10.3 Normalized potentials and minimal singularitiesF0B52F4D634F

Proof. Choose finitely many coordinate balls $B_ k$ covering $X$, smaller concentric balls $B'_ k$ still covering $X$, and smooth functions $h_ k$ near $\overline{B_ k}$ with $\mathrm{dd}^{\mathrm{c}}h_ k=\theta $ (lemma 6.14.2, applied on slightly larger concentric balls). For $\varphi \in \mathrm{PSH}(X,\theta )$ the functions $\varphi +h_ k$ are plurisubharmonic on $B_ k$.

Let $(\varphi _ j)$ be a sequence with $a\leq \sup _ X\varphi _ j\leq b$. The plurisubharmonic functions $\varphi _ j+h_ k$ are uniformly bounded above on each $B_ k$. Since $\varphi _ j$ is upper semicontinuous on the compact manifold $X$, it attains its supremum at a point $x_ j$, and after passing to a subsequence $x_ j\to x$, since $X$ is compact and metrizable. Choose $k$ with $x\in B'_ k$. On $B_ k$ the functions $\varphi _ j+h_ k$ cannot converge locally uniformly to $-\infty $, because their values at $x_ j$ are bounded below by $a+\inf _{B'_ k}h_ k$ for large $j$. By theorem 5.10.4, a further subsequence converges in $L^1_{\mathrm{loc}}(B_ k)$. If $B_ l$ meets $B_ k$, the functions $\varphi _ j+h_ l$ cannot tend to $-\infty $ locally uniformly on $B_ l$, since on the nonempty open set $B_ k\cap B_ l$ they differ by the smooth function $h_ l-h_ k$ from a sequence that converges in $L^1_{\mathrm{loc}}$; so again a subsequence converges in $L^1_{\mathrm{loc}}(B_ l)$. Since $X$ is connected, every ball is joined to $B_ k$ by a finite chain of overlapping balls, and after finitely many extractions the sequence converges in $L^1(B'_ l)$ for every $l$, hence in $L^1(X)$, to some $\varphi $. The limit satisfies $\theta +\mathrm{dd}^{\mathrm{c}}\varphi \geq 0$ as a distributional limit of positive currents, and its canonical representative belongs to $\mathrm{PSH}(X,\theta )$.

Now let $\varphi _ j\to \varphi $ in $L^1(X)$, all in $\mathrm{PSH}(X,\theta )$; we show that $\sup _ X\varphi _ j\to \sup _ X\varphi $. The submean inequality for $\varphi _ j+h_ k$ on small balls shows that the $\varphi _ j$ are uniformly bounded above. Let $x_ j\to x$; we claim that $\varlimsup _ j\varphi _ j(x_ j)\leq \varphi (x)$. Indeed, choose $k$ with $x\in B_ k$, let $c>\varphi (x)$ be real, and use upper semicontinuity to choose a closed ball $K\subset B_ k$ around $x$ on which $\varphi <c$. The function $c+h_ k$ is continuous near $K$ and $\varphi +h_ k<c+h_ k$ on $K$, so the second part of theorem 5.10.4 gives $\varphi _ j+h_ k<c+h_ k$ on $K$ for large $j$; hence $\varphi _ j(x_ j)<c$ for large $j$.

Choose a subsequence along which $\sup _ X\varphi _ j$ tends to $\varlimsup _ j\sup _ X\varphi _ j$ and points $x_ j$ with $\varphi _ j(x_ j)=\sup _ X\varphi _ j$ converge to some $x$; the claim gives $\varlimsup _ j\sup _ X\varphi _ j\leq \varphi (x)\leq \sup _ X\varphi $. Conversely, choose a subsequence along which $\sup _ X\varphi _ j$ tends to $\varliminf _ j\sup _ X\varphi _ j$ and $\sum _ j\| \varphi _ j-\varphi \| _{L^1}<\infty $; by monotone convergence $\sum _ j|\varphi _ j-\varphi |$ is integrable, hence finite almost everywhere, so $\varphi _ j\to \varphi $ almost everywhere along it. We get $\varphi \leq \varliminf _ j\sup _ X\varphi _ j$ almost everywhere, hence everywhere, as noted at the beginning of this section. The two inequalities prove that the supremum is continuous; in particular, the limit in the previous paragraph satisfies $a\leq \sup _ X\varphi \leq b$. Thus the set in question is sequentially compact, hence compact, in the metric space $L^1(X)$. The continuous function $\| \cdot \| _{L^1}$ is bounded on the compact set of normalized functions.

The arguments so far use only that $X$ is a connected compact complex manifold of positive dimension and that the volume form defining $L^1(X)$ is smooth and positive; hence they also prove the last assertion of the proposition.

For smoothly convergent backgrounds, choose $C\geq 0$ with $\theta _ j\leq C\omega $ for all $j$. Then $C\omega +\mathrm{dd}^{\mathrm{c}}\varphi _ j\geq 0$, so all $\varphi _ j$ lie in $\mathrm{PSH}(X,C\omega )$, and the first part, applied to the closed form $C\omega $, gives an $L^1$-convergent subsequence and the convergence of the suprema. Since $\theta _ j+\mathrm{dd}^{\mathrm{c}}\varphi _ j\to \theta +\mathrm{dd}^{\mathrm{c}}\varphi $ as distributions, the limit belongs to $\mathrm{PSH}(X,\theta )$. $\square$

Proof. (If $\mathrm{PSH}(X,\theta )$ is empty, there is nothing to prove.) Choose finitely many coordinate polydiscs $D_ k$ such that the polydiscs $\frac14D_ k$ with the same centers and a quarter of the radii cover $X$, and smooth functions $h_ k$ near $\overline{D_ k}$ with $\mathrm{dd}^{\mathrm{c}}h_ k=\theta $. For $u$ as in the statement, $v_ k=u+h_ k-\sup _{D_ k}h_ k$ is a nonpositive plurisubharmonic function on $D_ k$. By proposition 10.3.1 and the boundedness of the $h_ k$ and of the coordinate volume densities, the norms $\| v_ k\| _{L^1(D_ k)}$ are bounded independently of $u$. After rescaling $D_ k$ to the unit polydisc, (5.10.9) bounds $\| v_ k\| _{L^ p(\frac14D_ k)}$ independently of $u$. Since $u=v_ k-h_ k+\sup _{D_ k}h_ k$ on $D_ k$, the bounded functions $h_ k$ and the bounded volume densities give the assertion. $\square$

Proof. (If $\mathrm{PSH}(X,\theta )$ is empty, any $\alpha ,C>0$ will do.) Choose finitely many coordinate unit balls $B_ j$ such that the balls $\frac14B_ j$ with the same centers and a quarter of the radii cover $X$, and smooth functions $\rho _ j$ near $\overline{B_ j}$ with $\mathrm{dd}^{\mathrm{c}}\rho _ j=\theta $. For $u$ as in the statement put $v_ j=u+\rho _ j-\sup _{B_ j}\rho _ j$ on $B_ j$; then $v_ j$ is plurisubharmonic and $v_ j\leq u\leq 0$. By proposition 10.3.1 and comparison of the coordinate volume densities, $\| v_ j\| _{L^1(B_ j)}\leq M_ j$ for a constant $M_ j\geq 1$ independent of $u$. Applying proposition 5.10.12 to $v_ j/M_ j$, we see that $\int _{\frac14B_ j}\mathrm e^{-\alpha _ jv_ j}$ is bounded independently of $u$, where $\alpha _ j=\alpha _ n/M_ j$. Put $\alpha =\min _ j\alpha _ j>0$. Since $v_ j\leq u\leq 0$ and $\alpha \leq \alpha _ j$, we have $\mathrm e^{-\alpha u}\leq \mathrm e^{-\alpha _ ju}\leq \mathrm e^{-\alpha _ jv_ j}$ on $B_ j$. Summing over $j$ and comparing the coordinate volume densities with $\omega ^ n$ gives (10.3.4). $\square$

Proof. Suppose that no such constant exists. Since $\sup _ X\varphi -\sup _ K\varphi $ does not change when a constant is added to $\varphi $, there are $\varphi _ j\in \mathrm{PSH}(X,\theta )$ with $\sup _ X\varphi _ j=0$ and $\sup _ K\varphi _ j\leq -2^ j$. Choose $C\geq 0$ with $\theta \leq C\omega $ (lemma 6.14.1) and put $\psi _ N=\sum _{j=1}^ N2^{-j}\varphi _ j$. Since $\sum _{j\leq N}2^{-j}\leq 1$, we have

The sequence $(\psi _ N)$ is decreasing because $\varphi _ j\leq 0$, and $\| \psi _ N\| _{L^1}\leq \sup _ j\| \varphi _ j\| _{L^1}<\infty $ by proposition 10.3.1. By monotone convergence the limit $\psi $ is integrable, so it is not identically $-\infty $, and theorem 5.10.1, applied on coordinate balls after adding local potentials of $C\omega $, shows that $\psi \in \mathrm{PSH}(X,C\omega )$. On $K$ every term $2^{-j}\varphi _ j$ is at most $-1$, so $\psi =-\infty $ on $K$. On every coordinate ball $U$ with a smooth function $h$ such that $\mathrm{dd}^{\mathrm{c}}h=C\omega $, the function $\psi +h$ is plurisubharmonic and equals $-\infty $ on $K\cap U$. Hence $K$ is pluripolar, a contradiction. The last assertion follows: bounds for $\sup _ K\varphi $ give bounds for $\sup _ X\varphi $, and proposition 10.3.1 applies. $\square$

Proof. Fix an index $i_0$. For $i\geq i_0$ the suprema are bounded above by $\sup _ X\varphi _{i_0}$, so the tail of the net lies in a compact subset of $L^1(X)$ by proposition 10.3.1, and a subnet converges in $L^1$ to some $\psi \in \mathrm{PSH}(X,\theta )$. For each fixed $i$, the members of the subnet are eventually below $\varphi _ i$, so $\psi \leq \varphi _ i$ almost everywhere, and therefore everywhere, as noted at the beginning of this section. Hence the pointwise infimum $\varphi $ is at least $\psi $ and is not identically $-\infty $. Applying theorem 5.10.1 on the coordinate balls of a finite cover, after adding local potentials of $\theta $, we find that $\varphi \in \mathrm{PSH}(X,\theta )$ and $\varphi _ i\to \varphi $ in $L^1(X)$. The continuity of the supremum in proposition 10.3.1 gives the last assertion. $\square$

Proof. The family whose upper envelope defines $V_\theta $ in definition 10.3.7 is nonempty, because $\varphi -\sup _ X\varphi $ belongs to it for every $\varphi \in \mathrm{PSH}(X,\theta )$, and it is bounded above by $0$. Theorem 5.10.2, applied on coordinate balls after adding smooth local potentials of $\theta $, therefore gives $V_\theta \in \mathrm{PSH}(X,\theta )$ and $V_\theta \leq 0$; and $\sup _ XV_\theta =0$, because $V_\theta \geq \varphi -\sup _ X\varphi $ for every $\varphi $. The same inequality reads $\varphi \leq V_\theta +\sup _ X\varphi $, so $V_\theta $ has minimal singularities. Any two potentials with minimal singularities are each more singular than the other, so their difference is bounded on the set where both are finite. Replacing $\theta $ by $\theta +\mathrm{dd}^{\mathrm{c}}h$ with $h$ smooth replaces $\mathrm{PSH}(X,\theta )$ by $\mathrm{PSH}(X,\theta )-h$; since $h$ is bounded, the notion of minimal singularities depends only on the class $\{ \theta \} $. $\square$

Proof. The manifold $Y=\pi ^{-1}(X)$ is compact, because $\pi $ is proper. By the definition of bimeromorphic maps (section 9.4), there is a nowhere dense analytic subset $E_0\subset X$ such that $\pi ^{-1}(E_0)$ is nowhere dense and $\pi $ restricts to an isomorphism of $Y\setminus \pi ^{-1}(E_0)$ onto $X\setminus E_0$. The set $X\setminus E_0$ is connected by lemma 9.2.9, hence so is its dense preimage, and therefore $Y$ is connected. The image $\pi (Y)$ is closed, since proper maps are closed, and it contains the dense set $X\setminus E_0$; so $\pi $ is surjective. By proposition 9.8.3 there is an analytic subset $A\subset X$ of codimension at least two such that $\pi $ is an isomorphism over $X\setminus A$. Then $\pi ^{-1}(A)$ is an analytic subset of $Y$ different from $Y$, because $\pi $ is surjective. Since $Y$ is connected, $\pi ^{-1}(A)$ is locally contained in the zero set of a holomorphic function that does not vanish identically on a connected neighborhood (section 9.2), so it is nowhere dense and a null set (lemma 5.5.1). The same holds for $A$ in $X$.

If $\varphi \in \mathrm{PSH}(X,\theta )$, let $B$ be a coordinate ball and $h$ a smooth function on $B$ with $\mathrm{dd}^{\mathrm{c}}h=\theta $ (lemma 6.14.2). On every connected coordinate neighborhood $W\subset \pi ^{-1}(B)$ the function $(\varphi +h)\circ \pi $ is plurisubharmonic or identically $-\infty $, by proposition 5.9.3, and the second case is excluded because $\varphi +h$ is finite almost everywhere on the image under the biholomorphism $\pi $ of the nonempty open set $W\setminus \pi ^{-1}(A)$. Hence $\varphi \circ \pi =(\varphi +h)\circ \pi -h\circ \pi $ is quasi-plurisubharmonic, with $\pi ^*\theta +\mathrm{dd}^{\mathrm{c}}(\varphi \circ \pi )=\mathrm{dd}^{\mathrm{c}}((\varphi +h)\circ \pi )\geq 0$ on $\pi ^{-1}(B)$. Thus $\varphi \circ \pi \in \mathrm{PSH}(Y,\pi ^*\theta )$. If $\varphi \circ \pi =\varphi '\circ \pi $, then $\varphi =\varphi '$ on $X\setminus A$, hence almost everywhere, and therefore everywhere, as noted at the beginning of this section; this proves injectivity.

Conversely, let $\psi \in \mathrm{PSH}(Y,\pi ^*\theta )$. Over $X\setminus A$ it descends to the function $\psi \circ \pi ^{-1}$, which is $\theta $-plurisubharmonic there. It is bounded above, because $\psi $ is upper semicontinuous on the compact manifold $Y$. On a coordinate ball $B$ with $\mathrm{dd}^{\mathrm{c}}h=\theta $ on $B$, the plurisubharmonic function $\psi \circ \pi ^{-1}+h$ on $B\setminus A$ is therefore locally bounded above near $A$, and it extends to a plurisubharmonic function on $B$ by theorem 5.11.1. The extensions on two balls agree outside the null set $A$, hence everywhere, so they define $\varphi \in \mathrm{PSH}(X,\theta )$. The pullback $\varphi \circ \pi $ equals $\psi $ outside the null set $\pi ^{-1}(A)$, hence everywhere. This proves surjectivity.

For continuity, we use proposition 10.3.1 on $X$ and on $Y$; by its last assertion, its assertions for a fixed background hold on $Y$. Let $\varphi _ j\to \varphi $ in $L^1(X)$. By the continuity of the supremum, it suffices to treat normalized functions, $\sup _ X\varphi _ j=\sup _ X\varphi =0$, since subtracting the suprema preserves $L^1$ convergence. The pullbacks have the same suprema, because $\pi $ is surjective, so they are relatively compact in $L^1(Y)$, and every cluster value lies in $\mathrm{PSH}(Y,\pi ^*\theta )$. On compact subsets of $Y\setminus \pi ^{-1}(A)$, a change of variables shows that every $L^1$ cluster value of $(\varphi _ j\circ \pi )$ coincides almost everywhere with $\varphi \circ \pi $; since $\pi ^{-1}(A)$ is a null set, every cluster value is $\varphi \circ \pi $. A relatively compact sequence in a metric space with a single cluster value converges to it; hence $\varphi _ j\circ \pi \to \varphi \circ \pi $ in $L^1(Y)$.

Finally, $\nu (\varphi \circ \pi ,E)$ is the infimum over $y\in E$ of the point Lelong numbers $\nu (\varphi \circ \pi ,y)$. Near $y$ choose a smooth $g$ with $\mathrm{dd}^{\mathrm{c}}g=\pi ^*\theta $, so that the functions $\varphi \circ \pi +g$ are plurisubharmonic. By (6.7.10), applied to them, each $\nu (\varphi \circ \pi ,y)$ is upper semicontinuous for $L^1$ convergence of $\varphi \circ \pi $, hence, by the continuity just proved, for $L^1$ convergence of $\varphi $; and an infimum of upper semicontinuous functions is upper semicontinuous. Since Lelong numbers do not change when a constant is added, we may restrict to the normalized functions, which form a compact set by proposition 10.3.1. An upper semicontinuous function with finite values attains its maximum on a compact set, which gives the uniform bound. $\square$

Proof. Choose a ball $B''$ in the same coordinates with $\overline B\subset B''$ and $\overline{B''}\subset B'$. Write $\theta =\mathrm{dd}^{\mathrm{c}}h$ on $B''$, with $h$ smooth (lemma 6.14.2), and apply corollary 7.7.5 to the locally bounded plurisubharmonic function $\varphi +h$ on $B''$. Subtracting $h$ from the result on $B''$ and keeping $\varphi $ outside $B''$ gives $\psi $, since the result equals $\varphi +h$ on $B''\setminus B$; here $(\theta +\mathrm{dd}^{\mathrm{c}}\psi )^ n=(\mathrm{dd}^{\mathrm{c}}(\psi +h))^ n$ is the Bedford–Taylor product of the bounded plurisubharmonic function $\psi +h$. If $\psi '$ is another function with the stated properties, then by upper semicontinuity $\varlimsup _{z\to \zeta ,\, z\in B}\psi '(z)\leq \psi '(\zeta )=\varphi (\zeta )$ for every $\zeta \in \partial B$. Hence $\psi '+h$ belongs to the family whose upper envelope is $\psi +h$ in corollary 7.7.5, and $\psi '\leq \psi $ on $B$. If $\psi '$ is continuous on $\overline B$, then its boundary values $\varphi |_{\partial B}$ are continuous, and $\psi '+h$ is the solution of the homogeneous Dirichlet problem with the continuous boundary values $(\varphi +h)|_{\partial B}$; by the uniqueness in theorem 7.7.4, any two such continuous functions coincide. $\square$