10.8 Extension from a compact submanifold9264FE392E91

Proof. The Bochner–Kodaira identity (10.7.3) of lemma 10.7.2 holds for every smooth metric on $F$; we apply it to the metric $ah$. Let $\delta _ h=D''{}^*$ be the formal adjoint for the metric $h$ and the Kähler form $\sigma $, and $C_ a=(\bar\partial a\wedge )^*$ the contraction with the dual of $\bar\partial a$. In a frame, the weight of $ah$ is $p-\log a$, so

On $(n,1)$-forms, the commutator with the last term is $[\frac{\mathrm i}{a}\partial a\wedge \bar\partial a,\Lambda _\sigma ]=a^{-1}C_ a^*C_ a$. Indeed, where $\partial a\neq 0$, choose a $\sigma $-unitary coframe with $\partial a=|\partial a|\theta _1$, so that $\bar\partial a=|\partial a|\bar\theta _1$, and put $e=\theta _1\wedge \cdots \wedge \theta _ n$. By lemma 10.7.1(1), the left side multiplies the coefficient $v_1$ of $e\wedge \bar\theta _1$ by $|\partial a|^2/a$ and annihilates the other coefficients; so does the right side, because $\bar\partial a\wedge fe=(-1)^ n|\partial a|f\, e\wedge \bar\theta _1$, whence $C_ av=(-1)^ n|\partial a|v_1e$. Where $\partial a=0$, both sides vanish. Writing (10.7.3) for $ah$ in terms of $h$, expanding $a|\delta _ hv-a^{-1}C_ av|^2$, cancelling the term $a^{-1}|C_ av|^2$ that appears on both sides, and dropping the nonnegative term $\| D'{}^*v\| ^2$, we obtain for compactly supported smooth $v$, with all integrals taken with respect to $h$, $\sigma $ and $\mathrm dV_\sigma $,

The inequality $2\operatorname{Re}\langle A,B\rangle \geq -b|A|^2-b^{-1}|B|^2$ and the identity $b^{-1}|C_ av|^2=\langle [\frac{\mathrm i}{b}\partial a\wedge \bar\partial a,\Lambda _\sigma ]v,v\rangle $ consequently give

Consider the Hilbert spaces of $(n,0)$-, $(n,1)$- and $(n,2)$-forms with the respective weights $h/(a+b)$, $h$ and $ah$, and the operators $T=D''$ and $S=D''$ with maximal domains between them. Testing against compactly supported forms shows that $T^*=(a+b)\delta _ h$, so $\| T^*v\| ^2$ is the first term on the right of (10.8.3), and $\| Sv\| ^2$ is the second. Compactly supported smooth forms are dense in $\operatorname{Dom}T^*\cap \operatorname{Dom}S$ for the graph norm, by lemma 10.7.4, applied with the metrics $h/(a+b)$, $h$ and $ah$, whose ratios $a+b$ and $a$ are bounded.

The form $g$ lies in $\ker S$, and in the second space by lemma 10.7.1(3), because $\sigma \geq \omega /2$. The operator $[H,\Lambda _\sigma ]$ is positive definite on $(n,1)$-forms by lemma 10.7.1(1). As in the proof of lemma 10.7.6, the pointwise Cauchy–Schwarz inequality for this operator, (10.8.3) and the density of compactly supported smooth forms give the hypothesis of theorem 6.2.4, which yields a solution of $\bar\partial u=g$ whose squared norm in the first space is at most $\int \langle [H,\Lambda _\sigma ]^{-1}g,g\rangle \, \mathrm dV_\sigma $. This integral, and the squared norm of $(n,0)$-forms times the volume form, do not depend on the Hermitian form used, by lemma 10.7.1(3); hence the bound is $M_ g$, and we obtain (10.8.2). $\square$

Proof. (1) Since $\varepsilon +R<2\mathrm e^{-4}<\mathrm e^{-3}$, we have $\log \varepsilon <\psi _\varepsilon <-3$, and $\psi <\psi _\varepsilon $ because $\varepsilon >0$. Next, $a_\varepsilon =\varepsilon -\psi _\varepsilon +\log (1-\psi _\varepsilon )\geq -\psi _\varepsilon $, and $\mu '(t)=-1-(1-t)^{-1}$ lies in $[-2,-1]$ for $t<0$. With $s=-\psi _\varepsilon >3$ we have $a_\varepsilon \leq 1+s+\log (1+s)\leq 1+2s$ and $b_\varepsilon =(2+s)^2\leq 4s^2$, so $a_\varepsilon +b_\varepsilon \leq 1+2s+4s^2\leq 5s^2$, since $s^2-2s-1>0$ for $s\geq 3$; and $s^2\leq \psi ^2$ because $\psi <\psi _\varepsilon <0$. Finally, $\log \varepsilon <\psi _\varepsilon <-3$ gives $3<a_\varepsilon \leq \varepsilon -\log \varepsilon +\log (1-\log \varepsilon )$ and $25<b_\varepsilon \leq (2-\log \varepsilon )^2$.

(2) We have $\mu ''(t)=-(1-t)^{-2}$, hence $-\mu '(t)^2/\mu ''(t)=\bigl ((1-t)+1\bigr )^2=(2-t)^2$, so $b_\varepsilon =-\mu '(\psi _\varepsilon )^2/\mu ''(\psi _\varepsilon )$. By the chain rule, $\mathrm i\partial \bar\partial a_\varepsilon =\mu '(\psi _\varepsilon )\mathrm i\partial \bar\partial \psi _\varepsilon +\mu ''(\psi _\varepsilon )\mathrm i\partial \psi _\varepsilon \wedge \bar\partial \psi _\varepsilon $ and $b_\varepsilon ^{-1}\mathrm i\partial a_\varepsilon \wedge \bar\partial a_\varepsilon =b_\varepsilon ^{-1}\mu '(\psi _\varepsilon )^2\mathrm i\partial \psi _\varepsilon \wedge \bar\partial \psi _\varepsilon =-\mu ''(\psi _\varepsilon )\mathrm i\partial \psi _\varepsilon \wedge \bar\partial \psi _\varepsilon $. The gradient terms cancel, which proves (2).

(3) For $c=\varepsilon $ and for $c=0$ we have $\partial \bar\partial \log (c+R)=(c+R)^{-1}\partial \bar\partial R-(c+R)^{-2}\partial R\wedge \bar\partial R$. Hence $\partial \bar\partial \psi _\varepsilon -\frac{R}{\varepsilon +R}\partial \bar\partial \psi =\bigl (R^{-1}(\varepsilon +R)^{-1}-(\varepsilon +R)^{-2}\bigr )\partial R\wedge \bar\partial R=\frac{\varepsilon }{R}(\varepsilon +R)^{-2}\partial R\wedge \bar\partial R$, and $(\varepsilon +R)^{-2}\partial R\wedge \bar\partial R=\partial \psi _\varepsilon \wedge \bar\partial \psi _\varepsilon $. $\square$

Proof. If $f=0$, take $F=0$. Put $n=m+q$, and let $\beta $ be the Euclidean Kähler form on $\mathbb C^ n$. We first treat smooth weights: we cut off the trivial extension of $f$ near $\{ w=0\} $, correct it with lemma 10.8.1 on $D\setminus \{ w=0\} $, and let the width $\varepsilon $ of the cutoff tend to zero. So assume that $p$ is smooth near $\overline D$, and set $R=|w|^2$, $\psi =\log R$ and $\psi _\varepsilon =\log (\varepsilon +R)$ for small $\varepsilon >0$. The function $P=p+\pi |z|^2$, where $z=(z',w)$, satisfies $\mathrm{dd}^{\mathrm{c}}P\geq \beta $, and replacing $p$ by $P$ changes both sides of (10.8.7) by factors bounded in terms of $r'$ and $r''$ only; we prove the estimate for $P$.

The twisted form. On $D\setminus \{ w=0\} $ consider the trivial line bundle with the metric $\mathrm e^{-P-q\psi }$, and let $\mu $, $a_\varepsilon $ and $b_\varepsilon $ be the functions of lemma 10.8.4, for $0<\varepsilon <\mathrm e^{-4}$; that lemma applies because $0<R<\mathrm e^{-4}$ on $D\setminus \{ w=0\} $, since $r''<\mathrm e^{-2}$. By its part (1), $a_\varepsilon \geq 1$ and $-\mu '(\psi _\varepsilon )\geq 1$, and for fixed $\varepsilon $ the functions $a_\varepsilon $ and $b_\varepsilon $ are bounded above and below by positive constants. By its part (2), applied with $\Theta =\mathrm i\partial \bar\partial (P+q\psi )$, the curvature form of this metric times $2\pi $, the gradient terms cancel in the form $H$ of lemma 10.8.1, and

Both logarithmic Hessians $\mathrm i\partial \bar\partial \psi $ and $\mathrm i\partial \bar\partial \psi _\varepsilon $ are semipositive, so (10.8.5) gives $\mathrm i\partial \bar\partial \psi _\varepsilon \geq (\varepsilon /R)\, \mathrm i\partial \psi _\varepsilon \wedge \bar\partial \psi _\varepsilon $. Since also $\mathrm i\partial \bar\partial P=2\pi \mathrm{dd}^{\mathrm{c}}P\geq 2\pi \beta $, the bounds $a_\varepsilon \geq 1$ and $-\mu '(\psi _\varepsilon )\geq 1$ give $H_\varepsilon \geq 2\pi \beta +(\varepsilon /R)\, \mathrm i\partial \psi _\varepsilon \wedge \bar\partial \psi _\varepsilon $.

Complete cutoffs. The manifold $D\setminus \{ w=0\} $ carries a Kähler form and cutoffs as required in lemma 10.8.1. Indeed, put $\tau _0=-\log (r'^2-|z'|^2)-\log (r''^2-|w|^2)$, omitting the first term if $m=0$, and $\tau _1=-\log (-\psi )$. Each of the three functions $\tau $ formed by the two terms of $\tau _0$ and by $\tau _1$ is of the form $\tau =-\log g$ with $g>0$ smooth and $-\mathrm{dd}^{\mathrm{c}}g\geq 0$, namely $g=r'^2-|z'|^2$, $g=r''^2-|w|^2$ and $g=-\psi $; hence

Thus $\tau _0$ and $\tau _1$ are plurisubharmonic, and $\sigma =\beta +\mathrm{dd}^{\mathrm{c}}\tau _0+\mathrm{dd}^{\mathrm{c}}\tau _1\geq \beta $ is a Kähler form. Moreover $\sigma \geq \mathrm{dd}^{\mathrm{c}}\tau \geq \frac{\mathrm i}{2\pi }\partial \tau \wedge \bar\partial \tau $ for each of the three functions; the eigenvalue of the rank-one form $\frac{\mathrm i}{2\pi }\partial \tau \wedge \bar\partial \tau $ with respect to $\sigma $ is $|\partial \tau |_\sigma ^2/(2\pi )$, so $|\partial \tau |_\sigma ^2\leq 2\pi $, and these functions have bounded gradients for $\sigma $. The function $\tau _0-\tau _1$ tends to $+\infty $ at the boundary of $D$ and along $\{ w=0\} $, and its sublevel sets are compact in $D\setminus \{ w=0\} $. If $\lambda \in C^\infty (\mathbb R)$ equals one on $(-\infty ,1]$ and zero on $[2,\infty )$, the functions $\chi _ j=\lambda ((\tau _0-\tau _1)/j)$ are compactly supported, tend to one, and have gradients $O(1/j)$. The canonical bundle of $\mathbb C^ n$ is trivialized by $\mathrm dz_1\wedge \cdots \wedge \mathrm dz_ n$, whose norm for $\beta $ is constant, so we identify functions with $(n,0)$-forms, as in the proof of theorem 10.7.9; this multiplies all the pointwise norms below by the same constant factor.

The correction. Let $G(z',w)=f(z')$. Choose $\rho \in C^\infty (\mathbb R)$ with $0\leq \rho \leq 1$, equal to one on $(-\infty ,1/2]$ and to zero on $[1,\infty )$, and put $G_\varepsilon =\rho (R/\varepsilon )G$. Since $\bar\partial R/\varepsilon =(1+R/\varepsilon )\bar\partial \psi _\varepsilon $,

which is supported in $\{ \varepsilon /2\leq R\leq \varepsilon \} $. For an $(n,1)$-form $v$, the rank-one part of the lower bound for $H_\varepsilon $ and lemma 10.7.1(1),(2), applied with $\alpha =\partial \psi _\varepsilon $, give

where $|G|$ is the modulus of the function $G$ and the inner products are those of $\beta $ and the metric $\mathrm e^{-P-q\psi }=\mathrm e^{-P}R^{-q}$. Hence, on the support of $g_\varepsilon $, where $R/\varepsilon \leq 1$, we have $\langle [H_\varepsilon ,\Lambda _\beta ]^{-1}g_\varepsilon ,g_\varepsilon \rangle \leq C|G|^2\mathrm e^{-P}R^{-q}$, and the integral $M_{g_\varepsilon }$ of lemma 10.8.1 is at most

In polar coordinates in $w$, the integral of $R^{-q}$ over the shell $\varepsilon /2\leq |w|^2\leq \varepsilon $ is a fixed multiple of $\int _{\sqrt{\varepsilon /2}}^{\sqrt\varepsilon }\mathrm dt/t=\frac12\log 2$. Since $P$ is uniformly continuous on $\overline D$ and $G$ does not depend on $w$, the last integral tends to a fixed multiple of $\int _{B'}|f|^2\mathrm e^{-P(z',0)}\, \mathrm dV_{B'}$ as $\varepsilon \to 0$. Lemma 10.8.1 gives $u_\varepsilon $ on $D\setminus \{ w=0\} $ with $\bar\partial u_\varepsilon =g_\varepsilon $ and

For fixed $\varepsilon $ the denominator is bounded and $\mathrm e^{-P}R^{-q}$ is bounded below, so $u_\varepsilon $ is square integrable, and $F_\varepsilon =G_\varepsilon -u_\varepsilon $, which is holomorphic off $\{ w=0\} $, extends holomorphically to $D$ by lemma 10.7.8. The difference $u_\varepsilon =G_\varepsilon -F_\varepsilon $ is then smooth near $\{ w=0\} $, and it vanishes there: a nonzero value at a point $(z'_0,0)$ would make $\int |u_\varepsilon |^2|w|^{-2q}$ diverge near that point, since $\int |w|^{-2q}\, \mathrm dV(w)$ diverges logarithmically at $w=0$ in $\mathbb C^ q$. Hence $F_\varepsilon (z',0)=G_\varepsilon (z',0)=f(z')$.

The limit $\varepsilon \to 0$. By lemma 10.8.4(1), $a_\varepsilon +b_\varepsilon \leq 5(-\log (\varepsilon +R))^2$, so the correction estimate bounds $u_\varepsilon $ in the weaker weight $(\varepsilon +R)^{-q}(-\log (\varepsilon +R))^{-2}\mathrm e^{-P}$. In this weight the squared norm of $G_\varepsilon $ is at most $C_ P(-\log \varepsilon )^{-2}$, because its support has normal volume $O(\varepsilon ^ q)$ and $(\varepsilon +R)^{-q}\leq \varepsilon ^{-q}$ there; the constant $C_ P$ is finite for the fixed smooth weight $P$. Consequently the squared norms of $F_\varepsilon $ in this weight have upper limit, as $\varepsilon \to 0$, at most a constant, depending only on $m,q,r',r''$, times $\int _{B'}|f|^2\mathrm e^{-P(z',0)}$. Since $t^{-q}(-\log t)^{-2}$ is bounded below on $(0,\mathrm e^{-3}]$, the $F_\varepsilon $ are bounded in $L^2(D)$, and by lemma 10.6.2(1) a sequence $\varepsilon _ k\to 0$ gives functions converging locally uniformly to a holomorphic $F$ with $F(z',0)=f(z')$. By Fatou's lemma, $F$ satisfies (10.8.7) with $P$ in place of $p$, and removing the bounded factor $\mathrm e^{-\pi |z|^2}$ gives (10.8.7) for smooth $p$.

General $p$. Take smooth plurisubharmonic functions $p_ l\downarrow p$ on a neighborhood of $\overline D$, obtained by convolution: by (3.4.13) the convolutions $p*\rho _{1/l}$ decrease to $p$, they are smooth, and they are plurisubharmonic because $\mathrm{dd}^{\mathrm{c}}(p*\rho _{1/l})=(\mathrm{dd}^{\mathrm{c}}p)*\rho _{1/l}\geq 0$ (theorem 5.9.1). Their restrictions $p_ l(z',0)$ decrease to $p(z',0)$, so the right sides of (10.8.7) for $p_ l$ are at most the right side for $p$, and the smooth case gives extensions $F_ l$ satisfying (10.8.7) for $p_ l$ with the same constant. Since $p_ l\leq p_1$, the weights have a common positive lower bound, so the $F_ l$ are bounded in $L^2(D)$ and, by lemma 10.6.2(1), a subsequence converges locally uniformly on $D$, including on $\{ w=0\} $. The limit $F$ satisfies $F(z',0)=f(z')$, and Fatou's lemma gives (10.8.7). $\square$

Proof. Cover a neighborhood of $Y$ by finitely many coordinate charts with coordinates $(z',w)$, $w\in \mathbb C^ q$, in which $Y=\{ w=0\} $, and add one open set whose closure does not meet $Y$. Choose smooth functions $\chi _ i\geq 0$ with compact supports in these sets and $\sum _ i\chi _ i^2=1$ on $X$; for instance, divide a finite family of bump functions by the square root of the sum of their squares. On a normal chart put $K_ i=\sum _{l=1}^ q|w_ l|^2$, on the additional set put $K_0=1$, and define $R_ Y=\sum _ i\chi _ i^2K_ i$, each term being extended by zero outside its chart.

The functions $K_ i$ are uniformly comparable on the overlaps where they occur. Indeed, let $(z',w)$ and $(\tilde z',\tilde w)$ be the coordinates of two normal charts, and let $y\in Y$ lie in both. In the coordinates $(z',w)$ choose a polydisc $P'\times P''$ centered at $y$ and contained in both charts. Each $\tilde w_ l$ vanishes on $Y\cap (P'\times P'')=\{ w=0\} $, and since $P''$ is a polydisc centered at $0$,

where the functions $a_{lk}(z',w)=\int _0^1(\partial \tilde w_ l/\partial w_ k)(z',tw)\, \mathrm dt$ are holomorphic, by differentiation under the integral sign, and bounded on smaller polydiscs. Hence $\sum _ l|\tilde w_ l|^2\leq C\sum _ k|w_ k|^2$ near $y$, and by symmetry the reverse inequality holds. The compact set $\operatorname{Supp}\chi _ i\cap \operatorname{Supp}\chi _ j\cap Y$ is covered by finitely many such smaller polydiscs, and on the remaining part of $\operatorname{Supp}\chi _ i\cap \operatorname{Supp}\chi _ j$, a compact set disjoint from $Y$, the functions $K_ i$ and $K_ j$ are continuous and positive, hence bounded above and below; the same holds on $\operatorname{Supp}\chi _0\cap \operatorname{Supp}\chi _ i$, which does not meet $Y$. Hence there is one constant $C\geq 1$ with $K_ j\leq CK_ i$ on $\operatorname{Supp}\chi _ i\cap \operatorname{Supp}\chi _ j$ for all $i,j$. At every point some $\chi _ i^2\geq 1/J$, where $J$ is the number of sets in the cover, so $K_ j/R_ Y\leq JC$ wherever $K_ j$ contributes and $R_ Y>0$, and likewise $(K_ j+\varepsilon )/(R_ Y+\varepsilon )\leq JC$.

To prove (2), let $x\in X$. The terms with $x\notin \operatorname{Supp}\chi _ j$ vanish near $x$, and for the remaining indices $j$ the point $x$ lies in the chart of $K_ j$. On a small neighborhood of $x$ we may therefore apply the identity (6.17.2) of lemma 6.17.1(2) to these indices, with $K_ j+\varepsilon $ in place of the functions denoted there by $\mathrm e^{w_ j}$, so that $S=R_ Y+\varepsilon $. The term with pairwise differences is semipositive, and so is the term with the Hessians $\mathrm{dd}^{\mathrm{c}}\log (K_ j+\varepsilon )$, since $\log (\sum _ l|w_ l|^2+\varepsilon )$ is plurisubharmonic. The remaining term is

which is bounded below by $-b\omega $ because the cutoffs have bounded first and second derivatives and the ratios are bounded by $JC$. This proves the second inequality in (2), and the first follows as $\varepsilon \downarrow 0$, since $\log (R_ Y+\varepsilon )$ decreases to $\log R_ Y$.

To prove (1), let $y\in Y$, and let $\gamma $ be a smooth curve with $\gamma (0)=y$ and $\gamma '(0)=v$. In coordinates the chain rule gives $(R_ Y\circ \gamma )''(0)=D^2R_ Y(y)(v,v)$, because $DR_ Y(y)=0$, so $H_ y$ is well defined. The term $\chi _0^2K_0$ vanishes near $Y$, and a term $\chi _ i^2K_ i$ with $y\notin \operatorname{Supp}\chi _ i$ vanishes near $y$. For a normal chart with $y\in \operatorname{Supp}\chi _ i$ and normal coordinates $w$, we have $w(\gamma (t))=t\, \mathrm dw_ y(v)+O(t^2)$, hence $(\chi _ i^2K_ i)(\gamma (t))=t^2\chi _ i(y)^2|\mathrm dw_ y(v)|^2+O(t^3)$. Therefore

the sum running over the normal charts, with normal coordinates $w^{(i)}$. Since the kernel of $\mathrm dw^{(i)}_ y$ on $T_ yX$ is $T_ yY$ and $\sum _ i\chi _ i(y)^2=1$, we get $H_ y(v,v)>0$ for $v\notin T_ yY$.

To prove (3), let $(z',w)$ be coordinates on $U$ as in the statement, and let $y=(z',0)\in Y\cap U$. Since $\mathrm dw^{(i)}_ y$ is complex linear, the formula for $H_ y$ shows that $Q_{z'}(w)=\sum _ i\chi _ i(y)^2|\mathrm dw^{(i)}_ y(0,w)|^2$ is a Hermitian form in $w$, and it is positive definite because $(0,w)\notin T_ yY$ for $w\neq 0$. In the coordinates, $Q_{z'}(w)$ is half the second derivative of $R_ Y$ at $(z',0)$ in the direction $(0,w)$, so it depends continuously on $z'$. Let $V'\times V''$ be a product of closed balls in the coordinates, centered at a point of $Y\cap U$ and contained in $U$. Taylor's formula in the variable $w$ at $(z',0)$, where $R_ Y$ and its first derivatives vanish, gives $|R_ Y(z',w)-Q_{z'}(w)|\leq C|w|^3$ on $V'\times V''$, with $C$ depending on a bound for the third derivatives of $R_ Y$ there. If $c_0>0$ and $C_0$ are the minimum and the maximum of $Q_{z'}(w)$ for $z'\in V'$ and $|w|=1$, then $c_0|w|^2-C|w|^3\leq R_ Y\leq C_0|w|^2+C|w|^3$ on $V'\times V''$, and (10.8.9) follows on the set where $|w|\leq c_0/(2C)$.

Finally, for (4), note that on the blowup of $Y$, or on any log resolution of $\mathcal I_ Y$, every normal tuple is a local equation of the exceptional divisor times a tuple without common zeros, because the tuples generate $\mathcal I_ Y$; so $\log R_ Y$ is the logarithm of the squared modulus of that equation plus a smooth function. Since moreover $\exp (\log R_ Y)=R_ Y$ is smooth on $X$, $\log R_ Y$ has gentle analytic singularities of type $(1,\mathcal I_ Y)$. Multiplying $R_ Y$ by a constant $\lambda >0$ adds $\log \lambda $ to $\log R_ Y$ and replaces $\log (R_ Y+\varepsilon )$ by $\log \lambda +\log (R_ Y+\varepsilon /\lambda )$, so the inequalities persist. $\square$

Proof. We argue by induction on the number of blowups. On $Z$ itself, $A_ E=b_ E=1$, and every other prime divisor $D$ of $Z$ has $A_ D=1$ and $b_ D=0$. Strict transforms keep their log discrepancies and orders, hence their margins. Since every exceptional divisor lies in the boundary, a prime divisor of a model outside the boundary is the strict transform of a prime divisor of $Z$ other than $E$, so it has $A_ D=1$ and $b_ D=0$. Suppose that the statement holds on a model. Blowing up a center of codimension one is an isomorphism and creates no new divisor; it only adds the center to the boundary, and the center then has margin one if it was not a boundary component. It therefore suffices to show that the exceptional divisor $E_{\mathrm{new}}$ of the blowup of a center $C$ of codimension $c\geq 2$ has a positive margin.

Suppose that $C$ is contained in exactly $r$ boundary components, near its generic point. Since $C$ has normal crossings with the boundary, $r\leq c$. Near such a point choose coordinates $(z,w)$, $w=(w_1,\ldots ,w_ c)$, in which $C=\{ w=0\} $ and the boundary components through the point are coordinate hyperplanes; those containing $C$ are then hyperplanes $\{ w_ j=0\} $. By proposition 9.6.3, in the coordinates $(z,t,v)$ of (9.6.4) we have $E_{\mathrm{new}}=\{ t=0\} $, and the Jacobian determinant of the blowup is $\pm t^{c-1}$, so it vanishes to order $c-1$ along $E_{\mathrm{new}}$. A local equation of a component $\{ w_ j=0\} $ containing $C$ is a unit times $w_ j$ (lemma 6.12.1(1)), and $w_ j$ pulls back to $t$ or to $tv_ j$, which vanishes to order one along $E_{\mathrm{new}}$, because $v_ j$ does not vanish identically on $\{ t=0\} $ (lemma 6.12.1(2),(4)). Thus the pullback of a local equation of each component containing $C$ vanishes to order one along $E_{\mathrm{new}}$. Hence $A_{\mathrm{new}}=c+\sum _{D\supset C}(A_ D-1)$ and $b_{\mathrm{new}}=\sum _{D\supset C}b_ D$, the sums running over the boundary components containing $C$ (other divisors through $C$ have $A_ D=1$ and $b_ D=0$ and contribute nothing). Therefore

All terms on the right are nonnegative by the induction hypothesis. If one of the components containing $C$ has a positive margin, so does $E_{\mathrm{new}}$. Otherwise only the strict transform of $E$ can contain $C$, so $r\leq 1$ and $c-r\geq 1$, and again the margin of $E_{\mathrm{new}}$ is positive. $\square$

Proof. In a holomorphic frame of $L$ the local weight $p$ of $h$ is plurisubharmonic, and its restriction to $Y$ is plurisubharmonic, because the restriction is defined and $Y$ is connected. If $Y$ is a point, the value of $p$ there is finite and we take any $\kappa >0$. Otherwise, by the strong openness theorem (corollary 6.11.6), every point of $Y$ has a neighborhood in $Y$ on which $f$ is square integrable with the weight $\mathrm e^{-(1+\kappa )p|_ Y}$ for some $\kappa >0$; since $Y$ is compact, finitely many coordinate products $B'_ i\times B''_ i$ of the form of lemma 10.8.6, with tangential balls slightly larger than needed and each contained in a neighborhood on which (10.8.9) holds, cover $Y$, and one $\kappa >0$ works for all of them.

Lemma 10.8.6, applied in a frame with the plurisubharmonic weight $(1+\kappa )p$, gives holomorphic extensions $F_ i$ of $f$ on these products whose critical integrals with the logarithmic gain are finite. Let $U_ i$ be smaller concentric products that still cover $Y$. On $U_ i$, the smooth factors coming from the frames and the comparison $c|w|^2\leq R_ Y\leq C|w|^2$ of (10.8.9) preserve this finiteness, which gives (10.8.12); for the logarithmic factor we use that $-\log R_ Y\geq c'(-\log |w|^2)$ for a constant $c'>0$, since $R_ Y\leq C|w|^2$ and both $R_ Y$ and $|w|^2$ are less than $\mathrm e^{-4}$.

We now glue the local extensions. Choose smooth functions $\eta _ i$ with compact support in $U_ i$ whose sum is one on a neighborhood of $Y$; they are obtained by grouping the members of a partition of unity with compact supports subordinate to the cover of $X$ by the sets $U_ i$ and $X\setminus Y$ (proposition 5.2.6). Put $G=\sum _ i\eta _ iF_ i$, each term extended by zero. Then $G|_ Y=f$. Since all $F_ i$ are holomorphic with the same restriction $f$, we have $\bar\partial G=\sum _ i(\bar\partial \eta _ i)F_ i$, and on $Y$ this equals $f\, \bar\partial \sum _ i\eta _ i=0$. The bounds on the cutoffs and their derivatives and (10.8.12) give (10.8.13).

Put $\gamma =|\bar\partial G|_{\omega ,h_0}^2$. Since $\bar\partial G$ is smooth and vanishes on $Y$, we have $\gamma \leq C|w|^2$ in normal coordinates near each point of $Y$, hence $\gamma \leq CR_ Y$ near $Y$, by (10.8.9) and the compactness of $Y$. Let $A=\{ R_ Y<\varepsilon \} $; all integrals below are taken with $\mathrm dV$. Hölder's inequality with the exponents $1+\kappa $ and $(1+\kappa )/\kappa $ gives

The first factor is bounded by (10.8.13). In normal polar coordinates, using (10.8.9) and $\gamma =O(R_ Y)$, the integrability of $\gamma R_ Y^{-q}(-\log R_ Y)^{2/\kappa }$ near $Y$ reduces to $\int _0^ r(-\log t)^{2/\kappa }\, \mathrm dt<\infty $. Hence the second factor tends to zero as $\varepsilon \downarrow 0$, by the absolute continuity of the integral, which proves (10.8.14). $\square$

Proof. We compute on a log resolution adapted to $\Phi $ and $Y$; by lemma 10.8.10, only the strict transform of the exceptional divisor of the blowup of $Y$ contributes to the limit.

A resolution adapted to $\Phi $ and $Y$. Let $\operatorname {Bl}_ YX\to X$ be the blowup of $Y$, with exceptional divisor $E$; for $q=1$ this is the identity and $E=Y$. By proposition 9.6.3, its Jacobian determinant is $\pm t^{q-1}$ in the charts of (9.6.4), where $E=\{ t=0\} $, so its relative canonical divisor is $(q-1)E$. Let $\mathfrak a$ be the ideal in the type of $\Phi $. We apply theorem 9.6.7 on the compact manifold $\operatorname {Bl}_ YX$ to the pullback of $\mathfrak a$, which is nonzero, and to the divisor $E$. This gives a finite sequence of blowups with smooth centers, each having normal crossings with the accumulated divisor of theorem 9.6.7, which starts with $E$; this accumulated divisor is the boundary in the sense of lemma 10.8.10. On the final model the boundary has simple normal crossings, and the pullback of $\mathfrak a$ is the ideal of an effective divisor supported in the boundary. Denote the composite map to $X$ by $\pi $. It is a composite of blowups along closed submanifolds that principalizes $\mathfrak a$, that is, a log resolution of $\mathfrak a$ in the sense of section 6.14. Since definition 6.14.12 requires a smooth remainder on every log resolution, $\Phi \circ \pi $ has a smooth remainder.

Divisor orders. In a normal-crossing chart on the final model write

with $v$ smooth, $r$ smooth and strictly positive, and $a_ D,b_ D\geq 0$. The Jacobian factor of the pulled-back volume is $\prod _ D|s_ D|^{2(A_ D-1)}$ times a smooth positive function, where $A_ D=A_ X(D)$ is the log discrepancy. Let $F_ i$ be one of the holomorphic extensions used in $G$, and suppose that the chart lies over the open set $U_ i$ of lemma 10.8.11. Then $F_ i\circ \pi $ vanishes to some integral order $e_{i,D}\geq 0$ along each coordinate divisor $D$ of the chart. By (10.8.12), the integral of $|F_ i|_{h_0}^2\mathrm e^{-(1+\kappa )\Phi }R_ Y^{-q}(-\log R_ Y)^{-2}$ over $U_ i$ is finite, and this forces

Indeed, fix the other coordinates away from their divisors and integrate radially in $s_ D$: the integrand behaves like $|s_ D|^{2(e_{i,D}+A_ D-(1+\kappa )a_ D-qb_ D)-2}(-\log |s_ D|^2)^{-2}$ when $b_ D>0$, and the squared logarithm in the denominator makes the exponent $-1$ of the radial variable integrable but cannot compensate a smaller exponent. If $b_ D=0$, the logarithmic factor is bounded above and below near generic points of $D$, and the inequality must be strict. This is the computation of theorem 6.12.8 with one additional logarithmic factor.

Shrinking the chart, we may assume that for each $i$ either the chart lies over $U_ i$ or $\eta _ i\circ \pi $ vanishes on it, because $\operatorname{Supp}\eta _ i$ is a compact subset of the open set $U_ i$. Then, on a smaller chart, the pullback of $G=\sum _ i\eta _ iF_ i$ is the monomial $\prod _ Ds_ D^{e_ D}$, with $e_ D=\min _ ie_{i,D}$ over the indices $i$ of the first kind, times a smooth bounded function. The density of $|G|_ h^2R_ Y^{-q}\, \mathrm dV$ is therefore bounded by a constant times $\prod _ D|s_ D|^{2(\gamma _ D-1)}$, where

If $a_ D>0$, (10.8.17) gives $\gamma _ D\geq \kappa a_ D>0$, and if $a_ D=b_ D=0$, it gives $\gamma _ D>0$. If $a_ D=0$ and $b_ D>0$, the formula for the relative canonical divisor of the first blowup gives $A_ D=A_{\operatorname {Bl}_ YX}(D)+(q-1)\operatorname {ord}_ D(E)$, and $b_ D=\operatorname {ord}_ D(E)$ because $\mathcal I_ Y$ pulls back to the ideal of $E$; hence

By lemma 10.8.10, applied with $Z=\operatorname {Bl}_ YX$, this is positive unless $D$ is the strict transform of $E$. For that divisor $b_ E=1$ and $A_ E=q$, and $a_ E=0$: since the restriction of $h$ to $Y$ is defined, generic points of $Y$ lie outside the pole set of $\Phi $. Hence $\gamma _ E=e_ E\geq 0$, and the strict transform of $E$ is the only divisor along which $\gamma _ D$ can vanish.

The annular limit. By a finite partition of unity on the compact inverse image of $Y$, it suffices to consider the integral over a small chart. If no coordinate divisor of the chart is the strict transform of $E$ with $\gamma _ E=0$, the density is integrable, the shells $\{ \varepsilon /2\leq R_ Y\leq \varepsilon \} $ shrink to the null set $\pi ^{-1}(Y)$, and dominated convergence shows that the integral over the shell tends to zero. Otherwise let $t$ be the coordinate defining the strict transform of $E$. The density is $|t|^{-2}$ times a smooth bounded coefficient and the factors $|s_ D|^{2(\gamma _ D-1)}$, $D\ne E$, which are integrable since $\gamma _ D>0$. For fixed values of the remaining coordinates $s$, the shell is $\varepsilon /2\leq |t|^2m(s)r(t,s)\leq \varepsilon $, with $m(s)=\prod _{D\ne E}|s_ D|^{2b_ D}$. Since $r$ is bounded above and below, the shell lies in a range of $\log |t|$ of bounded length, so the integral of $|t|^{-2}$ over it is bounded independently of $\varepsilon $ and $s$; this gives an integrable dominating function. As $\varepsilon \to 0$, the shell shrinks to $t=0$, the smooth coefficient tends to its value at $t=0$, and the integral of $|t|^{-2}$ over the shell tends to $\pi \log 2$. By dominated convergence, only the restriction to $t=0$ of the coefficient of $|t|^{-2}$ in the density contributes to the limit.

Identification of the limit. The argument just given, applied after multiplying the density by a continuous function with compact support, shows that the integrals over the shells converge to $\pi \log 2$ times the total mass of a measure $\mu $ on the strict transform of $E$; in a chart as above, $\mu $ has the density obtained by restricting to $t=0$ the smooth coefficient of $|t|^{-2}$, multiplied by the factors $|s_ D|^{2(\gamma _ D-1)}$, $D\ne E$. Since $\mu $ is determined by the limits of integrals over the shells $\{ \varepsilon /2\leq R_ Y\leq \varepsilon \} $, it does not depend on the coordinates in which it is computed, and since its density in every chart is integrable, it gives no mass to proper analytic subsets. We may therefore compute $\mu $ on the dense open subset of the strict transform of $E$ lying over the complement of the pole set of $\Phi $ in $Y$ and over which the further blowups are isomorphisms, using coordinates of the first blowup $\operatorname {Bl}_ YX$.

Let $(z',w)$ be coordinates on $X$ in which $Y=\{ w=0\} $, and write $\mathrm dV=J(z',w)\, \mathrm dV(z')\, \mathrm dV(w)$ with $J$ smooth and positive. By lemma 10.8.8, $R_ Y=Q_{z'}(w)+O(|w|^3)$ locally uniformly, where $Q_{z'}$ is a positive definite Hermitian form on $\mathbb C^ q$ depending continuously on $z'$. Over this chart, $\operatorname {Bl}_ YX$ is covered by charts with coordinates $(z',t,\xi )$, $\xi \in \mathbb C^{q-1}$, in which, after renumbering the coordinates $w$, we have $w=t(1,\xi )$ and $E=\{ t=0\} $ (for $q=1$ there is no $\xi $, and $t=w$). There $\mathrm dV(w)=|t|^{2(q-1)}\, \mathrm dV(t)\, \mathrm dV(\xi )$ and $R_ Y=|t|^2\bigl (Q_{z'}(1,\xi )+O(|t|)\bigr )$, so away from the poles of $\Phi $ the density of $|G|_ h^2R_ Y^{-q}\, \mathrm dV$ is $|t|^{-2}$ times the smooth function

whose restriction to $t=0$ equals $|f(z')|_ h^2Q_{z'}(1,\xi )^{-q}J(z',0)$, because $G|_ Y=f$. The points of the projective normal fibers not covered by such a chart form a hyperplane in each fiber and are negligible. Integrating over $\xi $, we find that the image of $\mu $ under the projection to $Y$ has the density $c(z')|f(z')|_ h^2$ with respect to $\mathrm dV(z')$, where

Since $Q_{z'}(1,\xi )\geq c_0(1+|\xi |^2)$ with $c_0>0$ locally uniformly in $z'$, and $(1+|\xi |^2)^{-q}$ is integrable on $\mathbb C^{q-1}$, the function $c$ is locally bounded. It depends only on $\omega $, $R_ Y$ and the chart. Covering the compact manifold $Y$ by finitely many such charts and comparing $\mathrm dV(z')$ with $\mathrm dV_ Y$, we obtain (10.8.16). $\square$

Proof. If $Y=X$, take $s=f$, any $N$ and $C=1$. Suppose that $Y$ has codimension $q\geq 1$; if $f=0$, take $s=0$. As in the proof of theorem 10.7.9, we cut off a smooth extension of $f$ and correct the error, now with lemma 10.8.1; a pole of order $q$ along $Y$ in the weight makes the correction vanish on $Y$. Fix $R=R_ Y$ and the constant $b$ of lemma 10.8.8, with $\max _ XR<\mathrm e^{-4}$, and choose $b_ K\geq 0$ with $c_1(K_ X^{-1},h_{K^{-1},\omega })\geq -b_ K\omega $ (lemma 6.14.1). In Steps 1–4 we prove the theorem for a weight $\Phi $ with gentle analytic singularities, as for the approximations of theorem 6.17.6, under the curvature bound

In these steps $h$ denotes the metric $h_0\mathrm e^{-\Phi }$, and $A$ the pole set of $\Phi $. Step 5 treats arbitrary weights.

Step 1: the twisted curvature form. For the given $f$ choose $G$ and $\kappa $ by lemma 10.8.11. On $M=X\setminus (A\cup Y)$ consider $F=L\otimes K_ X^{-1}$ with the metric $h\otimes h_{K^{-1},\omega }\mathrm e^{-q\psi }$, where $\psi =\log R$; the canonical identification of $L$-valued functions with $F$-valued $(n,0)$-forms preserves the squared norms times the volume. The manifold $M$ carries a Kähler form and cutoffs as required in lemma 10.8.1, by lemma 10.7.5 applied to $\Phi +\psi $, which is quasi-plurisubharmonic, has pole set $A\cup Y$ and is smooth on $M$. The resulting Kähler form depends on $\Phi $, but the estimate of lemma 10.8.1 does not involve it.

Let $\psi _\varepsilon =\log (\varepsilon +R)$, $\mu $, $a_\varepsilon $ and $b_\varepsilon $ be the functions of lemma 10.8.4, applied on $M$ to the smooth function $R$, which satisfies $0<R<\mathrm e^{-4}$ there, for $0<\varepsilon <\mathrm e^{-4}$. For fixed $\varepsilon $, $a_\varepsilon $ and $b_\varepsilon $ are bounded above and below by positive constants, $a_\varepsilon \geq 1$ and $1\leq -\mu '(\psi _\varepsilon )\leq 2$. By part (2) of that lemma, applied with $2\pi $ times the curvature form of the metric of $F$,

Since $\mathrm i\partial \bar\partial \psi =2\pi \mathrm{dd}^{\mathrm{c}}\psi \geq -2\pi b\omega $, $0\leq R/(\varepsilon +R)\leq 1$ and $-\mu '\leq 2$, the identity (10.8.5) gives

where in the last step we used $a_\varepsilon \geq 1$ and $N_0-b_ K-qb=2b+2$. All constants in this display depend only on the geometry.

Step 2: the correction. Let $\rho \in C^\infty (\mathbb R)$ satisfy $0\leq \rho \leq 1$, $\rho =1$ on $(-\infty ,1/2]$ and $\rho =0$ on $[1,\infty )$, and put $G_\varepsilon =\rho (R/\varepsilon )G$. Since $\bar\partial R/\varepsilon =(1+R/\varepsilon )\bar\partial \psi _\varepsilon $, the $\bar\partial $-closed form $g_\varepsilon =\bar\partial G_\varepsilon $ splits as

regarded as $F$-valued $(n,1)$-forms. As in the proof of lemma 10.8.6, the rank-one part of (10.8.20) and lemma 10.7.1(1),(2) give

pointwise on the shell $\{ \varepsilon /2\leq R\leq \varepsilon \} $, where $R/\varepsilon \leq 1$, the norms being those of the metric of $F$. Hence the integral $M_{g_\varepsilon ^{(1)}}$ of lemma 10.8.1, formed with $H_\varepsilon $, is at most $C_1\int _{\{ \varepsilon /2\leq R\leq \varepsilon \} }|G|_ h^2R^{-q}\, \mathrm dV$, whose upper limit as $\varepsilon \to 0$ is at most $C_2\int _ Y|f|_ h^2\, \mathrm dV_ Y$ by lemma 10.8.15. Since $H_\varepsilon \geq 2\pi \omega $, lemma 10.7.1(3) bounds $M_{g_\varepsilon ^{(2)}}$ by

which tends to zero by (10.8.14).

Both forms are square integrable for the metric of $F$. For $g_\varepsilon ^{(1)}$, the derivative of the cutoff is bounded for fixed $\varepsilon $, and (10.8.13) bounds the integral of $|G|_ h^2R^{-q}$ over the shell, where the logarithm is bounded and $\mathrm e^{-\Phi }\leq \mathrm e^{\kappa \sup _ X\Phi }\mathrm e^{-(1+\kappa )\Phi }$; for $g_\varepsilon ^{(2)}$ this follows from (10.8.14). Since $M_{g'+g''}\leq 2(M_{g'}+M_{g''})$, by the Cauchy–Schwarz inequality for the positive form $\langle [H_\varepsilon ,\Lambda _\omega ]^{-1}\cdot ,\cdot \rangle $, lemma 10.8.1 gives $u_\varepsilon $ on $M$ with

The constants $C_1,\ldots ,C_4$ depend only on the geometry and the fixed cutoff $\rho $; neither the exponent $\kappa $ nor the derivatives of the local extensions $F_ i$ enter them.

Step 3: holomorphic sections with the trace $f$. For fixed $\varepsilon $ the denominator $a_\varepsilon +b_\varepsilon $ is bounded, and $h\geq \mathrm e^{-\sup _ X\Phi }h_0$ because $\Phi $ is bounded above on the compact manifold $X$. Hence $s_\varepsilon =G_\varepsilon -u_\varepsilon $, which is holomorphic on $M$, is square integrable for $h_0$, and it extends to a global holomorphic section by lemma 10.7.8. Then $u_\varepsilon =G_\varepsilon -s_\varepsilon $ is smooth on $X$.

It remains to show that $s_\varepsilon |_ Y=f$. Let $y\in Y\setminus A$, and choose holomorphic coordinates $(z',w)$ near $y$ with $Y=\{ w=0\} $. Near $y$ the weight $\Phi $ is smooth and $a_\varepsilon +b_\varepsilon $ is bounded, so $\int |u_\varepsilon |_{h_0}^2R^{-q}\, \mathrm dV$ is finite over a neighborhood of $y$. Since $R\leq C|w|^2$ near $y$ by lemma 10.8.8, and $\int _{|w|<r}|w|^{-2q}\, \mathrm dV(w)=\infty $ in $\mathbb C^ q$ for every $r>0$, by polar coordinates, the continuous function $u_\varepsilon $ vanishes at $y$. Hence $u_\varepsilon =0$ on $Y\setminus A$, and $s_\varepsilon =G=f$ there. Since the restriction of $h$ to $Y$ is defined, $A\cap Y$ is a proper analytic subset of the connected manifold $Y$, and the identity theorem gives $s_\varepsilon |_ Y=f$.

Step 4: the norm bound. For all sufficiently small $\varepsilon $,

because $a_\varepsilon +b_\varepsilon \leq 5\psi ^2$ by lemma 10.8.4(1), and because $R^ q(-\log R)^2$ is bounded on $X$. Consequently (10.8.21) bounds $\int _ X|u_\varepsilon |_ h^2\, \mathrm dV$ by $C_6$ times the integral there. The section $G$ has finite $h$-norm, since (10.8.13) controls it, $\Phi $ being bounded above and $R^ q(-\log R)^2$ bounded; therefore $\| G_\varepsilon \| _ h\to 0$ as $\varepsilon \to 0$, by absolute continuity of the integral on the shrinking supports. The triangle inequality gives

with $C_7=C_6C_4$. The sections $s_\varepsilon $ are bounded in $L^2$ for $h_0$, so, by lemma 10.6.2(2), a sequence $\varepsilon _ k\to 0$ gives sections converging uniformly to a holomorphic section $s$. Its restriction to $Y$ is $f$, and Fatou's lemma gives $\| s\| _ h^2\leq C_7\int _ Y|f|_ h^2\, \mathrm dV_ Y$. This proves the theorem for gentle weights with the constant $C_7$, which depends only on the geometry.

Step 5: arbitrary weights. Put $N=N_0+1$. By theorem 6.17.6, applied to $\varphi $ with the background $c_1(L,h_0)-N\omega $, there are $\Phi _ j\downarrow \varphi $ with gentle analytic singularities of the kind required in the gentle case and $c_1(L,h_0\mathrm e^{-\Phi _ j})\geq (N-\delta _ j)\omega $, where $\delta _ j\to 0$; for large $j$ the curvature is at least $N_0\omega $. Since $\Phi _ j\geq \varphi $, the restrictions of the metrics $h_0\mathrm e^{-\Phi _ j}$ to $Y$ are defined, and the norms of $f$ for them are at most its norm for $h$. The gentle case gives $s_ j\in H^0(X,L)$ with $s_ j|_ Y=f$ and $\int _ X|s_ j|_{h_0}^2\mathrm e^{-\Phi _ j}\, \mathrm dV\leq C_7\int _ Y|f|_ h^2\, \mathrm dV_ Y$. Since $\Phi _ j\leq \Phi _1$, the $s_ j$ are bounded in $L^2$ for $h_0$, and by lemma 10.6.2(2) a subsequence converges uniformly to a holomorphic section $s$ with $s|_ Y=f$. The weights $\mathrm e^{-\Phi _ j}$ converge to $\mathrm e^{-\varphi }$ wherever $\varphi $ is finite, hence almost everywhere, and Fatou's lemma gives (10.8.19) with $C=C_7$. $\square$

Proof. The metric $h_ T\otimes h_0^ k\mathrm e^{-k\varphi }$ on $T\otimes L^ k$ is plurisubharmonic, its curvature current is $c_1(T,h_ T)+k(\theta +\mathrm{dd}^{\mathrm{c}}\varphi )\geq (k\delta -b_ T)\omega \geq N\omega $, and its restriction to $Y$ is defined. Hence theorem 10.8.18 applies. Note that the finiteness of the norm on $Y$ means that $f$ lies locally in the multiplier ideal of $k\varphi |_ Y$, and that the constant $C$ is multiplied by $n!/(\dim Y)!$ if the volumes $\omega ^ n$ and $\omega |_ Y^{\dim Y}$ are used instead of $\mathrm dV$ and $\mathrm dV_ Y$. $\square$

Proof. (1) Choose a closed ball $N$ in the $y$-coordinates, centered at $y(p)$, whose preimage lies in $V$. The map $x\circ y^{-1}$ is smooth on the convex compact set $N$, so by the mean value inequality it is Lipschitz on $N$ with some constant $L$. Moreover $N$ has some diameter $D$, and $t^\alpha \leq D^{\alpha -\alpha '}t^{\alpha '}$ for $0\leq t\leq D$. Hence, for $a,b$ in the preimage of $N$, which is a neighborhood of $p$,

(2) By (1), applied with $y=x$, every point of $M$ has a coordinate neighborhood on which $f$ is Hölder continuous with some exponent, and on which it remains so with every smaller exponent. Finitely many of these neighborhoods cover $M$, and the smallest of their exponents serves on all of them.

(3) A smooth function is Lipschitz on every closed coordinate ball, by the mean value inequality, so it is Hölder continuous with exponent one. Let $f$ and $g$ be Hölder continuous and $p\in M$. By (1) there are coordinates $x$ on a neighborhood $V$ of $p$ and $\alpha \in (0,1]$ such that $f$ and $g$ are Hölder continuous with exponent $\alpha $ on $V$ in these coordinates, say with the constant $C$; shrinking $V$, we may assume that $|f|,|g|\leq B$ on $V$. For $a,b\in V$ we have $|(f+g)(a)-(f+g)(b)|\leq 2C|x(a)-x(b)|^\alpha $,

and $|\max \{ f,g\} (a)-\max \{ f,g\} (b)|\leq \max \{ |f(a)-f(b)|,|g(a)-g(b)|\} $.

For the power, note first that $(s+t)^\gamma \leq s^\gamma +t^\gamma $ for $s,t\geq 0$: if $s+t>0$, divide by $(s+t)^\gamma $ and use $\lambda ^\gamma \geq \lambda $ for $\lambda \in [0,1]$. Hence, if $s\geq t\geq 0$, then $s^\gamma \leq (s-t)^\gamma +t^\gamma $, that is, $|s^\gamma -t^\gamma |\leq |s-t|^\gamma $ for all $s,t\geq 0$. If $f\geq 0$ has exponent $\alpha $ on $V$ with the constant $C$, it follows that $|f(a)^\gamma -f(b)^\gamma |\leq C^\gamma |x(a)-x(b)|^{\alpha \gamma }$.

(4) Let $p'\in M'$, let $V$ be a coordinate neighborhood of $\Phi (p')$, with coordinates $x$, on which $f$ is Hölder continuous with exponent $\alpha $ and constant $C$, and let $N$ be a closed coordinate ball around $p'$, in coordinates $x'$, with $\Phi (N)\subset V$. The map $x\circ \Phi $, written in the coordinates $x'$, is smooth on the convex compact set $N$, hence Lipschitz on $N$ with some constant $L$. Therefore $|f(\Phi (a))-f(\Phi (b))|\leq CL^\alpha |x'(a)-x'(b)|^\alpha $ for $a,b\in N$. $\square$

Proof. We begin with the inequalities, which are pointwise. Fix a point and suppose that $a\geq b$ there; the other case is symmetric. If $a=-\infty $, then $b=-\infty $ as well, and all three expressions equal $-\infty $. Otherwise $\log (\mathrm e^ a+\mathrm e^ b)=a+\log (1+\mathrm e^{b-a})$, and $0\leq \log (1+\mathrm e^{b-a})\leq \log 2$.

Put $H(x,y)=\log (\mathrm e^ x+\mathrm e^ y)$ for $x,y\in [-\infty ,\infty )$, with the same conventions. On $\mathbb R^2$ the function $H$ is smooth and increasing in each variable: its partial derivatives are $p$ and $1-p$, where $p=\mathrm e^ x/(\mathrm e^ x+\mathrm e^ y)$, and its Hessian matrix is $p(1-p)\left(\begin{smallmatrix}1& -1\\ -1& 1\end{smallmatrix}\right)$. On $[-\infty ,\infty )^2$ it is still increasing in each variable, and it is continuous, with values in $[-\infty ,\infty )$. Indeed, by the inequalities just proved, $\max \{ x,y\} \leq H(x,y)\leq \max \{ x,y\} +\log 2$, so $H(x,y)\to -\infty $ when $\max \{ x,y\} \to -\infty $; and $H(x,y)=y+\log (1+\mathrm e^{x-y})\to y_0$ when $x\to -\infty $ and $y\to y_0\in \mathbb R$.

The first assertion is local. Let $B\Subset U$ be a coordinate ball, let $s$ be a smooth real function on $B$ with $\mathrm{dd}^{\mathrm{c}}s=\theta $ (lemma 6.14.2), and let $B'\Subset B$ be a smaller concentric ball. Then $a+s$ and $b+s$ are plurisubharmonic on $B$, by lemma 6.14.4(2), and $\log (\mathrm e^ a+\mathrm e^ b)+s=H(a+s,b+s)$. If $f,g$ are smooth plurisubharmonic functions, the chain rule gives

with $p$ evaluated at $(f,g)$; hence $H(f,g)$ is plurisubharmonic, by proposition 5.9.2(1). In general, let $f_ k$ and $g_ k$ be the regularizations of $a+s$ and $b+s$ by radial mollifiers whose radii decrease to zero and are smaller than the distance from $B'$ to $\partial B$. By proposition 5.9.2(2), they are smooth plurisubharmonic functions on $B'$, and they decrease to $a+s$ and $b+s$. Since $H$ is continuous and increasing in each variable on $[-\infty ,\infty )^2$, the functions $H(f_ k,g_ k)$ decrease to $H(a+s,b+s)$. This limit is at least $a+s$, so it is not identically $-\infty $, and it is plurisubharmonic on $B'$ by theorem 5.10.1. Consequently $\log (\mathrm e^ a+\mathrm e^ b)=H(a+s,b+s)-s$ is quasi-plurisubharmonic on $B'$ and $\theta +\mathrm{dd}^{\mathrm{c}}\log (\mathrm e^ a+\mathrm e^ b)=\mathrm{dd}^{\mathrm{c}}H(a+s,b+s)\geq 0$ there. Every point of $U$ is the center of such a ball $B'$, so the first assertion follows. $\square$

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 $X$. $\square$