Theorem 10.8.26853DA86626EB

Proof. We construct local extensions near $Y$ (Steps 1 and 2), compare them on overlaps (Step 3), glue them (Step 4), extend the result from a neighborhood of $Y$ to $X$ (Step 5), and finally treat analytic singularities (Step 6).

Step 0. Reductions. If $Y=\varnothing $, the function $\widetilde\varphi =0$ and $\varepsilon =1$ have all the stated properties; $0=\log |1|^2$ has analytic singularities. Let $Y\neq \varnothing $. The connected components of $Y$ are open in the compact set $Y$, so there are finitely many of them, say $Y_1,\ldots ,Y_ r$; each $Y_ k$ is closed in $X$, hence a compact connected complex submanifold, and different components may have different dimensions. If some $Y_ k$ has dimension $n$, then it is open in $X$, because near each of its points it is a coordinate plane of full dimension; being also closed, it equals $X$, as $X$ is connected. Then $Y=X$, and $\widetilde\varphi =\varphi $ with $\varepsilon =\delta $ has the stated properties. We may therefore assume that every component of $Y$ has positive codimension; components of dimension zero are allowed.

Fix $\varepsilon $ with $0<\varepsilon <\min \{ \delta /4,1/4\} $. By lemma 10.8.24(2), $\mathrm e^\varphi $ is Hölder continuous with one exponent $\alpha \in (0,1]$ on finitely many coordinate neighborhoods covering $Y$. By lemma 10.8.24(1), every point $p\in Y$ then has, in every system of coordinates of $Y$ near $p$, a neighborhood on which $\mathrm e^\varphi $ is Hölder continuous with exponent $\alpha $.

Step 1. A small normal pole. For each $k$ let $R_ k=R_{Y_ k}$ be the function of lemma 10.8.8 for the submanifold $Y_ k$, and let $\kappa _ k\geq 0$ be the constant called $b$ there, so that $\mathrm{dd}^{\mathrm{c}}\log R_ k\geq -\kappa _ k\omega $. Put

Then $R\geq 0$ is smooth and vanishes exactly on $Y$. The function $\log R=\sum _ k\log R_ k$ is quasi-plurisubharmonic, satisfies $\mathrm{dd}^{\mathrm{c}}\log R\geq -\kappa \omega $, and has analytic singularities by proposition 6.14.8, as a sum of functions with analytic singularities. Choose a rational number $c$ with $0<c<\alpha $ and $c\kappa /2\leq 1-4\varepsilon $; this is possible because $1-4\varepsilon >0$. Put $F=(c/2)\log R$. Then

The function $F$ is smooth on $X\setminus Y$, equals $-\infty $ on $Y$, and has analytic singularities, since $c/2$ is rational. Its exponential $\mathrm e^ F=R^{c/2}$ is Hölder continuous on $X$ by lemma 10.8.24(3).

We need a lower bound for $R$ near $Y$. Let $p\in Y_ k$, and let $(z',w)$ be holomorphic coordinates near $p$ in which $Y=\{ w=0\} $. The function $P=\prod _{l\neq k}R_ l$ is smooth and positive near $p$, since the closed sets $Y_ l$, $l\neq k$, do not contain $p$. On a neighborhood of $p$ that does not meet these sets we have $Y_ k=Y=\{ w=0\} $, so the second inequality in (10.8.9) of lemma 10.8.8, applied to $Y_ k$, bounds $R_ k$ below by a positive multiple of $|w|^2$ near $p$. Since $R=R_ kP$ and $P$ is bounded away from zero near $p$, there are a closed ball $K$ centered at $p$ and a constant $c_0>0$ such that

Moreover, at the points $(z',0)$ of the interior of $K$, the real Hessian $D_ w^2R(z',0)$ of $w\mapsto R(z',w)$ at $w=0$ is positive definite. Indeed, $R\geq 0$ attains its minimum $0$ at such a point, so its first derivatives vanish there, and Taylor's formula gives $R(z',t\xi )=\frac{t^2}2D_ w^2R(z',0)[\xi ,\xi ]+O(t^3)$ as $t\to 0$, for every $\xi $; comparing with $R(z',t\xi )\geq c_0t^2|\xi |^2$ for small $t$, we obtain $D_ w^2R(z',0)[\xi ,\xi ]\geq 2c_0|\xi |^2$.

Step 2. Local extensions. Every point of $Y$ has holomorphic coordinates $(z',w)\in \mathbb C^ m\times \mathbb C^{n-m}$, centered at that point, in which $Y=\{ w=0\} $ on the coordinate domain, where $m<n$ is the dimension of the component of $Y$ through the point. A product of two small balls centered at the origin is then a chart $\widehat W=\widehat B'\times \widehat B''$ with $Y\cap \widehat W=\widehat B'\times \{ 0\} $. The products $B'\times B''$ of the concentric balls of half the radii cover $Y$, and since $Y$ is compact, finitely many of them suffice. We thus obtain finitely many charts $W_ j=B'_ j\times B''_ j$, relatively compact in charts $\widehat W_ j=\widehat B'_ j\times \widehat B''_ j$ of the same form, with $Y\cap W_ j=B'_ j\times \{ 0\} $ and with dimensions $m_ j<n$. Let $\pi _ j\colon \widehat W_ j\to Y$, $\pi _ j(z',w)=(z',0)$, be the holomorphic projection.

Below we shrink the normal factors $B''_ j$ several times, keeping them balls centered at the origin; this does not change $Y\cap W_ j$, so the charts keep covering $Y$. If a property holds on an open set $O\subset \widehat W_ j$ containing the compact set $\overline{B'_ j}\times \{ 0\} $, then this compact set has positive distance from the complement of $O$ in the coordinates, so $O$ contains $W_ j$ after the normal factor has been shrunk. We claim that there is $A>0$ such that, after such a shrinking,

belong to $\mathrm{PSH}(W_ j,(1-2\varepsilon )\omega )$ for every $j$.

Since $4\varepsilon \leq \delta $, the hypothesis gives $\mathrm{dd}^{\mathrm{c}}\varphi \geq -(1-\delta )\omega |_ Y\geq -(1-4\varepsilon )\omega |_ Y$. Choose a ball $\widetilde B'_ j$, concentric with $B'_ j$, with $\overline{B'_ j}\subset \widetilde B'_ j\Subset \widehat B'_ j$, and a smooth function $s$ on $\widetilde B'_ j\times \{ 0\} $ with $\mathrm{dd}^{\mathrm{c}}s=(1-4\varepsilon )\omega |_ Y$ there (lemma 6.14.2; if $m_ j=0$, take $s=0$). Then $\varphi +s$ is plurisubharmonic on $\widetilde B'_ j\times \{ 0\} $, by lemma 6.14.4(2). By proposition 5.9.3, $(\varphi +s)\circ \pi _ j$ is plurisubharmonic or identically $-\infty $ on the connected set $\widetilde B'_ j\times \widehat B''_ j$, and the second alternative is excluded because $\pi _ j$ maps this set onto $\widetilde B'_ j\times \{ 0\} $, where $\varphi +s$ is not identically $-\infty $. Since $\mathrm{dd}^{\mathrm{c}}(s\circ \pi _ j)=\pi _ j^*\mathrm{dd}^{\mathrm{c}}s$ for the smooth function $s$, we obtain that $\varphi \circ \pi _ j$ is quasi-plurisubharmonic on $\widetilde B'_ j\times \widehat B''_ j\supset \overline{W_ j}$ and

For $a_ j$ it therefore suffices that the smooth form

be positive on $W_ j$.

We first examine $Q_ j$ at the points of $Y$. Write a real $(1,1)$-form on $W_ j$ as $\mathrm i\sum _{\mu ,\nu }g_{\mu \nu }\, \mathrm d\zeta _\mu \wedge \mathrm d\bar\zeta _\nu $, where $\zeta =(z',w)$, and split its Hermitian coefficient matrix $(g_{\mu \nu })$ into blocks with respect to the tangential coordinates $z'$ and the normal coordinates $w$; the form is positive definite at a point if and only if this matrix is. The form $\pi _ j^*(\omega |_ Y)$ involves only $\mathrm dz'$ and $\mathrm d\bar z'$, and at a point $(z',0)$ its coefficients are those of $\omega |_ Y$ at that point. Hence its mixed and normal blocks vanish, and at points of $Y$ its tangential block is the tangential block $\Omega _ T$ of $\omega $.

The function $R$ vanishes on $Y$ together with its first derivatives. Differentiating the identities $R(z',0)=0$ and $\partial R/\partial \bar w_\nu (z',0)=0$ with respect to $z'$ and $\bar z'$ shows that $\mathrm{dd}^{\mathrm{c}}R$ has zero tangential and mixed blocks along $Y$. Its normal block is $(2\pi )^{-1}$ times the matrix $N_ R=(\partial ^2R/\partial w_\mu \partial \bar w_\nu )$, which is positive definite at points of $Y$: for $\xi \in \mathbb C^{n-m_ j}\setminus \{ 0\} $, applying the identity $\Delta =4\partial _ t\partial _{\bar t}$ to $t\mapsto R(z',t\xi )$ at $t=0$ gives

by Step 1.

Consequently, at a point of $Y$, the tangential block of $Q_ j$ is $2\varepsilon \Omega _ T$, which is positive definite, its mixed block is the mixed block $\Omega _ M$ of $(1-2\varepsilon )\omega $, and its normal block is $\Omega _ N+AN_ R/(2\pi )$, where $\Omega _ N$ is the normal block of $(1-2\varepsilon )\omega $. By the Schur complement criterion for Hermitian block matrices whose upper left block is positive definite, $Q_ j$ is positive definite at a point of $Y$ if and only if

is positive definite there. (If $m_ j=0$, there is no tangential block, and the condition reads $\Omega _ N+AN_ R/(2\pi )>0$.) All blocks depend continuously on the point, $N_ R$ is positive definite, and the sets $\overline{B'_ j}\times \{ 0\} $ are compact and finite in number. Hence the smallest eigenvalue of $N_ R$ on them has a positive lower bound and the other two terms are bounded, so one $A$ makes this matrix positive definite at every point of every $\overline{B'_ j}\times \{ 0\} $. Fix such an $A$. Since the coefficients of $Q_ j$ are continuous on $\widehat W_ j$ and positive definiteness is an open condition, $Q_ j$ is positive definite on an open set containing $\overline{B'_ j}\times \{ 0\} $, and we shrink the normal factor $B''_ j$ so that $W_ j$ lies in this set. This proves the claim for $a_ j$.

Along $Y$ the form $\mathrm{dd}^{\mathrm{c}}R$ is positive semidefinite, so $A\mathrm{dd}^{\mathrm{c}}R+\varepsilon \omega $ is positive definite on an open set containing $\overline{B'_ j}\times \{ 0\} $; after shrinking the normal factors once more we have $A\mathrm{dd}^{\mathrm{c}}R\geq -\varepsilon \omega $ on every $W_ j$. Together with $\mathrm{dd}^{\mathrm{c}}F\geq -(1-4\varepsilon )\omega $ this gives

which proves the claim for $b_ j$.

By lemma 10.8.25, applied with $\theta =(1-2\varepsilon )\omega $, the functions

belong to $\mathrm{PSH}(W_ j,(1-2\varepsilon )\omega )$. On $Y$ we have $F=-\infty $, $R=0$ and $\varphi \circ \pi _ j=\varphi $, so $\varphi _ j=\varphi $ on $Y\cap W_ j$, also at the poles of $\varphi $. Off $Y$ the function $\mathrm e^ F$ is positive and continuous and $\mathrm e^{\varphi \circ \pi _ j}$ is continuous, so $\varphi _ j$ is continuous on $W_ j\setminus Y$. Finally $\mathrm e^{\varphi _ j}=(\mathrm e^\varphi \circ \pi _ j+\mathrm e^ F)\, \mathrm e^{AR}$ is Hölder continuous on $W_ j$ by lemma 10.8.24(3),(4).

Step 3. Comparison on overlaps. We claim that for every $p\in Y\cap W_ i\cap W_ j$ and every $\eta >0$ there is a neighborhood of $p$ on which

Work in the coordinates $(z',w)$ of $W_ i$, and write $p=(z'_ p,0)$. The coordinate $z'$ restricts to coordinates of $Y$ on $Y\cap W_ i$, so by Step 0 there are a neighborhood $V_ p$ of $p$ in $Y\cap W_ i$ and a constant $C_ H$ with $|\mathrm e^{\varphi (y_1)}-\mathrm e^{\varphi (y_2)}|\leq C_ H|z'(y_1)-z'(y_2)|^\alpha $ for $y_1,y_2\in V_ p$. Since $\pi _ i$ and $\pi _ j$ are continuous and fix $p$, there is a closed ball $K\subset W_ i\cap W_ j$ centered at $p$ such that $\pi _ i(K)\cup \pi _ j(K)\subset V_ p$ and (10.8.27) holds on $K$ in these coordinates. The map $z'\circ \pi _ j$ is smooth on the convex compact set $K$, so it is Lipschitz there with some constant $L$. Let $x=(z',w)\in K$, and put $y=(z',0)=\pi _ i(x)$. Then $|y-p|\leq |x-p|$, so $y\in K\cap Y$, and $\pi _ j(y)=y$ because $\pi _ j$ is the identity on $Y\cap W_ j$. Hence

and therefore

Since $c<\alpha $, we have $C_ HL^\alpha |w|^\alpha \leq (\mathrm e^\eta -1)c_0^{c/2}|w|^ c$ whenever $|w|$ is sufficiently small, and then (10.8.27) gives $C_ HL^\alpha |w|^\alpha \leq (\mathrm e^\eta -1)\mathrm e^{F(x)}$. At such points $x\in K$, including the points of $Y$,

The logarithm is increasing on $[0,\infty )$, with $\log 0=-\infty $, so taking logarithms and adding $AR$ to both sides gives (10.8.28) on the neighborhood of $p$ formed by the interior points $x$ of $K$ with $|w(x)|$ sufficiently small.

Step 4. Gluing near $Y$. Every point of $Y$ has an open neighborhood whose closure is contained in some $W_ j$, and finitely many of these neighborhoods cover $Y$; let $W'_ j$ be the union of those chosen inside $W_ j$. Then $W'_ j\Subset W_ j$ and the $W'_ j$ cover $Y$. Choose open sets $U_ j$ and $T_ j$ with $W'_ j\Subset U_ j\Subset T_ j\Subset W_ j$, for instance sets of points whose coordinate distance from $\overline{W'_ j}$ is less than suitable small constants. The compact set $\overline{T_ j}\setminus U_ j$ is contained in the open set $W_ j\setminus \overline{W'_ j}$, so by proposition 5.2.6 there is a smooth function $f_ j\colon X\to [0,1]$ with compact support in $W_ j\setminus \overline{W'_ j}$ which equals one on a neighborhood of $\overline{T_ j}\setminus U_ j$; in particular $f_ j=0$ on the neighborhood $X\setminus \operatorname{Supp}f_ j$ of $\overline{W'_ j}$. Fix $\eta >0$ so small that $\eta \mathrm{dd}^{\mathrm{c}}f_ j\leq \varepsilon \omega $ for every $j$. Since $(1-\varepsilon )\omega +\mathrm{dd}^{\mathrm{c}}\varphi _ j\geq \varepsilon \omega $, the functions

belong to $\mathrm{PSH}(W_ j,(1-\varepsilon )\omega )$. On $W=\bigcup _ jT_ j$ define

The set of indices in this maximum changes with $x$. We show that near $Y$ the maximum is nevertheless taken over a fixed set of indices. Let $p\in Y$, and choose $j$ with $p\in W'_ j$; then $f_ j=0$ near $p$, and $j$ occurs in the maximum near $p$. Let $l$ be an index with $p\in \partial T_ l$. Then $p\in \overline{T_ l}\setminus U_ l$, so $f_ l=1$ near $p$. Since $p\in Y\cap W_ l\cap W_ j$, the inequality (10.8.28) with $\eta /2$ in place of $\eta $ gives, near $p$,

These are inequalities between functions with values in $[-\infty ,\infty )$, valid also at common poles. As there are finitely many indices, $p$ has a neighborhood $O_ p$ on which the indices $l$ with $p\in \partial T_ l$ never affect the maximum, the indices with $p\notin \overline{T_ l}$ do not occur, and the indices with $p\in T_ l$ occur at every point. Thus, on $O_ p$, the function $\psi $ is the maximum of the $\widehat\varphi _ l$ over the fixed finite set of indices $l$ with $p\in T_ l$.

Let $W_0=\bigcup _{p\in Y}O_ p$. A finite maximum of functions in $\mathrm{PSH}(O_ p,(1-\varepsilon )\omega )$ belongs to this class, by lemma 6.14.4(3). Hence $\psi \in \mathrm{PSH}(W_0,(1-\varepsilon )\omega )$; moreover $\psi $ is continuous on $W_0\setminus Y$, and $\mathrm e^\psi $, which is locally a finite maximum of the functions $\mathrm e^{\widehat\varphi _ l}=\mathrm e^{\varphi _ l}\mathrm e^{-\eta f_ l}$, is Hölder continuous on $W_0$ by lemma 10.8.24(3). On $Y$ every $\widehat\varphi _ l$ restricts to $\varphi -\eta f_ l\leq \varphi $, while $\widehat\varphi _ j=\varphi $ near $p$ for the index $j$ chosen above. Consequently $\psi |_ Y=\varphi $.

Step 5. The global extension. Choose an open neighborhood $G$ of $Y$ with $\overline G\subset W_0$, for instance a finite union of coordinate balls with closures in $W_0$ covering the compact set $Y$. Its boundary $\partial G$ is compact and disjoint from $Y$, and both $\psi $ and $F$ are finite and continuous near $\partial G$. Choose a constant $M$ with $M>\max _{\partial G}(\psi -F)$ (if $\partial G=\varnothing $, every $M$ will do); since $F+M-\psi $ is continuous on $W_0\setminus Y$ and positive on $\partial G$, it is positive on an open neighborhood $G_1\subset W_0\setminus Y$ of $\partial G$. Set

The two expressions agree on $G_1$, so $\widetilde\varphi =F+M$ on the open set $G_1\cup (X\setminus \overline G)$, which contains $X\setminus G$.

The function $F+M$ belongs to $\mathrm{PSH}(X,(1-4\varepsilon )\omega )\subset \mathrm{PSH}(X,(1-\varepsilon )\omega )$, and $\psi \in \mathrm{PSH}(W_0,(1-\varepsilon )\omega )$. Let $B$ be a coordinate ball whose closure lies in a chart, and $h$ a smooth function on $B$ with $\mathrm{dd}^{\mathrm{c}}h=(1-\varepsilon )\omega $ (lemma 6.14.2). Then $F+M+h\in \mathrm{PSH}(B)$ and $\psi +h\in \mathrm{PSH}(G\cap B)$. Every point $x$ of the boundary of $G\cap B$ in $B$ lies in $\partial G$, and $\varlimsup _{G\ni y\to x}(\psi +h)(y)\leq F(x)+M+h(x)$, because $\psi <F+M$ on $G_1$ and $F+M+h$ is continuous at $x$. By proposition 5.9.5, $\widetilde\varphi +h$ is plurisubharmonic on $B$. Since such balls cover $X$, $\widetilde\varphi $ is quasi-plurisubharmonic and $(1-\varepsilon )\omega +\mathrm{dd}^{\mathrm{c}}\widetilde\varphi \geq 0$, that is, $\widetilde\varphi \in \mathrm{PSH}(X,\omega )$ and $\omega +\mathrm{dd}^{\mathrm{c}}\widetilde\varphi \geq \varepsilon \omega $. Since $F=-\infty $ on $Y$, we have $\widetilde\varphi |_ Y=\psi |_ Y=\varphi $. The function $\widetilde\varphi $ is continuous on $X\setminus Y$, because $\psi $ and $F$ are continuous on $W_0\setminus Y$ and $X\setminus Y$, respectively. Its exponential equals $\max \{ \mathrm e^\psi ,\mathrm e^ M\mathrm e^ F\} $ on $G$ and $\mathrm e^ M\mathrm e^ F$ on the open set $G_1\cup (X\setminus \overline G)$; these two open sets cover $X$, so $\mathrm e^{\widetilde\varphi }$ is Hölder continuous on $X$ by lemma 10.8.24(3).

Step 6. Analytic singularities. Finally suppose that $\varphi $ has analytic singularities (definition 6.14.5). By proposition 6.14.8, so does the holomorphic pullback $\varphi \circ \pi _ j$, which is not identically $-\infty $, and the function $F$ has analytic singularities by Step 1. By the same proposition, $\max \{ \varphi \circ \pi _ j,F\} $ has analytic singularities; adding the smooth function $AR$ changes the bounded remainder in its local presentations (6.14.6) by a locally bounded function, so $\max \{ \varphi \circ \pi _ j,F\} +AR$ has analytic singularities. By lemma 10.8.25,

Hence a local presentation (6.14.6) of $\max \{ \varphi \circ \pi _ j,F\} +AR$ yields one of $\varphi _ j$, with the same holomorphic functions and a bounded remainder changed by at most $\log 2$; so $\varphi _ j$ has analytic singularities. Subtracting the smooth function $\eta f_ j$ changes the remainders by bounded functions, and the locally fixed finite maxima of Step 4 preserve analytic singularities by proposition 6.14.8; so $\psi $ has analytic singularities on $W_0$. On $G$, the function $\widetilde\varphi $ is the maximum of $\psi $ and $F+M$, and on the open set $G_1\cup (X\setminus \overline G)$ it equals $F+M$; both functions have analytic singularities. Therefore $\widetilde\varphi $ has analytic singularities on $X$. $\square$