_{1}\cap W_{2}$. Consider now a neighborhood of $M_{2}\backslash P$ in $X$, small enough so that $\theta_{1}=1$ and $\varphi_{1},\varphi_{2}$ are finite on it. Since $\varphi_{1},\varphi_{2}$ agree on $V$ and are smooth on this neighborhood, we see that there exists a possibly smaller such neighborhood where

so that $\tilde{\varphi}_{0}=\tilde{\varphi}_{2}$ there. Similarly, on any sufficiently small neighborhood of $M_{1}\backslash P$ we have $\tilde{\varphi}_{0}=\tilde{\varphi}_{1}$. Therefore there is an open neighborhood $W_{0}$ of $(K\backslash((M_{1}\cap P)\cup(M_{2}\cap P)))\cap\overline{U_{1}}\cap\overline{U_{2}}$ in $X$ such that $\tilde{\varphi}_{0}=\tilde{\varphi}_{1}$ on $W_{0}\backslash\overline{U_{2}}$ and $\tilde{\varphi}_{0}=\tilde{\varphi}_{2}$ on $W_{0}\backslash\overline{U_{1}}$. Therefore we can define

which is a neighborhood of $K\backslash((M_{1}\cap P)\cup(M_{2}\cap P))$ in $X$, and define a function $\varphi^{\prime}$ on $W^{\prime}$ to be equal to $\tilde{\varphi}_{0}$ on $W_{0}$, equal to $\tilde{\varphi}_{1}$ on $U_{1}\backslash\overline{U_{2}}$ and equal to $\tilde{\varphi}_{2}$ on $U_{2}\backslash\overline{U_{1}}$. Then $\varphi^{\prime}$ satisfies $\omega+i\partial\overline{\partial}\varphi^{\prime}\geqslant\varepsilon\omega$ and equals $\varphi$ on $W^{\prime}\cap V$. Clearly, $W^{\prime}$ contains $(W_{1}^{\prime}\cup W_{2}^{\prime})\cap(V\backslash P)$. We refer to $W^{\prime}$ as a pinched neighborhood of $K$, since in general it is not a neighborhood of the whole of $K$ and it might pinch off at points in $M_{1}\cap P$ and $M_{2}\cap P$. We will later need to decrease the value of $\eta$, which might change the set $W^{\prime}$ slightly, but it will still remain a pinched neighborhood of $K$.

We now deal with points in $M_{1}\cap P$ and $M_{2}\cap P$. By symmetry, it suffices to consider a point $p\in M_{2}\cap P$. Recall that $R=\omega+i\partial\overline{\partial}F$ is a Kähler current on $X$ with analytic singularities exactly along $V$. Choose a small coordinate neighborhood $U\subset X$ centered at $p$, small enough so that $\theta_{1}=1$ and $\theta_{2}=0$ on $U$. In particular at points in $W^{\prime}$ sufficiently near $p$ we have $\varphi^{\prime}=\tilde{\varphi}_{0}=\tilde{\varphi}_{2}=\varphi_{2}$. Since $\tilde{\varphi}_{1},\tilde{\varphi}_{2},F$ have analytic singularities, they can be expressed as (recall that $\tilde{\varphi}_{1}=\tilde{\varphi}_{2}$ on $V\cap U$)

near $p$, where $\sigma_{k}$, $k=1,2,3$ are local smooth functions, and $f_{i},\tilde{f}_{i},g_{j}$ are local holomorphic functions, with the functions $g_{j}$ locally defining $V$. Moreover, when restricted to $V$, we have

since $\varphi_{1},\varphi_{2}$ both extend $\varphi$. Then by the above argument, the function $\varphi^{\prime}$ is defined at least on the set $\mathcal{S}\subset U$ given by

where the strict inequality in particular requires $\tilde{\varphi}_{2}$ to be finite. The idea is to show that the singularities of $F$ are comparable to those of $\tilde{\varphi}_{2}$ on $U\backslash\mathcal{S}$. That is, for $0<\nu<1$ we consider the subset of $U$ given by

where we now allow points where both sides of the inequality are $-\infty$. In particular, we always have that $p\in E_{\nu}$. We can also subtract a constant to $F$ so that $\sup_{X}F\leqslant 0$, and then we see that the sets $E_{\nu}$ are decreasing in $\nu$.

We refer the reader to figure 2 for the geometry of this local gluing problem.

The proof of Lemma 2.1 requires several additional lemmas. The main technique we use is Hironaka’s resolution of singularities. Define an analytic set $G\subset U$ by

and let $\mathcal{I}_{G}$ be its defining ideal sheaf. By shrinking $U$ if necessary, we may assume that every irreducible component of $G$ passes through $p$. Let $\pi:\tilde{U}\rightarrow U$ be a log resolution of $\mathcal{I}_{G}$ obtained by blowing up smooth centers. In order to simplify the notation, we assume that we first blow up $V$, to obtain a divisor $D$, and then resolve the strict transform of $G$. After resolving, we have that

is a sum of smooth divisors with simple normal crossings and $\tilde{V}$ is the irreducible divisor containing $\pi^{-1}(v)$ for a generic point $v\in V$ ($\tilde{V}$ is the strict transform of $D$). If we can show that $\pi^{-1}(E_{\nu}\cup\mathcal{S})$ contains an open neighborhood of $\tilde{V}+\sum_{\ell}D_{\ell}$, then it would follow immediately that $E_{\nu}\,\cup\mathcal{S}$ contains an open neighborhood of $p$.

By Lemma 2.2, it suffices to work on a compact set away from $\tilde{V}$. Fix a point $q\in D_{\ell}\cap\overline{\pi^{-1}(\mathcal{S}^{c})}$ and an open set $Z\subset\tilde{U}$ disjoint from $\tilde{V}$, with a coordinate system $(w_{1},\dots,w_{n})$ centered at $q$ so that

where $U_{F}(w)$ and $U_{2}(w)$ are smooth, positive functions on $\overline{Z}$, and $\beta_{i},\gamma_{i}$ are nonnegative real numbers. Our goal is to find $0<\nu<1$ such that

contains a neighborhood of $0\in\mathbb{C}^{n}$. First, we prove a lemma.

With these results in place, we can easily complete the proof of Lemma 2.1.

Furthermore, Lemma 2.1 holds with $\nu_{0}$ independent of the value of $\eta>0$, which we are then free to decrease later on. We pick $\nu\leqslant\nu_{0}$ so that

holds as currents on $X$. Since $\omega$ is a Kähler metric, we have

Recall that $E_{\nu}=\{\nu F\geqslant\tilde{\varphi}_{2}\}\cap U$ and $\mathcal{S}=\{\tilde{\varphi}_{2}>\tilde{\varphi}_{1}\}\cap U$. It follows that on $\mathcal{S}$ we have $\tilde{\varphi}_{2}>-\infty$, and so if $U^{\prime}\subset U$ is a slightly smaller open neighborhood of $p$, then on $(\partial E_{\nu})\cap\mathcal{S}\cap U^{\prime}$ we have $F>-\infty$ and $\nu F=\tilde{\varphi}_{2}$. In particular, $\nu F>\tilde{\varphi}_{2}$ in a neighborhood of $(\partial E_{\nu})\cap\mathcal{S}\cap U^{\prime}$ and so the function

is defined in a neighborhood $U_{p}$ of $p$ and satisfies $\omega+i\partial\overline{\partial}\varphi_{\nu}\geqslant\nu^{2}\omega$ (and the value of $\nu$ does not change if we decrease $\eta$, since the cutoff functions $\theta_{1}$ and $\theta_{2}$ are constant on $U$). Furthermore, since $F$ goes to $-\infty$ on $V$ while $\varphi^{\prime}$ is finite on $(V\backslash P)\cap W^{\prime}$, we have that $\varphi_{\nu}$ equals $\varphi^{\prime}=\varphi$ on $U_{p}\cap(V\backslash P)$.

Repeating this argument at every point $p\in M_{2}\cap P$, as well as every point $p\in M_{1}\cap P$, taking a finite covering given by the resulting open sets $U_{p}$, and taking the smallest $\nu$ of the resulting ones, we conclude that there exists $\nu>0$ sufficiently small such that $\varphi_{\nu}$ is defined in a whole neighborhood $W_{\nu}$ of $K$ in $X$, and satisfies the same properties. This completes the first step.

We then fix slightly smaller open sets $W^{\prime}_{\nu}\Subset U_{\nu}\Subset W_{\nu}$ such that $\cup_{j\geqslant 3}W_{j}^{\prime}\cup W^{\prime}_{\nu}$ still covers $V$. We replace $W_{1}$ and $W_{2}$ with $W_{\nu}$, and replace $\varphi_{1}$ and $\varphi_{2}$ with $\varphi_{\nu}$, and repeat the same procedure with two other open sets in this new covering. The only difference is that while the functions $\varphi_{1},\varphi_{2}$ have analytic singularities, this is not the case for $\varphi_{\nu}$, which is instead locally given as the maximum of finitely many functions with analytic singularities. We now explain what modifications are needed in the arguments above.

At any subsequent step, we will have two open sets $W_{a},W_{b}$ with $W_{a}^{\prime}\Subset U_{a}\Subset W_{a}$ and $W_{b}^{\prime}\Subset U_{b}\Subset W_{b}$ and with $W^{\prime}_{a}\cap W^{\prime}_{b}\cap V$ nonempty. On $W_{a}$ we have a function $\varphi_{a}$ with $\omega+i\partial\overline{\partial}\varphi_{a}\geqslant\varepsilon\omega$, with $\varphi_{a}=\varphi$ on $W_{a}\cap V$, and similarly for $W_{b}$. Then exactly as before we obtain a function $\varphi^{\prime}$ on a neighborhood of $K\backslash((M_{a}\cap P)\cup(M_{b}\cap P))$, which is equal to $\varphi^{\prime}=\max\{\tilde{\varphi}_{a},\tilde{\varphi}_{b}\}$ on $W_{a}\cap W_{b}$, where we picked cutoff functions $\theta_{a},\theta_{b}$ as before and defined $\tilde{\varphi}_{a}=\varphi_{a}-\eta^{\prime}\theta_{a},\tilde{\varphi}_{b}=\varphi_{b}-\eta^{\prime}\theta_{b}$, where $\eta^{\prime}>0$ is small enough so that $\omega+i\partial\overline{\partial}\tilde{\varphi}_{a}$ and $\omega+i\partial\overline{\partial}\tilde{\varphi}_{b}$ are larger than $\varepsilon^{\prime}\omega$ for some $\varepsilon^{\prime}>0$. Because of the construction we just did, near a point $x\in M_{b}\cap P$ we can write

for some $p>0$, some $0<\nu_{a}<1$ and a smooth function $\rho_{a}$ (in general the maximum will contain several terms of the form $\nu_{j}F+\rho_{j}$, but since $F(x)=-\infty$, up to shrinking the open set where we work on, only one of them contributes to the maximum). Here the functions $\tilde{\varphi}_{i_{k}}$ are defined in (2.1), so they have analytic singularities, and so does $F$, and we can also assume that their values of the parameter $\eta$ are all equal and smaller than $\eta^{\prime}/2$. Similarly, we can write

We work again on a small coordinate neighborhood $U$ centered at $x$ where $\theta_{a}=1$ and $\theta_{b}=0$. We proved earlier that $\varphi^{\prime}$ is defined at least on $\mathcal{S}=\{\tilde{\varphi}_{b}>\tilde{\varphi}_{a}\}\cap U$. For $0<\nu<1$ we let $E_{\nu}=\{\nu F\geqslant\tilde{\varphi}_{b}\}\cap U$. If we can show that there exists $\nu$ such that $E_{\nu}\cup\mathcal{S}$ contains a neighborhood of $x$, then we can complete this step exactly as before. For simplicity, we write

To prove this, we pick a log resolution $\pi:\tilde{U}\to U$ of the ideal sheaf of

with $\pi^{-1}(G)=\tilde{V}+\sum_{\ell}D_{\ell}$ as before. Now note that $\mathcal{S}$ contains the set $\mathcal{A}\cap\mathcal{B},$ where

The set $\mathcal{A}$ equals

Since $\tilde{\varphi}_{j_{\ell}}=\varphi_{j_{\ell}}-\eta\theta_{j_{\ell}}$ with $0\leqslant\theta_{j_{\ell}}\leqslant 1$, and similarly for $\tilde{\varphi}_{i_{k}}$, we see that

Thanks to Lemma 2.2, each of the sets $\pi^{-1}(\{\varphi_{j_{\ell}}>\varphi_{i_{k}}-\eta^{\prime}/2\}\cap U)$ contains a neighborhood of $\tilde{V}$, and therefore so does $\pi^{-1}(\mathcal{A})$. On the other hand, we have that $E_{\nu}=\{\nu F\geqslant\varphi_{b}\}\cap U$ equals

If we choose $\nu$ small, then $\{\nu F\geqslant\nu_{b}F+\rho_{b}\}\cap U=U$. Lemma 2.3 together with the proof of Lemma 2.1 shows that there exists $\nu>0$ small such that each set $\pi^{-1}(\{\nu F\geqslant\tilde{\varphi}_{j_{m}}\}\cap U)$ contains a neighborhood of

Therefore, $E_{\nu}$ contains a neighborhood of

This means that $\pi^{-1}(\mathcal{A}\cup E_{\nu})$ contains a whole neighborhood of $\tilde{V}+\sum_{\ell}D_{\ell}$. On the other hand, the set $E_{\nu}\cup\mathcal{B}$ equals

and if we pick $\nu$ small enough then this set equals $U$. This finally proves that $\pi^{-1}(E_{\nu}\cup\mathcal{S})$ contains a neighborhood of $\tilde{V}+\sum_{\ell}D_{\ell}$, which implies that $E_{\nu}\cup\mathcal{S}$ contains a neighborhood of $x$, and this step is complete.

After at most $N$ such steps, we end up with an open neighborhood $W$ of $V$ in $X$ with a function $\varphi^{\prime\prime}$ defined on $W$ which satisfies $\omega+i\partial\overline{\partial}\varphi^{\prime\prime}\geqslant\varepsilon^{\prime}\omega$ for some $\varepsilon^{\prime\prime}>0$, which equals $\varphi$ on $V$.

Now we have a Kähler current defined on $W$. On $\partial W$ the function $F$ is smooth, so we can choose a large constant $A>0$ such that $F>\varphi^{\prime\prime}-A$ in a neighborhood of $\partial W$. Therefore we can finally define

which is defined on the whole of $X$, it satisfies $\omega+i\partial\overline{\partial}\Phi\geqslant\varepsilon^{\prime}\omega$ for some $\varepsilon^{\prime}>0$. Since $F$ goes to $-\infty$ on $V$, while $\varphi^{\prime\prime}$ is continuous near the generic point of $V$, it follows that $\Phi$ equals $\varphi$ on $V.$ This completes the proof of Theorem 1.1. ∎