Proposition 6.14.111A99E612D41C

Proof. By proposition 9.6.3, $Y$ is a complex manifold and $E$ is a smooth hypersurface with $\mathcal I_ W\mathcal O_ Y=\mathcal O_ Y(-E)$; moreover $Y$ is compact, since $\pi $ is proper (proposition 9.6.2), and $\pi $ restricts to a biholomorphism $Y\setminus E\to X\setminus W$. The last assertion follows from the preceding ones by induction: since $X$ is compact, a log resolution is a composite of finitely many blowups along closed submanifolds of positive codimension, and each intermediate manifold is compact Kähler by what precedes.

We construct $\eta $ as the curvature of a metric on $\mathcal O_ Y(-E)$ and show that it is positive definite along the fibers of $E\to W$ (Step 1). Since $\pi ^*\omega $ is positive in the remaining directions, $\pi ^*\omega +\varepsilon \eta $ is then positive definite near $E$ for small $\varepsilon >0$ (Step 2), and also away from $E$ (Step 3). We use the charts of proposition 9.6.3. Near a point of $W$ at which $W$ has codimension $r$, choose coordinates $(z,w)$ on $X$, $w=(w_1,\ldots ,w_ r)$, with $W=\{ w=0\} $. For each pivot $i$ the blowup has a chart with coordinates $(z,t,v)$, $v=(v_ j)_{j\neq i}$, in which

in it $E=\{ t=0\} $, and the fibers of $E\to W$ are the sets $\{ z=\text{const},\ t=0\} $. If $r=1$ there is no variable $v$, and $\pi $ is the identity near the point.

Step 1: a closed $(1,1)$-form that is positive along the fibers of $E$. Choose finitely many charts $V_1,\ldots ,V_ N$ of $X$ covering $W$, with coordinates $(z^\alpha ,w^\alpha )$ such that $W\cap V_\alpha =\{ w^\alpha =0\} $, and put $V_0=X\setminus W$. Grouping the members of a partition of unity subordinate to this cover (proposition 5.2.6) according to a set $V_\alpha $ containing their supports, we obtain a smooth partition of unity $(\lambda _\alpha )_{0\leq \alpha \leq N}$ on $X$ with $\operatorname{Supp}\lambda _\alpha \subset V_\alpha $. Put

each term extended by zero outside $V_\alpha $. This is a smooth nonnegative function on $X$ that vanishes on $W$, because $\lambda _0=0$ there, and is positive on $X\setminus W$: at a point $x\notin W$ some $\lambda _\alpha (x)$ is positive, and $w^\alpha (x)\neq 0$ if $\alpha \geq 1$.

We describe $\rho $ near a point $x_0\in W$, in coordinates $(z,w)$ as above on a polydisc $\Delta $ centered at $x_0$. Shrinking $\Delta $, we may assume that $\Delta $ does not meet $\operatorname{Supp}\lambda _0$, nor $\operatorname{Supp}\lambda _\alpha $ for the indices $\alpha $ with $x_0\notin V_\alpha $, and that $\Delta \subset V_\alpha $ for the others. For such an $\alpha $, each component $w^\alpha _ k$ vanishes on $W=\{ w=0\} $, so

that is, $w^\alpha =A_\alpha w$ with a holomorphic $r\times r$ matrix $A_\alpha $ on $\Delta $. At points of $W$ the matrix $A_\alpha (z,0)=(\partial w^\alpha _ k/\partial w_ l)(z,0)$ is invertible: since $w^\alpha $ vanishes on $W$, its derivatives in the $z$-directions vanish there, and the Jacobian matrix of the coordinate change $(z,w)\mapsto (z^\alpha ,w^\alpha )$ is invertible. Hence, on $\Delta $,

where the sums run over the indices $\alpha \geq 1$ with $x_0\in V_\alpha $, and $w^*$ is the conjugate transpose of the column vector $w$. The Hermitian matrix $\mathcal Q$ is smooth and positive semidefinite, and it is positive definite at $x_0$, because the coefficients $\lambda _\alpha (x_0)$ of the invertible matrices $A_\alpha (x_0)$ add up to one; shrinking $\Delta $, we may assume that $\mathcal Q$ is positive definite on $\Delta $. In a chart $(z,t,v)$ over $\Delta $ we obtain

and $h_ E$ is smooth and positive, since $\tilde v\neq 0$.

Define a real $(1,1)$-form $\eta $ on $Y$ by $\eta =\mathrm{dd}^{\mathrm{c}}\log (\rho \circ \pi )$ on $Y\setminus E$ and $\eta =\mathrm{dd}^{\mathrm{c}}\log h_ E$ on each chart $(z,t,v)$ as above. These definitions agree. Off $E$ we have $\log (\rho \circ \pi )=\log |t|^2+\log h_ E$, and $\log |t|^2$ is pluriharmonic there. On the overlap of two charts, the local equations $t$ and $t'$ of $E$ generate the same ideal $\mathcal O_ Y(-E)$, so $t'=ut$ with $u$ holomorphic and nowhere zero. Off $E$ the corresponding functions $h_ E$ and $h'_ E$ satisfy $\log h_ E-\log h'_ E=\log |u|^2$. Both sides are continuous on the overlap and $E$ is nowhere dense, so this identity holds on the whole overlap, and $\log |u|^2$ is pluriharmonic. Thus $\eta $ is a smooth closed real $(1,1)$-form on $Y$. It is the curvature, in the sense of definition 6.14.9, of the metric on the line bundle $\mathcal O_ Y(-E)$ with the frame $1$ on $Y\setminus E$, the frames $t$ on the charts and the transition functions $t$ and $t'/t$, for which $|1|^2=1/(\rho \circ \pi )$ and $|t|^2=1/h_ E$. These squared norms are smooth and positive, and they satisfy the compatibility condition of that definition because $\rho \circ \pi =|t|^2h_ E$ and $h'_ E=|t/t'|^2h_ E$.

We show that $\eta $ is positive definite along the fibers of $E\to W$. Fix a point $(z_0,0,v_0)$ of $E$ in a chart, and put $\mathcal Q_0=\mathcal Q(z_0,0)$. On the fiber $\{ z=z_0,\ t=0\} $ we have $h_ E=\tilde v^*\mathcal Q_0\tilde v$. Write $\mathcal Q_0=\Gamma ^*\Gamma $ with an invertible matrix $\Gamma $, for instance the positive square root of $\mathcal Q_0$, and let $e_1,\ldots ,e_ r$ be the standard basis of $\mathbb C^ r$. Then $h_ E=|F(v)|^2$ on the fiber, where $F(v)=\Gamma \tilde v=\Gamma e_ i+\sum _{j\neq i}v_ j\Gamma e_ j$ is an affine map whose linear part $\Lambda \xi =\sum _{j\neq i}\xi _ j\Gamma e_ j$ is injective, and $F(v)$ never lies in the image of $\Lambda $, because the columns $\Gamma e_1,\ldots ,\Gamma e_ r$ are linearly independent. Writing $\langle a,b\rangle =\sum _ ka_ k\bar b_ k$ on $\mathbb C^ r$, we have $\partial _{\bar v_ l}|F|^2=\langle F,\Gamma e_ l\rangle $ and $\partial _{v_ k}|F|^2=\langle \Gamma e_ k,F\rangle $, so for $\xi \in \mathbb C^{r-1}$

By the Cauchy–Schwarz inequality this is nonnegative, and it vanishes only if $\Lambda \xi =\mu F$ for some $\mu \in \mathbb C$. Since $F$ does not lie in the image of $\Lambda $, this forces $\mu =0$, so $\Lambda \xi =0$ and $\xi =0$. By the matrix criterion, the restriction of $\eta $ to each fiber of $E\to W$ is positive definite.

Step 2: positivity near $E$. The form $P=\pi ^*\omega $ is a smooth closed semipositive $(1,1)$-form on $Y$. We regard the real $(1,1)$-forms $P$ and $\eta $ in a chart $(z,t,v)$ as Hermitian forms on holomorphic tangent vectors, given by their coefficient matrices, so that $P(\tau ,\tau )=\omega (\mathrm d\pi \, \tau ,\mathrm d\pi \, \tau )$ by the chain rule; norms of tangent vectors are taken in the coordinates of the chart. Let $V$ be spanned by the vectors $\partial /\partial v_ k$ and $H$ by the vectors $\partial /\partial z_ l$ and $\partial /\partial t$, and write $P$ and $\eta $ in block form with respect to the decomposition of the tangent space into $V$ and $H$. At points of $E$ we have $\mathrm d\pi (\partial /\partial v_ k)=t\, \partial /\partial w_ k=0$, while $\mathrm d\pi (\partial /\partial z_ l)=\partial /\partial z_ l$ and $\mathrm d\pi (\partial /\partial t)=\sum _ j\tilde v_ j\, \partial /\partial w_ j$ are linearly independent. Since $\omega $ is positive definite, $P_{HV}=0$ and $P_{HH}$ is positive definite at points of $E$, while $\eta _{VV}$ is positive definite there by Step 1.

Choose finitely many compact sets $K_\iota $ whose interiors cover $E$, each contained in a chart $(z,t,v)$. The smallest eigenvalues of $P_{HH}$ and of $\eta _{VV}$ are continuous and positive on the compact sets $E\cap K_\iota $, so there are $a,c>0$ such that they exceed $2c$ and $2a$, respectively, on $E\cap K_\iota $ for every $\iota $. Let $M>0$ bound the operator norm of $\eta $ on every $K_\iota $, so that $|\eta (\tau ,\tau ')|\leq M|\tau ||\tau '|$, and choose $\ell \in (0,1]$ with $2M\ell +M\ell ^2\leq a/2$. Where $P_{HH}$ is invertible, the linear map $L=P_{HH}^{-1}P_{HV}\colon V\to H$ depends continuously on the point, and it vanishes on $E$. Hence every point of $E\cap K_\iota $ has a neighborhood on whose intersection with $K_\iota $ we have $P_{HH}\geq c$, $\eta _{VV}\geq a$ and $\| L\| \leq \ell $. Let $O_\iota $ be the union of these neighborhoods. Then

is an open neighborhood of $E$ with $\mathcal N\cap K_\iota \subset O_\iota $ for every $\iota $. On $\mathcal N\cap K_\iota $ we have, for $\xi \in V$,

Write a tangent vector at a point of $\mathcal N\cap K_\iota $ as $\tau =\xi +\zeta $ with $\xi \in V$ and $\zeta \in H$, and put $\sigma =\zeta +L\xi \in H$. Completing the square,

and $S\geq 0$ because $P\geq 0$ (take $\zeta =-L\xi $, so that $\sigma =0$). Hence $P(\tau ,\tau )\geq c|\sigma |^2$. Since $\tau =(\xi -L\xi )+\sigma $,

Therefore, for $\varepsilon >0$,

The right-hand side is a quadratic form in $(|\sigma |,|\xi |)$, which is positive definite as soon as $\varepsilon M<c$ and $(c-\varepsilon M)a>8\varepsilon M^2$; and $(\sigma ,\xi )\neq 0$ for $\tau \neq 0$. Thus $\pi ^*\omega +\varepsilon \eta $ is positive definite on the fixed neighborhood $\mathcal N$ of $E$ for all sufficiently small $\varepsilon >0$.

Step 3: positivity away from $E$. On the compact set $Y\setminus \mathcal N\subset Y\setminus E$ the map $\pi $ is a local biholomorphism, so $\pi ^*\omega $ is positive definite at every point. Cover $Y\setminus \mathcal N$ by finitely many compact sets contained in charts. On each of them the smallest eigenvalue of the coefficient matrix of $\pi ^*\omega $ is bounded below by a positive constant $c'$, and the operator norm of the coefficient matrix of $\eta $ is bounded by a constant $M'$, with $c'$ and $M'$ independent of the set. Hence $\pi ^*\omega +\varepsilon \eta $ is positive definite on $Y\setminus \mathcal N$ if $\varepsilon M'<c'$. Together with Step 2, this shows that $\pi ^*\omega +\varepsilon \eta $ is positive definite on all of $Y$ for every sufficiently small $\varepsilon >0$. It is closed, since both terms are, so it is a Kähler form on $Y$. $\square$