On the extension of holomorphic adjoint sections from reduced unions of strata of divisors

Abstract

In this paper we study the problem of extension of holomorphic sections of adjoint line bundles/vector bundles from reduced unions of strata of divisors. We prove a qualitative extension theorem for adjoint bundles. The main technical result is an extension theorem of the Ohsawa–Takegoshi type.

Appendices

Appendix A: Preliminaries for solving variants of the $\overline{\partial }$-equation

Theorem 2

((6.1) Theorem in [4]) Let (X, g) be an n-dimensional Kähler manifold admitting a complete Kähler metric (which is not necessarily g), (E, h) a holomorphic vector bundle on X with a smooth hermitian metric, and $\lambda$ and $\mu$ two strictly positive bounded smooth functions. If

$$\begin{aligned} R=R_{h,\lambda ,\mu }:=\sqrt{-1}\Bigg (\lambda \Theta _{h} - \bigg (\partial \overline{\partial } \lambda - \frac{1}{\mu }\partial {\lambda }\wedge \overline{\partial }{\lambda }\bigg )\otimes \mathrm {id}_E\Bigg )\geqslant _{\mathrm {Nak}} 0, \end{aligned}$$

then for any constant $C\geqslant 0$ and any $\eta \in L^2_{g,h}(X, \wedge ^{n,q}T^*X\otimes E)$ (with $q\geqslant 1$) such that $\overline{\partial }\eta =0$ and

$$\begin{aligned} |(\eta , v)_{L^2_{g,h}}|^2 \leqslant C \int _X R[v,v]_{g,h}\, dV_g\, \text { for every }v\in \mathcal D^{n,q}(X,E), \end{aligned}$$

(5.1)

there exist a locally Lebesgue integrable E-valued $(n,q-1)$-form $\gamma$ such that

$$\begin{aligned} \overline{\partial }\gamma = \eta \ \text { in the sense of current and } \int _X \frac{1}{\lambda +\mu }\,|\gamma |^2_{g,h}\,dV_g\leqslant C. \end{aligned}$$

We list two important auxiliary results due to Demailly.

Lemma 5.1

([5] Lemma 1.5) Given an analytic set Z and a relatively compact open set $\Omega$ in a Kähler manifold X, if $\Omega$ admits a complete Kähler metric, so does $\Omega \setminus Z$.

Lemma 5.2

([5] Lemma 6.9) Let $\Omega$ be an open set in $\mathbf{C}^n$ and Z an analytic subset of $\Omega$. For any $L^1_{\mathrm loc}$ (p, q)-form u and any $L^2_{\mathrm loc}$ $(p,q-1)$-form v on $\Omega$, if $\overline{\partial }u=v$ (as currents) on $\Omega \setminus Z$, then $\overline{\partial }u=v$ on $\Omega$.

We need the following variant of Lemma 4.6 of [11]:

Lemma 5.3

Let Z be an analytic subset of a hermitian manifold (M, g). Suppose that there are

• (i) open subsets $O_1\Subset O_2\Subset \cdots \Subset O_m\Subset \cdots$ of M such that $M\setminus Z=\bigcup \limits _m O_m$,

• (ii) a sequence $U_m$ of Lebesgue measurable sections of $K_M\otimes E$, E being a hermitian holomorphic vector bundle equipped with a smooth hermitian metric h, for every point $p\in M$ there exists an open neighborhood $V_p$ of p and an index $m_p$ such that $U_m|_{V_p}\, (m\geqslant m_p)$ are all holomorphic, and

• (iii) a sequence $w_m$ of positive Lebesgue measurable functions on M such that for every compact subset K of $M\setminus Z$ there exists an index $m_K$ such that the family of functions $w_m\, (m\geqslant m_K)$ are uniformly bounded away both from 0, and

If $w_m$ converges to a function w almost everywhere, and if $\liminf \nolimits _{m\rightarrow \infty } \int _{O_m} w_m\langle U_m\rangle ^2_h$ exists as a real number, then $U_m$ admits a subsequence which converges uniformly on every compact subset of M to a section $U\in \Gamma \big (M,\mathcal O_M(K_M\otimes E)\big )$ such that

$$\begin{aligned} \int _M w\langle U\rangle ^2_h \leqslant \liminf \nolimits _{m\rightarrow \infty } \int _{O_m} w_m\langle U_m\rangle ^2_h. \end{aligned}$$

Proof

We may assume that $\int _{O_m}w_m\langle U_m\rangle ^2_h$ actually converges as $m\rightarrow \infty$ by passing to subsequences, and that Z is a submanifold by considering $(M\setminus Z_{\mathrm {sing}},Z\setminus Z_{\mathrm {sing}})$ instead of (M, Z) and then extending the obtained section on $M\setminus Z_{\mathrm {sing}}$ to M by Riemann’s extension theorem. Besides, it suffices to show, as we will do in the next paragraph, that every point $p\in M$ admits an open neighborhood $N_p$ on which $U_m$ with m sufficiently large form a normal family, since an application of the diagonal method yields a subsequence $U_{m_k}$ which converges uniformly on every compact subset of M to a limit U, which is clearly a holomorphic section of $K_M\otimes E$ on M by (ii). Then for every j we have

$$ \int _{O_j} w\langle U\rangle ^2_h\leqslant \lim \limits _{k\rightarrow \infty }\int _{O_{m_k}} w_{m_k}\langle U_{m_k}\rangle ^2_h $$

by Fatou’s lemma, and hence

$$ \int _M w\langle U\rangle ^2_h=\int _{M\setminus Z} w\langle U\rangle ^2_h=\lim \limits _{j\rightarrow \infty }\int _{O_j} w\langle U\rangle ^2_h \leqslant \lim \limits _{m\rightarrow \infty }\int _{O_m} w_m\langle U_m\rangle ^2_h. $$

Therefore it remains to find the desired neighborhood $N_p$ for every $p\in M$.

Since the statement is purely local, we may assume that M is an open subset of $\mathbf{C}^n$. It suffices to find a neighborhood $N_p$ for every $p\in M$ such that $U_m$ for m sufficiently are uniformly bounded with respect to the euclidean $L^2$ norms on $N_p$. By (i), (ii), and (iii), for any $k\in \mathbf{N}$ there exists some $m_k\in \mathbf{N}$ such that

• $\inf \limits _{m\geqslant m_k}\inf \limits _{O_k}w_m>0$ and

• $U_m\, (m\geqslant m_k)$ are all holomorphic on $O_k$.

For $p\in M\setminus Z$, we may simply take $N_p$ to be any $O_k$ containing p. For $p\in Z$, we may further assume that

$$\begin{aligned} (M, Z,p)=\big (D_2(0)^n, \{0\}\times D_2(0)^{n-1}, (0,\dots ,0)\big ) \end{aligned}$$

where $D_R(0):=\{z\in \mathbf{C}\,|\, |z|<R\}\big )$, and that E is the trivial bundle. Without any control on $w_m$, we need to apply the following elementary fact (cf. Lemma 4.4 of [11]): for fixed $0<r<1$ and for every holomorphic function F on $D_2(0)^n$, we have

$$\begin{aligned} \int _{D_1(0)^n} |F|^2\, d\lambda \leqslant \frac{1}{1-r^2} \int _{\big (D_1(0)\setminus D_r(0)\big )\times D_1(0)^{n-1}} |F|^2\, d\lambda . \end{aligned}$$

For m sufficiently large $\big (D_1(0)\setminus D_r(0)\big )\times D_1(0)^{n-1}$ is contained by some $O_k$, and hence

$$\begin{aligned} \int _{\big (D_1(0)\setminus D_r(0)\big )\times D_1(0)^{n-1}} |U_m|^2\, d\lambda \end{aligned}$$

is bounded by a fixed multiple of $\int _{O_k}w_m\langle U_m\rangle _h$. This completes the proof. $\square$

The following lemma will be used to regularize singular metrics on line bundles over manifolds which admits projective morphisms to Stein spaces.

Lemma 5.4

Let X be an analytic subset of $\mathbf{C}^l\times \mathbf{P}^r$

and let $L_1,\dots ,L_s$ be holomorphic line bundles on X. Consider an open ball $B\subset \mathbf{C}^l$ and let $X_B=X\cap (B\times \mathbf{P}^r)$. For any analytic subset S of $X_B$ and any finite set $T\subset X_B$ with $S\cap T=\emptyset$, there exists a nonwhere dense analytic subset H of $X_B$ such that

• (i) $S\subseteq H$,

• (ii) $T\cap H=\emptyset$,

• (iii) $X_B\setminus H$ is Stein, and

• (iv) $L_j|_{X_B\setminus H}\, (j=1,\dots ,s)$ are all trivial.

Proof

Step 1. We first show that there is an integer $m_0>0$ such that, for every $t\in T$, there exists a holomorphic section $Q_t\in \Gamma \big (B\times \mathbf{P}^r, \mathcal O_{\mathbf{C}^l\times \mathbf{P}^r}(m_0)\big )$ with

$$\begin{aligned} Q_t|_{S\cup T\setminus \{t\}}=0\ \text { and }\ Q_t(t)\ne 0. \end{aligned}$$

Let be the map induced by the standard quotient map from $\mathbf{C}^{r+1}\setminus \{0\}$ to $\mathbf{P}^r$. For any $t\in T$, by a relative version of Chow’s theorem [12] (cf. [9] 4.3), there is an analytic cone⁹$K_t$ in $B\times \mathbf{C}^{r+1}$ over B such that

$$\begin{aligned} \pi ^{-1}(S\cup T\setminus \{t\})=K_t\cap \big (B\times (\mathbf{C}^{r+1}\setminus \{0\})\big ). \end{aligned}$$

Since $B\times \mathbf{C}^{r+1}$ is Stein and $t\notin S\cup T\setminus \{t\}$, there exists a holomorphic function $Q_t$ on $B\times \mathbf{C}^{r+1}$ such that $Q_t|_{K_t}=0$ and $Q_t$ is not identically zero on $\pi ^{-1}(t)$. We may write $Q_t$ in the form

$$\begin{aligned} Q_t(z_1,\dots ,z_l,w_0,\dots ,w_r)=\sum \limits _{\alpha =(\alpha _0,\dots ,\alpha _r)\in (\mathbf{N}\cup \{0\})^{r+1}}(Q_t)_{\alpha _0,\dots ,\alpha _r}(z_1,\dots ,z_l)w_0^{\alpha _0}\cdots w_r^{\alpha _r}, \end{aligned}$$

with uniquely determined $(Q_t)_{\alpha _0,\dots ,\alpha _r}\in \Gamma (B,\mathcal O)$. Let

$$\begin{aligned} Q_{t,m}(z_1,\dots ,z_l,w_0,\dots ,w_r)=\sum \limits _{\alpha _0+\cdots +\alpha _r=m}(Q_t)_{\alpha _0,\dots ,\alpha _r}(z_1,\dots ,z_l)w_0^{\alpha _0}\cdots w_r^{\alpha _r}. \end{aligned}$$

Since $K_t$ is a cone, $Q_{t,m}$ vanishes on $K_t$ for every m; there exists an integer $m_t>0$ such that $Q_{t,m_t}$ is nonvanishing along $\pi ^{-1}(t)$. Consequently, $Q_{t,m_t}$ determines a holomorphic section of the line bundle $\mathcal O_{\mathbf{C}^l\times \mathbf{P}^r}(m_t)|_{B\times \mathbf{P}^r}$ which vanishes along $S\cup T\setminus \{t\}$ and takes a nonzero value at the point t. It suffices to let

$$\begin{aligned} m_0=\prod \nolimits _{t\in T}m_t\ \text { and }\ Q_t=(Q_{t,m_t})^{\otimes \frac{m_0}{m_t}}\quad (t\in T). \end{aligned}$$

Step 2. Now we show that there exists a nowhere dense analytic subset H of $X_B$ satisfying the required conditions (i) - (iv). Suppose that $m_0$ and $Q_t\, (t\in T)$ are as in Step 1. We let $\mathcal F_j(m) := \mathcal O_X(L_j)\otimes \iota ^*\mathcal O_{\mathbf{C}^l\times \mathbf{P}^r}(m).$ By a standard result due to Grauert and Remmert [10] (cf. [2] Chapter IV, Theorem 2.1) there exists $m_1\in \mathbf{N}$ (depending on B) such that the natural morphisms

are surjective for every $m\geqslant m_1$. For every $t\in T$ we fix a section

$$\begin{aligned} f_{t,j}\in \Gamma \big (X_B,\mathcal F_j(m_0m_1)\big ) \end{aligned}$$

which generates $\mathcal F_j(m_0m_1)$ at t. Thus, the holomorphic sections

$$\begin{aligned} Q:=\Big (\sum \nolimits _{t\in T} Q_t\Big )^{\otimes {(m_1+1)}} \in \Gamma \big (X_B,\mathcal O_X(m_0m_1+m_0)\big ) \end{aligned}$$

and

$$\begin{aligned} f_j:=\sum \nolimits _{t\in T} f_{t,j}\otimes Q_t \in \Gamma \big (X_B,\mathcal F_j(m_0m_1+m_0)\big )\quad (j=1,\dots ,s) \end{aligned}$$

are all nonvanishing on T and all vanish along S. Finally we let

$$\begin{aligned} G:=Q\otimes f_1\otimes \cdots \otimes f_s \in \Gamma \big (X_B,\mathcal O_X(L_1\otimes \cdots \otimes L_s)(m_2)\big ) \end{aligned}$$

($m_2=(s+1)(m_0m_1+m_0)$) and let

$$\begin{aligned} H:=\text {the zero set of }G. \end{aligned}$$

Then H fulfils the statements (i) and (ii); (iv) holds since the meromorphic section $Q^{\otimes {(-1)}}\otimes f_j$ of $L_j$ is both holomorphic and nonvanishing on $X_B\setminus H$ for every j. It remains to verify (iii). Let h be the hermitian metric $q^*(h_0^{\otimes (-m_2)})$ on $\mathcal O_{\mathbf{C}^l\times \mathbf{P}^r}(m_2)$, where $h_0$ is the hermitian metric on $\mathcal O_{\mathbf{P}^r}(-1)$ induced by the eucliding metric via the canonical subbundle embedding

Then $\sqrt{-1}\Theta _{h}=m_2(q^*\omega _{FS})$, and hence $-\log |G|_{h}^2$ is a smooth psh function on $(B\times \mathbf{P}^r)\setminus H$, which is strongly psh along the direction of $\mathbf{P}^r$. On the other hand, fix a strongly psh exhaustion function $\phi$ on $B=B_R(z_0)$, e. g., take $\phi (z)=\frac{1}{R^2-|z-z_0|^2}$. It is direct to verify that

$$\exp (\phi \circ \mathrm {pr}_1)+|G|_h^{-2}$$

is a strongly psh exhaustion function on $(B\times \mathbf{P}^r)\setminus H$. By the solution to the Levi problem, $(B\times \mathbf{P}^r)\setminus H$, and hence its closed submanifold $X_B\setminus H$, is Stein. $\square$

Lemma 5.5

Let $X'$ be a Stein manifold and fix a sequence of relatively compact strongly pseudoconvex open sets $\Omega _n\, (n\in \mathbf{N})$ with $\Omega _n\nearrow X'$ as $n\nearrow \infty$. Given finitely many trivial holomorphic line bundles $(L_j, h_j)\, (j=1,\dots ,s)$ on $X'$ with hermitian metrics with $L_{\mathrm {loc}}^1$ weights, there exists a smooth metric $\widehat{h}_j^{(n)}$ on $L_j|_{\Omega _n}$ for every n such that

• (i) if $h_j$ is smooth on $X'$, then $\frac{1}{e}h_j\leqslant \widehat{h}_j^{(n)}\leqslant eh_j$ on $\Omega _n$ for every $n\in \mathbf{N}$,

• (ii) if $\sqrt{-1}\,\Theta _{h_j}\geqslant 0$, then $\sqrt{-1}\,\Theta _{\widehat{h}_j^{(n)}}\geqslant 0$ and $\widehat{h}_j^{(n)}\leqslant \widehat{h}_j^{(n+1)}\leqslant h_j$ on $L|_{\Omega _n}$ for every n,

• (iii) if $\sigma$ is a holomorphic section of L and $|\sigma |_h^2< e^A$ for some constant A, then $|\sigma |_{\widehat{h}^{(n)}}^2< e^A$ on $\Omega _n$ for every n, and

• (iv) for any trivial holomorphic line bundles $(L_j, h_j)\, (j=1,\dots ,s)$ on $X'$ with hermitian metrics with $L_{\mathrm {loc}}^1$ weights, if

  $$\begin{aligned} c_1\sqrt{-1}\,\Theta _{h_1}+\cdots + c_s\sqrt{-1}\,\Theta _{h_s}\geqslant 0 \end{aligned}$$

  as (1, 1)-currents, then for some constants $c_1,\dots ,c_s\in \mathbf{R}$, then

  $$\begin{aligned} c_1\sqrt{-1}\,\Theta _{\widehat{h}_1^{(n)}}+\cdots + c_s\sqrt{-1}\,\Theta _{\widehat{h}_s^{(n)}}\geqslant 0 \end{aligned}$$

  for every n.

Proof

We exploit the regularization argument used in [8]. We may simply assume $X'$ to be a Stein closed submanifold of $\mathbf{C}^N$. By [22], $X'$ admits an open Stein neighborhood U in $\mathbf{C}^N$ together with a holomorphic retract from W to $'$.¹⁰

Consider a general trivial line bundle L over $X'$, which can be viewed as the restriction of the trivial line bundle $U\times \mathbf{C}$ on U. A hermitian metric h on L is now expressed as a functions $e^{-\varphi }$ on $X'$, which admit the natural extension $e^{-\varphi \,\circ \, r}$ to W. Fix a smooth nonnegative radially symmetric function $\chi$ on $\mathbf{C}^N$ supported in the closed ball $\overline{B_1(0)}$ such that $\int _{\mathbf{C}^N}\chi =1$. For every $\varepsilon >0$ we define

$$\begin{aligned} \widehat{\varphi }_{\varepsilon }:=(\widetilde{\varphi \circ r})*\chi _\varepsilon , \end{aligned}$$

where $\widetilde{(\cdot )}$ means extension by the value 0 on $\mathbf{C}^N\setminus U$. For every $\varepsilon >0$, $\widehat{\varphi }_{\varepsilon }$ is a smooth function on

$$\begin{aligned} U_\varepsilon :=\left\{ p\in U\!\ \left| \!\ \mathrm {dist}(p,\mathbf{C}^N\setminus U)>\varepsilon \right. \right\} . \end{aligned}$$

We will choose a sequence $\varepsilon _n\searrow 0$ such that $\Omega _n\subseteq U_{\varepsilon _n}$ for all n. Then we let $\widehat{h}^{(n)}$ be the hermitian metric with weight $\widehat{\varphi }_{\varepsilon _n}|_{\Omega _n}$ on $L|_{\Omega _n}$ for every n.

(i) If $h_j$ is smooth on $X'$ with weight $\varphi _j$, then $\widehat{(\varphi _j)}_{\varepsilon }|_{\Omega _n}$ converges uniformly to $\varphi _j|_{\Omega _n}$ as $\varepsilon \searrow 0$. In particular, we may have chosen $\varepsilon _n\searrow 0$ so that

$$\begin{aligned} \varphi _j-1\leqslant \widehat{(\varphi _j)}_{\varepsilon _n}\leqslant \varphi _j+1 \ \text { on }\Omega _n. \end{aligned}$$

(ii) If $\sqrt{-1}\,\Theta _{h_j}\geqslant 0$, the weight $\varphi _j$ is equivalent to a psh function modulo a null function, and basic theory of psh functions implies that $\widehat{(\varphi _j)}_{\varepsilon _n}$ is a smooth psh function on $\Omega _n$ and $\widehat{(\varphi )}_{\varepsilon _n}\searrow \varphi \circ r$ as $n\rightarrow \infty$.

(iii) Let $\varphi$ be the weight of h. Then

$$\begin{aligned} \log |\sigma |^2 -\varphi \leqslant A. \end{aligned}$$

Since $\sigma$ is a holomorphic function on $X'$, $\log |\sigma |^2$ is psh, and hence we have on $U_{\varepsilon _n}$ that

$$\begin{aligned} \log |\sigma \circ r|^2 -(\widetilde{\varphi \circ r})*\chi _{\varepsilon _n} \leqslant (\widetilde{\log |\sigma \circ r|^2})*\chi _{\varepsilon _n} -(\widetilde{\varphi \circ r})*\chi _{\varepsilon _n} \leqslant A. \end{aligned}$$

Restricting the above to $\Omega _n$ and taking $\exp$ yield the desired inequality.

(iv) We let $\varphi _j$ be the weight function of the metric $h_j\, (j=1,\dots ,s)$. Then

$$\begin{aligned} \sqrt{-1}\,\partial \overline{\partial }(c_1\varphi _1+\cdots + c_s\varphi _s)=c_1\sqrt{-1}\,\Theta _{h_1}+\cdots + c_s\sqrt{-1}\,\Theta _{h_s}\geqslant 0 \end{aligned}$$

as (1, 1)-currents, and hence $\psi :=c_1\varphi _1+\cdots + c_s\varphi _s$ is equivalent to a psh function modulo a null function. In particular, $c_1\widehat{\varphi }_1^{(n)}+\cdots + c_s\widehat{\varphi }_s^{(n)}=\widehat{\psi }^{(n)}$ is a psh function on $\Omega _n$ for every n, and hence

$$\begin{aligned} c_1\sqrt{-1}\,\Theta _{\widehat{h}_1^{(n)}}+\cdots + c_s\sqrt{-1}\,\Theta _{\widehat{h}_s^{(n)}} = \sqrt{-1}\,\partial \overline{\partial }(c_1\widehat{\varphi }_1^{(n)}+\cdots + c_s\widehat{\varphi }_s^{(n)})\geqslant 0. \end{aligned}$$

$\square$

Lemma 5.6

Let $\upsilon _1,\dots ,\upsilon _q$ be strictly positive smooth functions on a complex manifold X. For an integer $1\leqslant l\leqslant q$ we have

$$\begin{aligned} \sqrt{-1}\,\partial \overline{\partial }\log \sum \limits _{i=1}^k \upsilon _i \geqslant \ \sum _{i=1}^k\frac{\upsilon _i}{\sum _\bullet \upsilon _\bullet }\, \sqrt{-1}\,\partial \overline{\partial }\log \upsilon _i. \end{aligned}$$

Proof

We let $\lambda _i:=\frac{\upsilon _i}{\sum _\cdot \upsilon _\cdot }$.

$$\begin{aligned} \begin{aligned}&\sqrt{-1}\,\partial \overline{\partial }\log \,\sum \limits _{i=1}^k \upsilon _i=\sqrt{-1}\,\left( \frac{\sum _i\partial \overline{\partial }\, \upsilon _i}{\sum _\cdot \upsilon \cdot } - \frac{\sum _r\partial \, \upsilon _r}{\sum _\cdot \upsilon \cdot }\wedge \frac{\sum _s\overline{\partial }\, \upsilon _s}{\sum _\cdot \upsilon \cdot }\right) \\&\quad = \sqrt{-1}\,\left( \sum _i\frac{\upsilon _i}{\sum _\cdot \upsilon \cdot }\frac{\partial \overline{\partial }\, \upsilon _i}{\upsilon _i} - \left( \sum _r\frac{\upsilon _r}{\sum _\cdot \upsilon \cdot }\frac{\partial \, \upsilon _r}{\upsilon _r}\right) \wedge \left( \sum _s\frac{\upsilon _s}{\sum _\cdot \upsilon \cdot }\frac{\overline{\partial }\, \upsilon _s}{\upsilon _s}\right) \right) \\&\quad = \sqrt{-1}\,\left( \sum _i\lambda _i\frac{\partial \overline{\partial }\, \upsilon _i}{\upsilon _i} - \left( \sum _r\lambda _r\frac{\partial \, \upsilon _r}{\upsilon _r}\right) \wedge \left( \sum _s\lambda _s\frac{\overline{\partial }\, \upsilon _s}{\upsilon _s}\right) \right) \\&\quad = \sum _i\lambda _i\sqrt{-1}\,\left( \frac{\partial \overline{\partial }\, \upsilon _i}{\upsilon _i}-\frac{\partial \, \upsilon _i}{\upsilon _i}\wedge \frac{\overline{\partial }\, \upsilon _i}{\upsilon _i}\right) + \sqrt{-1}\,\sum _{r,s}\big (\lambda _r\delta _{rs}-\lambda _r\lambda _s\big )\frac{\partial \, \upsilon _r}{\upsilon _r}\wedge \frac{\overline{\partial }\, \upsilon _s}{\upsilon _s}\\&\quad = \sum _i\lambda _i\, \sqrt{-1}\,\partial \overline{\partial }\log \upsilon _i + \sqrt{-1}\,\sum _{r,s}\big (\lambda _r\delta _{rs}-\lambda _r\lambda _s\big )\frac{\partial \, \upsilon _r}{\upsilon _r}\wedge \frac{\overline{\partial }\, \upsilon _s}{\upsilon _s}. \end{aligned} \end{aligned}$$

The second sum in the last line can be seen to be a semipositive (1, 1)-form. More precisely, for any $(a_1,\dots ,a_m)\in \mathbf{C}^k$ we have

$$\begin{aligned} \sum _{r,s}\big (\lambda _r\delta _{rs}-\lambda _r\lambda _s\big )a_r\overline{a}_s = \Big (\sum _i \lambda _i|a_i|^2\Big )\Big (\sum _i \lambda _i\Big )-\Big |\sum _r\lambda _r a_r\Big |^2\geqslant 0 \end{aligned}$$

by the Cauchy-Schwarz inequality. $\square$

Appendix B: An example illustrating Definition 1.1

A basic example of $S=S_1+S_2+S_3$ and $\mathcal W$, where $S_1$, $S_2$, and $S_3$ are exactly the coordinate planes in $X=\mathbf{C}^3$, respectively, and $\mathcal W=\{S_1\cap S_2,\, S_2\cap S_3, \, S_1\cap S_3\}$ consists of the three coordinate axes.

We give another example below. Let $X=\mathbf{C}^3$ and let $\mathcal S$ be given by taking $(S_j, h_j)\, (j=1,2,3,4,5)$ to be the trivial bundle with the standard euclidean metric, and by taking

$$\begin{aligned}&\sigma _1=x^3-y^2,\ \ \sigma _2=y-8,\ \ \sigma _3 = yz-1,\\&\sigma _4=(x^2-y^2)\big ((x-4)(y^2+y)(yz-1)+1\big )^2, \end{aligned}$$

and

$$\begin{aligned} \sigma _5= \begin{array}{l} \text {a polynomial which is nonvanishing on }V(\sigma _1\sigma _2\sigma _3\sigma _4)\\ \text {and such that }V(\sigma _5)\text { is smooth (scheme-theoretically).} \end{array} \end{aligned}$$

Then

$$\begin{aligned} \sigma ^{\{1,2,3\}}=\sigma _4\sigma _5,\, \sigma ^{\{2,3\}}=\sigma _1\sigma _4\sigma _5,\, \sigma ^{\{4\}}=\sigma _1\sigma _2\sigma _3\sigma _5,\, \sigma ^{\emptyset }=\sigma _1\sigma _2\sigma _3\sigma _4\sigma _5 \end{aligned}$$

(the symbol $\otimes$ being omitted for simplicity),

$$\begin{aligned} S_{\{1\}}= & {} V(x^3-y^2),\, S_{\{2\}}=V(y-8),\, S_{\{3\}}=V(yz-1),\,\\ S_{\{4\}}= & {} V(x^2-y^2)\cup V\big ((x-4)(y^2+y)(yz-1)+1\big ), \end{aligned}$$

and $S_{\{5\}}$ is a smooth surface disjoint from $S_{\{1\}}\cup S_{\{2\}}\cup S_{\{3\}}\cup S_{\{4\}}$. Note that $S_1=\mathrm {div}(\sigma _1)$ is nonsmooth and $S_4=\mathrm {div}(\sigma _4)$ is not irreducible nor reduced, and hence also nonsmooth. We have

$$\begin{aligned} S_{\{1,2\}}=S_{\{1\}}\cap S_{\{2\}}=\{(4,8),\,(4e^{\frac{2\pi }{3}},8),\, (4e^{\frac{4\pi }{3}},8)\}\times \mathbf{C},\, \end{aligned}$$

a union of three parallel complex lines. None of the two irreducible components

$$\begin{aligned} \{(4e^{\frac{2k\pi }{3}},8)\}\times \mathbf{C}\ (k=1,2) \end{aligned}$$

of $S_{\{1,2\}}$ is a $\mathcal S$-snc stratum since they both have nonempty intersection with the nonreduced part of $S_4$ defined by the square factor; the rest component $\{(4,8)\}\times \mathbf{C}$ of $S_{\{1,2\}}$ is a $\mathcal S$-snc stratum since it intersects $S_3$ transversally and is disjoint from $S_4$. Similarly, it is direct to see that the three lines

$$\begin{aligned} \{(0,0)\}\times \mathbf{C},\ \{(1,1)\}\times \mathbf{C},\ \text { and }\ \{(1,-1)\}\times \mathbf{C} \end{aligned}$$

are irreducible components of $S_{\{1,4\}}$. $\{(0,0)\}\times \mathbf{C}$ is not a $\mathcal S$-snc strtum since it lies in the singular locus of $S_1$; $\{(1,1)\}\times \mathbf{C}$ is not $\mathcal S$-snc strata since it has nonempty intersection with the nonreduced part of $S_4$; $\{(1,-1)\}\times \mathbf{C}$ is a $\mathcal S$-snc stratum since it intersects $S_3$ transversally and is disjoint from $S_2$ and the nonreduced part of $S_4$. Examples of $\mathcal S$-snc strata are given by

$$\begin{aligned} W_1= & {} S_{\{3\}},\ W_2=\{(4,8)\}\times \mathbf{C},\ W_3=\{(1,-1)\}\times \mathbf{C},\\ W_4= & {} W_1\cap W_2=\{(4,8,1/8)\}, \, W_5=S_{\{5\}},\, \text {etc.}, \end{aligned}$$

with $J_{W_1}=\{3\}$, $J_{W_2}=\{1,2\}$, $J_{W_3}=\{1,4\}$, $J_{W_4}=\{1,2,3\}$, and $J_{W_5}=\{5\}$. Examples of $\mathcal S$-snc family are

$$\begin{aligned} \{W_2\},\, \{W_1,W_2\},\, \{W_1,W_2,W_3\},\, \{W_1,W_3, W_5\},\, \{W_3,W_4\},\, \text {etc.} \end{aligned}$$

$\{W_1, W_4\}$ and $\{W_2, W_4\}$ are not $\mathcal S$-snc families due to the inclusions $W_4\subseteq W_1$ and $W_4\subseteq W_2$. Consider, for example, $\mathcal W=\{W_1,W_3,W_5\}$. Then

$$\begin{aligned} J(\mathcal W)=J_{W_1}\cup J_{W_3}\cup J_{W_5}=\{1,3,4,5\},\ \mathrm {J}_0(\mathcal W)=\{\{3\},\ \{1,4\},\ \{5\}\}, \end{aligned}$$

and

$$\begin{aligned} \underline{\mathcal W}= W_1\cup W_3\cup W_5= V(yz-1)\cup \big (\{(1,-1)\}\times \mathbf{C}\big )\cup V(\sigma _5), \end{aligned}$$

which is the union of two disjoint smooth surfaces and a line intersecting exactly one of them, transversally into a point. We have

$$\begin{aligned} \begin{aligned} S^{W_1}&=S^{J_{W_1}}|_{W_1}=S^{\{3\}}|_{W_1}=(S_1+S_2+S_4+S_5)|_{W_1} \\&=\mathrm {div}(x^3-y^2)|_{W_1} + \mathrm {div}(y-8)|_{W_1} + \big (\mathrm {div}(x-y)|_{W_1}+\mathrm {div}(x+y)|_{W_1}+0\big ) + 0,\\ S^{W_3}=&S^{J_{W_3}}|_{W_3}=(S_2+S_3+S_5)|_{W_3} = 0+ \mathrm {div}(-z-1)|_{W_3}+0, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} S^{W_5}=S^{J_{W_5}}|_{W_5}=(S_1+S_2+S_3+S_4)|_{W_5} =0. \end{aligned}$$

Therefore the derived family is

$$\begin{aligned} \mathcal W' = \{W_{11},\, W_{12},\, W_{13},\, W_{14}\}, \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} W_{11}&=S_{\{1\}}\cap W_1=V(x^3-y^2, yz-1),\\ W_{12}&=S_{\{2\}}\cap W_1=V(y-8, yz-1),\\ W_{13}&=V(x-y)\cap W_1=V(x-y, yz-1),\text { and }\\ W_{14}&=V(x+y)\cap W_1=V(x+y, yz-1) \end{aligned} \end{aligned}$$

with $J_{W_{11}}= \{1,3\}$, $J_{W_{12}}= \{2,3\}$, and $J_{W_{13}}= \{3, 4\}= J_{W_{14}}$. (According to the maximality condition in the definition of $\mathcal W'$, we omit $S_{\{3\}}\cap W_3=\{(1,-1,-1)\}$ since it is contained in $W_1\cap S_{\{1\}}=V(x^3-y^2, yz-1)$. Similarly, we have

$$\begin{aligned} \begin{aligned} S^{W_{11}}&=S^{J_{W_{11}}}|_{W_{11}}= S^{\{1,3\}}|_{W_{11}}=(S_2+S_4+S_5)|_{W_{11}}\\&= \big ([(4,8,1/8)] + [(4e^{\frac{2\pi }{3}},8,1/8)] + [(4e^{\frac{4\pi }{3}},8,1/8)]\big )\\&\quad + \big ( [(1,1,1)] + [(1,-1,-1)] \big ) + 0,\\ S^{W_{12}}&=S^{J_{W_{12}}}|_{W_{12}}= S^{\{2,3\}}|_{W_{12}}=(S_1+S_4+S_5)|_{W_{12}}\\&= \big ([(4,8,1/8)] + [(4e^{\frac{2\pi }{3}},8,1/8)] + [(4e^{\frac{4\pi }{3}},8,1/8)]\big )\\&\quad + \big ([(8,8,1/8)] + [(-8,8,1/8)] \big ) +0,\\ S^{W_{13}}&=S^{J_{W_{13}}}|_{W_{13}}= S^{\{3,4\}}|_{W_{13}}=(S_1+S_2+S_5)|_{W_{13}}\\&= [(1,1,1)] + [(8,8,1/8)] + 0, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} S^{W_{14}}&=S^{J_{W_{14}}}|_{W_{14}}= S^{\{3,4\}}|_{W_{14}}=(S_1+S_2+S_5)|_{W_{14}}\\&= [(1,-1,-1)] + [(-8,8,1/8)] +0. \end{aligned} \end{aligned}$$

Therefore,

$$\begin{aligned} \mathcal W^{(2)}= \left\{ \begin{array}{c} \{(4,8,1/8)\},\, \{(4e^{\frac{2\pi }{3}},8,1/8)\},\, \{(4e^{\frac{4\pi }{3}},8,1/8)\},\\ \{(1,1,1)\}, \{(-1,1,1)\},\\ \{(8,8,1/8)\},\, \{(-8,8,1/8)\} \end{array} \right\} \ \text { and }\ \mathcal W^{(3)}=\emptyset . \end{aligned}$$

Finally, we have

$$\begin{aligned} \mathcal W_{\mathrm {min}} = \{ W_5 \}\cup \mathcal W^{(2)}. \end{aligned}$$