Lemma 10.8.8B903D314198B

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. $\square$