6.14 Smooth backgrounds and analytic singularitiesA4223F8F8B02

Proof. If $K=\varnothing $ we take $C=0$. Otherwise cover $K$ by finitely many compact sets $K_\iota $, each contained in a chart and in the domain of $\alpha $, and write $\alpha =\mathrm i\sum _{j,k}a_{jk}\, \mathrm dz_ j\wedge \mathrm d\bar z_ k$ and $\omega =\mathrm i\sum _{j,k}w_{jk}\, \mathrm dz_ j\wedge \mathrm d\bar z_ k$ there, with Hermitian matrices $(a_{jk})$ and $(w_{jk})$. On $K_\iota $ the smallest eigenvalue of $(w_{jk})$, a continuous positive function, is at least some $c_\iota >0$, and the largest eigenvalue of $-(a_{jk})$ is at most some $M_\iota \geq 0$. Hence $(a_{jk})\geq -M_\iota I\geq -(M_\iota /c_\iota )(w_{jk})$ on $K_\iota $. By the matrix criterion for positivity (section 5.4), it follows that $\alpha \geq -C\omega $ on $K$ with $C=\max _\iota M_\iota /c_\iota $. $\square$

Proof. In dimension zero there is nothing to prove, so let $n\geq 1$. Choose concentric balls $B\Subset B'\Subset B''$ such that $\theta $ is defined on a neighborhood of $\overline{B''}$. By the last assertion of lemma 5.4.2, applied to the closed real $2$-form $\theta $ on the ball $B''$, there is a smooth real one-form $\alpha $ on $B''$ with $\mathrm d\alpha =\theta $. Since $\theta $ has type $(1,1)$, the $(0,2)$-part $\bar\partial \alpha ^{0,1}$ of $\mathrm d\alpha $ vanishes. The ball $B'$ is pseudoconvex by corollary 6.3.5, and $\alpha ^{0,1}$ is bounded on $B'$. Hence theorem 6.4.9, applied on $B'$ with $p=0$ and $q=|z|^2$, whose complex Hessian is the identity, gives a function $f\in L^2(B')$ with $\bar\partial f=\alpha ^{0,1}$.

The function $f$ is smooth. Indeed, let $y\in B'$ and choose polydiscs $P'\Subset P\Subset B'$ centered at $y$. By lemma 5.4.2, applied to the smooth $\bar\partial $-closed form $\alpha ^{0,1}$ on $P$, there is a smooth function $f_0$ on $P'$ with $\bar\partial f_0=\alpha ^{0,1}$. Then $\bar\partial (f-f_0)=0$ on $P'$ in the sense of distributions, so $\Delta (f-f_0)=4\sum _ j\partial _{z_ j}\partial _{\bar z_ j}(f-f_0)=0$ there, and by theorem 3.1.11 the locally integrable function $f-f_0$ agrees almost everywhere on $P'$ with a smooth function. Thus $f$ agrees almost everywhere near each point with a smooth function; two such smooth functions coincide on the overlap of their domains, being continuous and equal almost everywhere, so we may take $f$ smooth on $B'$. Since $\alpha $ is real, $\alpha ^{1,0}=\overline{\alpha ^{0,1}}=\partial \bar f$, and the $(1,1)$-part of $\mathrm d\alpha =\theta $ gives

Thus $h=4\pi \operatorname{Im}f$ has the required property on $B\subset B'$.

For the last assertion, let $h$ be real and pluriharmonic on a coordinate ball. Then $\partial h$ is a $(1,0)$-form with holomorphic coefficients, since $\bar\partial \partial h=0$, and it is closed. By the last assertion of lemma 5.4.2, applied to the closed one-form $2\partial h$ on this ball, there is a smooth function $F$ on the whole ball with $\mathrm dF=2\partial h$. As $\mathrm dF$ has type $(1,0)$, $F$ is holomorphic, and $\mathrm d(\operatorname{Re}F)=\partial h+\bar\partial h=\mathrm dh$. Hence $h-\operatorname{Re}F$ is constant. $\square$

Proof. At points where $X$ has dimension zero, which are isolated, all functions are smooth and $\mathrm{dd}^{\mathrm{c}}$ vanishes, so every assertion is trivial there; we may therefore assume that $X$ has positive dimension.

(1) Near each point, $\varphi =u+f$ with $u$ plurisubharmonic and $f$ smooth, and $u$ is locally integrable by theorem 5.9.1. Moreover $u$ is subharmonic, by the same theorem, so at every point it is the limit of its averages over small balls, by (3.4.13); the continuous function $f$ has the same property, and hence so has $\varphi $. Now let $\varphi $ and $\psi $ be quasi-plurisubharmonic on $V$ with $\varphi \leq \psi $ almost everywhere, and let $x\in V$. Since both functions are locally integrable, their averages over small coordinate balls centered at $x$, in a chart at $x$, are finite, and those of $\varphi $ are at most those of $\psi $; letting the radius tend to zero gives $\varphi (x)\leq \psi (x)$. Applying this in both directions, we see that two quasi-plurisubharmonic functions that agree almost everywhere agree everywhere.

(2) Let $\varphi $ be quasi-plurisubharmonic on $V$. If $\varphi +h$ is plurisubharmonic, then $\theta +\mathrm{dd}^{\mathrm{c}}\varphi =\mathrm{dd}^{\mathrm{c}}(\varphi +h)\geq 0$ by theorem 5.9.1. Conversely, suppose that $\mathrm{dd}^{\mathrm{c}}(\varphi +h)\geq 0$ on $V$. On a coordinate ball $B\subset V$, theorem 5.9.1 gives a plurisubharmonic function on $B$ that agrees almost everywhere with $\varphi +h$; both are quasi-plurisubharmonic on $B$, so they agree everywhere on $B$ by (1). Hence $\varphi +h$ is plurisubharmonic on $V$.

(3) Let $B$ be a coordinate ball and $h$ a smooth real function on $B$ with $\mathrm{dd}^{\mathrm{c}}h=\theta $ (lemma 6.14.2). By (2) the functions $\varphi _ i+h$ are plurisubharmonic on $B$, hence so is $\max _ i\varphi _ i+h=\max _ i(\varphi _ i+h)$, by proposition 5.9.5. Thus $\max _ i\varphi _ i$ is quasi-plurisubharmonic on $B$, and $\theta +\mathrm{dd}^{\mathrm{c}}\max _ i\varphi _ i\geq 0$ there by (2). Such balls cover $X$.

(4) By compactness there are finitely many open sets $V_\kappa \Subset V_\kappa ^+$ such that the $V_\kappa $ cover $X$ and $\varphi =u_\kappa +f_\kappa $ on $V_\kappa ^+$, with $u_\kappa $ plurisubharmonic and $f_\kappa $ smooth. By lemma 6.14.1, applied to the compact set $\overline{V_\kappa }$ and the form $\mathrm{dd}^{\mathrm{c}}f_\kappa $, there are constants $C_\kappa \geq 0$ with $\mathrm{dd}^{\mathrm{c}}f_\kappa \geq -C_\kappa \omega $ on $V_\kappa $, and $\mathrm{dd}^{\mathrm{c}}u_\kappa \geq 0$ by theorem 5.9.1. Hence $\mathrm{dd}^{\mathrm{c}}\varphi \geq -C\omega $ with $C=\max _\kappa C_\kappa $.

(5) Replacing $C$ by $\max \{ C,0\} $, we may assume that $C\geq 0$. Let $B$ be a coordinate ball whose closure lies in a chart. By lemma 6.14.1, applied in the chart to the compact set $\overline B$, the Hermitian form $\mathrm{dd}^{\mathrm{c}}|z|^2$ and the form $-\omega $, there is $C'$ with $\omega \leq C'\mathrm{dd}^{\mathrm{c}}|z|^2$ on $B$, so $\mathrm{dd}^{\mathrm{c}}(\varphi +A|z|^2)\geq 0$ on $B$ for $A=CC'$. By theorem 5.9.1, $\varphi +A|z|^2$ agrees almost everywhere on $B$ with a plurisubharmonic function, and subtracting $A|z|^2$ gives a quasi-plurisubharmonic function $\varphi _ B$ on $B$ that agrees almost everywhere with $\varphi $. If $B'$ is a second such ball, then $\varphi _ B$ and $\varphi _{B'}$ agree almost everywhere on $B\cap B'$, and both are quasi-plurisubharmonic, so they agree everywhere on $B\cap B'$ by (1). Since such balls cover $X$, the functions $\varphi _ B$ define a quasi-plurisubharmonic function on $X$ that agrees almost everywhere with $\varphi $. $\square$

Proof. Choose a connected coordinate neighborhood $V$ of $x$ on which the $g_ i$ are defined and generate $\mathfrak a$, on which $w=v+f$ with $v$ plurisubharmonic and $f$ smooth and bounded, and on which $|w-c\log \sum _ i|g_ i|^2|\leq M$ for a constant $M$. Let $I$ be the set of indices $i$ for which the germ of $g_ i$ at $x$ is not zero. It is not empty, since these germs generate $\mathfrak a_ x\neq 0$, and for $i\notin I$ the function $g_ i$ vanishes identically on $V$, by the identity theorem. Since $0\leq \log \sum _{i\in I}|g_ i|^2-\max _{i\in I}\log |g_ i|^2\leq \log s$, the function $v$ differs from $c\max _{i\in I}\log |g_ i|^2$ by a bounded function on $V$. The latter function is plurisubharmonic by proposition 5.10.3, proposition 5.9.5, so proposition 6.7.1 and proposition 6.7.9 give

where the first equality is the definition of Lelong numbers of quasi-plurisubharmonic functions (corollary 6.9.3) and the last one is the description of $\operatorname {ord}_ x\mathfrak a$ by generators in section 6.9. $\square$

Proof. Multiplication by $q\in \mathbb Q_{>0}$ replaces $c$ by $qc$. Sums and finite maxima of quasi-plurisubharmonic functions are quasi-plurisubharmonic. Indeed, let $\varphi _ i=u_ i+\sigma _ i$ near a point, with $u_ i$ plurisubharmonic and $\sigma _ i$ smooth. On a small coordinate ball about the point, the smooth functions $\sigma _ i+N|z|^2$ have positive semidefinite complex Hessians for a large constant $N$, hence are plurisubharmonic by proposition 5.9.2(1). By proposition 5.9.5, $\varphi _1+\varphi _2$ is a plurisubharmonic function minus $2N|z|^2$, and $\max _ i\varphi _ i$ is a plurisubharmonic function minus $N|z|^2$. For a sum or a maximum of two functions with analytic singularities, choose presentations about a given point, with coefficients $c$ and $d$, and a connected coordinate neighborhood $V$ of the point contained in both domains; by the identity theorem, neither tuple vanishes identically on $V$. Since $c$ and $d$ are rational, we can choose $c_0\in \mathbb Q_{>0}$ such that $c/c_0$ and $d/c_0$ are positive integers. For an integer $k\geq 1$ the multinomial formula gives

again a sum of squared moduli of holomorphic functions, not all identically zero on $V$. We may therefore assume that both presentations have the coefficient $c_0$, say $c_0\log A+b$ and $c_0\log B+b'$ with $A=\sum _ i|g_ i|^2$ and $B=\sum _ l|g'_ l|^2$. The sum is $c_0\log (AB)+b+b'$, and $AB=\sum _{i,l}|g_ ig'_ l|^2$, where some product $g_ ig'_ l$ is not identically zero on the connected set $V$. For the maximum we use

which shows that $\max (c_0\log A+b,c_0\log B+b')$ differs from $c_0\log (A+B)$ by a bounded function.

For the pullback, let $f\colon Y\to X$ be holomorphic, and suppose that $\varphi \circ f$ is not identically $-\infty $ on any connected component of $Y$. Near each point of $Y$, write $\varphi \circ f=u\circ f+\sigma \circ f$ on a connected coordinate neighborhood $W$, where $\varphi =u+\sigma $ near the image point with $u$ plurisubharmonic and $\sigma $ smooth; by proposition 5.9.3, $u\circ f$ is plurisubharmonic or identically $-\infty $ on $W$. Let $S$ be the open set of points of $Y$ near which $\varphi \circ f$ is identically $-\infty $. If $y$ lies in the closure of $S$ and $W$ is as above for $y$, then $u\circ f=-\infty $ on the nonempty open set $W\cap S$; since plurisubharmonic functions are locally integrable, $u\circ f$ is identically $-\infty $ on $W$, so $y\in S$. Thus $S$ is open and closed, and by hypothesis it contains no component of $Y$; hence $S=\varnothing $, and $\varphi \circ f$ is quasi-plurisubharmonic. If $\varphi =c\log \sum _ i|g_ i|^2+b$ on $V$, then on every connected coordinate neighborhood $W\subset f^{-1}(V)$ we have $\varphi \circ f=c\log \sum _ i|g_ i\circ f|^2+b\circ f$, with $b\circ f$ bounded, and the functions $g_ i\circ f$ do not all vanish identically on $W$, because $S=\varnothing $.

Now let $\pi $ be a log resolution of $\mathfrak a$ as in the statement. If $g_1,\ldots ,g_ r$ generate $\mathfrak a$, their pullbacks generate $\mathcal O_ Y(-D)$, so locally $g_ i\circ \pi =s_ Dh_ i$ with holomorphic $h_ i$, and $s_ D=\sum _ ik_ i(g_ i\circ \pi )$ for holomorphic $k_ i$. Since $s_ D$ is not a zero divisor, $\sum _ ik_ ih_ i=1$, so the $h_ i$ have no common zero and $\log \sum _ i|h_ i|^2$ is smooth. Hence

which proves both assertions about the remainder. $\square$

Proof. (1) By proposition 5.2.6 there is a smooth partition of unity $(\chi _ a)$ subordinate to the cover $(V_ i)$; for each $a$ choose an index $k(a)$ with $\operatorname{Supp}\chi _ a\subset V_{k(a)}$, and put $\rho _ k=\sum _{k(a)=k}\chi _ a$. Since the family $(\chi _ a)$ is locally finite, each $\rho _ k$ is smooth, the family $(\rho _ k)$ is locally finite, $\sum _ k\rho _ k=1$, and $\operatorname{Supp}\rho _ k\subset V_ k$, because a locally finite union of closed sets is closed. For each $i$ put

where the $k$th term, defined on $V_ i\cap V_ k$, is extended by zero to the rest of $V_ i$. Each term is smooth on $V_ i$, since it is smooth on $V_ i\cap V_ k$ and vanishes on the open set $V_ i\setminus \operatorname{Supp}\rho _ k$, and these two open sets cover $V_ i$. By local finiteness, $|e_ i|^2$ is smooth on $V_ i$, and it is positive, because the $\rho _ k$ are nonnegative with sum one and every $g_{ik}$ is nowhere zero. On $V_ i\cap V_ j$, the cocycle relation $g_{ik}=g_{ij}g_{jk}$ holds on $V_ i\cap V_ j\cap V_ k$, hence wherever $\rho _ k\neq 0$, so

Thus these squared norms form a smooth quadratic Hermitian metric on $L$.

(2) On $V_ i\cap V_ j$ we have $\log |e_ i|^2-\log |e_ j|^2=\log |g_{ij}|^2$, which is pluriharmonic, since $g_{ij}$ is holomorphic and nowhere zero (section 5.4). Hence the forms $-\mathrm{dd}^{\mathrm{c}}\log |e_ i|^2$ on $V_ i$ agree on the overlaps and define a $(1,1)$-form $\eta $ on $X$. It is smooth, because each $\log |e_ i|^2$ is a smooth real function. Moreover $\eta =-\mathrm d(\mathrm d^{\mathrm c}\log |e_ i|^2)$ on $V_ i$, by (5.2.10), and $\mathrm d^{\mathrm c}$ maps real forms to real forms; hence $\eta $ is real, and it is closed because $\mathrm d^2=0$.

(3) The functions $|e_ i|_{h_0}^2\mathrm e^{-f}$ are smooth and positive on $V_ i$, and they satisfy the compatibility condition of definition 6.14.9 because the functions $|e_ i|_{h_0}^2$ do. The curvature of $h_0\mathrm e^{-f}$ is $-\mathrm{dd}^{\mathrm{c}}\log \bigl (|e_ i|_{h_0}^2\mathrm e^{-f}\bigr )=-\mathrm{dd}^{\mathrm{c}}\log |e_ i|_{h_0}^2+\mathrm{dd}^{\mathrm{c}}f=\eta _0+\mathrm{dd}^{\mathrm{c}}f$ on $V_ i$. Conversely, let $h_1$ be a smooth quadratic Hermitian metric on $L$. On $V_ i$ the function $|e_ i|_{h_1}^2/|e_ i|_{h_0}^2$ is smooth and positive, and on $V_ i\cap V_ j$ it equals $|e_ j|_{h_1}^2/|e_ j|_{h_0}^2$, since both numerator and denominator are multiplied by $|g_{ij}|^2$. Hence these quotients define a smooth positive function on $X$, which we write as $\mathrm e^{-f}$ with $f$ smooth and real; then $h_1=h_0\mathrm e^{-f}$. The function $f$ is unique, since $\mathrm e^{-f}$ must equal these quotients. $\square$

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 $Y$. $\square$