3.1 Smooth canonical Hermitian metric of direct image sheaves

Let X be a Kähler manifold of dimension $m+n$ , and let Y be a complex manifold of dimension m. We consider a proper holomorphic submersion $f:X\to Y$ . The relative canonical bundle $K_{X/Y}$ corresponding to the map f is

$$ \begin{align*} K_{X/Y}=K_X\otimes f^*K_Y^{-1}. \end{align*} $$

When restricted to a generic fiber $X_t$ of t, we get $K_{X/Y}|_{X_t}\cong K_{X_t}$ .

Let L be a holomorphic line bundle over X equipped with a smooth semi-positive Hermitian metric h, that is, $i\Theta _{L,h}\geq 0$ . In this subsection, we discuss the complex structure of the direct image sheaf $f_{*}(K_{X/Y}\otimes L)$ on Y and the smooth canonical Hermitian metric H of this sheaf induced by h (cf. [Reference Berndtsson1]). Fixed a point $t\in Y$ , any section $u\in H^0(X_t,K_{X_t}\otimes L|_{X_t})$ extends in the sense that there is a holomorphic section

$$ \begin{align*} U\in H^0(f^{-1}(\Omega),K_X\otimes L|_{f^{-1}(\Omega)})\cong H^0(\Omega,K_Y\otimes f_{*}(K_{X/Y}\otimes L)) \end{align*} $$

such that $U|_{X_t}=u\wedge dt$ for some neighborhood $\Omega $ of t from the Ohsawa–Takegoshi $L^2$ -extension theorem (cf. [Reference Ohsawa and Takegoshi22]) and Kähler-ness of X. Here, we abusively denote by $dt$ the inverse image of a local generator $dt_1\wedge \cdots \wedge dt_m$ of $K_Y$ . In [Reference Berndtsson1], it was claimed that the total space

$$ \begin{align*} F:=\bigcup_{t\in Y}H^0(X_t,K_{X_t}\otimes L|_{X_t}) \end{align*} $$

has a natural structure of holomorphic vector bundle of rank $r:=h^0(X_t,K_{X_t}\otimes L_{X_t})$ over Y and coincides with the direct image $f_{*}(K_{X/Y}\otimes L)$ . Therefore, the space of local smooth sections of $F|_\Omega $ are simply the sections of the bundle $K_{X/Y}\otimes L|_{f^{-1}(\Omega )}$ whose restriction to each fiber of f is holomorphic.

The vector bundle $F=f_{*}(K_{X/Y}\otimes L)$ admits a natural complex structure as follows. Let u be a local section of E, then u is holomorphic if

$$ \begin{align*} \overline{\partial}u\wedge dt=0. \end{align*} $$

This is equivalent to saying that the section $u\wedge dt$ of $K_X\otimes L$ is holomorphic.

Note that u is holomorphic, that is, $\overline {\partial }u\wedge dt=0$ , which means that $\overline {\partial }u$ can be written

$$ \begin{align*} \overline{\partial}u=\sum \eta^j\wedge dt_j, \end{align*} $$

with $\eta ^j$ smooth forms of bidegree $(n-1,1)$ . Here, the following relationship is known (see [Reference Berndtsson1]) between $\eta ^j$ and the Kodaira–Spencer map $\rho _t:T^{1,0}_{Y,t}\to H^{0,1}(X_t,T^{1,0}_{X_t})$ :

$$ \begin{align*} \eta^j=\theta_j \rfloor u, \end{align*} $$

on each fiber where the classes $\rho _t(\partial /\partial t_j)$ can be represented by Kodaira–Spencer forms $\theta _j$ , that is, $\{\theta _j\}\in \rho _t(\partial /\partial t_j)$ .

The smooth Hermitian metric h of L induces the smooth canonical Hermitian metric H of F as follows. Let $u,v$ be two local sections of F. We denote by $(u_t)$ the family of L-twisted holomorphic $(n,0)$ -forms on fibers $K_{X_t}$ induced by u. The restriction of $u_t$ to $X_t$ is unique and denoted simply as u. Then the canonical Hermitian metric H is defined by

$$ \begin{align*} (u,v)_H(t):=\int_{X_t}c_nu_t\wedge\overline{v}_te^{-\varphi}=\int_{X_t}c_nu\wedge\overline{v}e^{-\varphi}, \end{align*} $$

where $h=e^{-\varphi }$ on locally and $c_n=i^{n^2}$ . This metric is smooth by Ehresmann’s fibration theorem and compactness of each fiber. This inner product of H is a function of t, and it will be convenient to write this function as

$$ \begin{align*} (u,v)_H=f_{*}(c_nu\wedge\overline{v}e^{-\varphi}), \end{align*} $$

where u and v are forms on X that represent the sections. Here $f_{*}$ denotes the direct image of form defined by

$$ \begin{align*} \int_Yf_{*}(\alpha)\wedge\beta=\int_X\alpha\wedge f^*(\beta), \end{align*} $$

if $\alpha $ is a form on X and $\beta $ is a form on Y.

3.2 Berndtsson calculation and Nakano positivity

Let $(t_1,\ldots ,t_m)$ be a local coordinate whose center is fixed point $y\in Y$ . Let $u_j$ be an m-tuple of local holomorphic sections to F that satisfy $D^{\prime H}u_j=0$ at y, that is, $t=0$ . Represent the $u_j$ by smooth forms on X and put

$$ \begin{align*} \hat{u}:=\sum u_j\wedge\widehat{dt_j}, \end{align*} $$

then we get

$$ \begin{align*} T_u^H=c_Nf_{*}(\hat{u}\wedge\overline{\hat{u}}e^{-\varphi}), \end{align*} $$

where $N=n+m-1$ and $\widehat {dt_j}$ is the wedge product of all differentials $dt_k$ except $dt_j$ such that $dt_j\wedge \widehat {dt_j}=dt=dt_1\wedge \cdots \wedge dt_m$ .

Using the following proposition, Berndtsson computed $i\partial \overline {\partial } T^H_u$ at fixed points.

Proposition 3.1 (cf. [Reference Berndtsson1, Prop. 4.2]).

Let u be a section of F over an open set U containing the origin such that $\overline {\partial }u=0$ in U, that is, holomorphic, and $D^{\prime H}u=0$ at $t=0$ . Then u can be represented by a smooth $(n,0)$ -form, still denoted u such that

$$ \begin{align*} \overline{\partial}u=\sum\eta^k\wedge dt_k, \end{align*} $$

where $\eta ^k$ is $\mathit {primitive}$ on $X_0$ , that is, satisfies $\eta ^k\wedge \omega =0$ on $X_0$ , and furthermore

$$ \begin{align*} \partial^\varphi u\wedge\widehat{dt_j}=0, \end{align*} $$

at $t=0$ for all j. Here, $\partial ^\varphi \cdot =e^\varphi \partial (e^{-\varphi }\cdot )$ .

Let $u_j\in \mathcal {O}(F)$ such that $D^{\prime H}u_j=0$ at $t=0$ , then we have

$$ \begin{align*} i\partial\overline{\partial} T^H_u=-c_Nf_{*}(\hat{u}\wedge\overline{\hat{u}}\wedge i\partial\overline{\partial}\varphi e^{-\varphi}) -\Bigl(\int_{X_0}|\eta|^2e^{-\varphi}dV_z\Bigr)dV_t, \end{align*} $$

at $t=0$ , where $\overline {\partial }u_j=\sum \eta ^k_j\wedge dt_k$ and $\eta =\sum \eta ^j_j$ .

From this calculation and Proposition 2.1, the following theorem is obtained.

3.3 Calculation of $\widetilde {T}^H_u$ for the canonical Hermitian metric on $f_{*}(K_{X/Y}\otimes L)$

Represent the $u_j$ by smooth forms on X and put

$$ \begin{align*} \tilde{u}:=\sum \overline{u}_j\wedge\widehat{dt_j}, \end{align*} $$

then we have the equality

$$ \begin{align*} \widetilde{T}_u^H=(-1)^nc_Nf_{*}(\tilde{u}\wedge\overline{\tilde{u}}e^{-\varphi}), \end{align*} $$

where using $ic_N=(-1)^N(-1)^{nm}c_nc_m$ and $ic_m(-1)^m\widehat {dt_j}\wedge \widehat {d\overline {t}_k}=\widehat {dt_j\wedge d\overline {t}_k}$ .

In this subsection, we show the following proposition.

Proposition 3.3. Let $u_j\in \mathcal {O}(F)$ such that $D^{\prime H}u_j=0$ at $t=0$ , then we have that

$$ \begin{align*} i\partial\overline{\partial} \widetilde{T}^H_u =-c_Nf_{*}(\hat{v}\wedge\overline{\hat{v}}\wedge i\partial\overline{\partial}\varphi e^{-\varphi})+c_n\Bigl(\int_{X_0}\sum\eta^j_k\wedge\overline{\eta}^k_j e^{-\varphi}\Bigr)dV_t \end{align*} $$

at $t=0$ , where $u_j=U_jdz,\, \hat {v}=\sum \overline {U}_j\wedge dz \wedge \widehat {dt_j}$ and $\overline {\partial }u_j=\sum \eta ^k_j\wedge dt_k$ . Here $c_Nf_{*}(\hat {v}\wedge \overline {\hat {v}}\wedge i\partial \overline {\partial }\varphi e^{-\varphi })\geq 0$ if $\varphi $ is plurisubharmonic.

In particular, if $\eta ^j_k$ is primitive on $X_0$ , then we can write $\eta ^j_k=\sum \eta _{jkl}\widehat {dz_l}\wedge d\overline {z}_l$ and get

$$ \begin{align*} c_n\Bigl(\int_{X_0}\sum\eta^j_k\wedge\overline{\eta}^k_j e^{-\varphi}\Bigr)dV_t=-\Bigl(\int_{X_0}\sum\eta_{jkl}\overline{\eta}_{kjl} e^{-\varphi}dV_z\Bigr)dV_t. \end{align*} $$

Proof. By $\partial ^\varphi \cdot =e^\varphi \partial (e^{-\varphi }\cdot )$ , we get

$$ \begin{align*} (-1)^n\partial \widetilde{T}^H_u =c_Nf_{*}(\partial\tilde{u}\wedge\overline{\tilde{u}}e^{-\varphi})+(-1)^Nc_Nf_{*}(\tilde{u}\wedge\partial^\varphi\overline{\tilde{u}}e^{-\varphi}). \end{align*} $$

From the equation

$$ \begin{align*} \partial\tilde{u}=\sum \partial\overline{u}_j\wedge\widehat{dt_j}=\sum \overline{\overline{\partial}u_j}\wedge\widehat{dt_j}=\sum \overline{\eta}^l_j\wedge d\overline{t}_l\wedge\widehat{dt_j}, \end{align*} $$

the form

$$ \begin{align*} \partial\tilde{u}\wedge\overline{\tilde{u}}&=\sum \overline{\eta}^l_j\wedge d\overline{t}_l\wedge\widehat{dt_j}\wedge u_k\wedge\widehat{d\overline{t}_k}\\ &=(-1)^N\sum \overline{\eta}^l_j\wedge \widehat{dt_j}\wedge u_k\wedge d\overline{t}_l\wedge\widehat{d\overline{t}_k}\\ &=(-1)^N\sum \overline{\eta}^k_j\wedge \widehat{dt_j}\wedge u_k\wedge d\overline{t} \end{align*} $$

contains a factor $d\overline {t}$ . On the other hand, the push forward of an $(n+m,n+m-1)$ -form is of bidegree $(m,m-1)$ . Therefore, it follows that

$$ \begin{align*} f_{*}(\partial\tilde{u}\wedge\overline{\tilde{u}}e^{-\varphi})=0.\\[-36pt] \end{align*} $$

Thus, the following is derived:

$$ \begin{align*} (-1)^n\overline{\partial}\partial\widetilde{T}^H_u =(-1)^Nc_Nf_{*}(\overline{\partial}^\varphi\tilde{u}\wedge\partial^\varphi\overline{\tilde{u}}e^{-\varphi})+c_Nf_{*}(\tilde{u}\wedge\overline{\partial}\partial^\varphi\overline{\tilde{u}}e^{-\varphi}). \end{align*} $$

By the equation $\overline {\partial }\partial ^\varphi +\partial ^\varphi \overline {\partial }=\partial \overline {\partial }\varphi $ , we obtain

$$ \begin{align*} c_Nf_{*}(\tilde{u}\wedge\overline{\partial}\partial^\varphi\overline{\tilde{u}}e^{-\varphi}) =c_Nf_{*}(\tilde{u}\wedge\overline{\tilde{u}}\wedge\partial\overline{\partial}\varphi e^{-\varphi})-c_Nf_{*}(\tilde{u}\wedge\partial^\varphi\overline{\partial\tilde{u}}e^{-\varphi}), \end{align*} $$

and by the vanishing $f_{*}(\tilde {u}\wedge \overline {\partial \tilde {u}}e^{-\varphi })=0$ , we obtain

$$ \begin{align*} 0=\partial f_{*}(\tilde{u}\wedge\overline{\partial\tilde{u}}e^{-\varphi}) =f_{*}(\partial\tilde{u}\wedge\overline{\partial\tilde{u}}e^{-\varphi})+(-1)^Nf_{*}(\tilde{u}\wedge\partial^\varphi\overline{\partial\tilde{u}}e^{-\varphi}). \end{align*} $$

Hence, the following calculation result is obtained:

$$ \begin{align*} (-1)^n\overline{\partial}\partial\widetilde{T}^H_u &=(-1)^Nc_Nf_{*}(\overline{\partial}^\varphi\tilde{u}\wedge\partial^\varphi\overline{\tilde{u}}e^{-\varphi})\\ &\qquad+c_Nf_{*}(\tilde{u}\wedge\overline{\tilde{u}}\wedge\partial\overline{\partial}\varphi e^{-\varphi})+(-1)^Nc_Nf_{*}(\partial\tilde{u}\wedge\overline{\partial\tilde{u}}e^{-\varphi}). \end{align*} $$

Note that with the choice of representatives of our sections $u_j$ furnished by Proposition 3.1, we have that $\overline {\partial }^\varphi \tilde {u}\wedge \partial ^\varphi \overline {\tilde {u}}=0$ at $t=0$ . In fact, $\overline {\partial }^\varphi \tilde {u}=\sum \overline {\partial ^\varphi u_j}\wedge \widehat {dt_j}$ and

$$ \begin{align*} \overline{\partial}^\varphi\tilde{u}\wedge\partial^\varphi\overline{\tilde{u}}=\sum\overline{\partial^\varphi u_j}\wedge\widehat{dt_j}\wedge \partial^\varphi u_k\wedge\widehat{d\overline{t}_k}=0 \end{align*} $$

at $t=0$ , where $\partial ^\varphi u_j\wedge \widehat {dt_k}=0$ at $t=0$ for all k.

Lemma 3.4. We obtain that

$$ \begin{align*} (-1)^Nc_Nf_{*}(\partial\tilde{u}\wedge\overline{\partial\tilde{u}}e^{-\varphi})=ic_n(-1)^n\Bigl(\int_{X_0}\sum \eta^j_k\wedge\overline{\eta}^k_je^{-\varphi}\Bigr)dV_t \end{align*} $$

at $t=0$ . In particular, if $\eta ^j_k$ is primitive on $X_0$ , that is, $\eta ^j_k\wedge \omega =0$ on $X_0$ , then we can write $\eta ^j_k=\sum \eta _{jkl}\widehat {dz_l}\wedge d\overline {z}_l$ and this integral value is

$$ \begin{align*} -i(-1)^n\Bigl(\int_{X_0}\sum \eta_{jkl}\overline{\eta}_{kjl}e^{-\varphi}dV_z\Bigr) dV_t. \end{align*} $$

Proof. Here $\partial \tilde {u}=\sum \overline {\eta }^l_j\wedge d\overline {t}_l\wedge \widehat {dt_j}$ , then

$$ \begin{align*} \partial\tilde{u}\wedge\overline{\partial\tilde{u}} =-(-1)^{nm}(-1)^{n^2}\sum \eta^j_k\wedge \overline{\eta}^k_j\wedge dt\wedge d\overline{t}. \end{align*} $$

Therefore, we get

$$ \begin{align*} (-1)^Nc_N\partial\tilde{u}\wedge\overline{\partial\tilde{u}}&=-(-1)^Nc_N(-1)^{nm}(-1)^{n^2}\sum \eta^j_k\wedge \overline{\eta}^k_j\wedge dt\wedge d\overline{t}\\ &=ic_nc_m(-1)^n\sum \eta^j_k\wedge \overline{\eta}^k_j\wedge dt\wedge d\overline{t}\\ &=ic_n(-1)^n\sum \eta^j_k\wedge \overline{\eta}^k_j\wedge dV_t, \end{align*} $$

where $ic_N=(-1)^N(-1)^{nm}c_nc_m$ .

If $\eta ^j_k$ is primitive, we can write $\eta ^j_k=\sum \eta _{jkl}\widehat {dz_l}\wedge d\overline {z}_l$ . Then we have

$$ \begin{align*} \eta^j_k\wedge \overline{\eta}^k_j&=\sum \eta_{jkl}\widehat{dz_l}\wedge d\overline{z}_l \wedge \overline{\eta}_{kj\mu}\widehat{d\overline{z}_\mu}\wedge dz_\mu\\ &=\sum \eta_{jkl}\overline{\eta}_{kjl}\widehat{dz_l}\wedge d\overline{z}_l \wedge \widehat{d\overline{z}_l}\wedge dz_l\\ &=(-1)^{2n-1}\sum \eta_{jkl}\overline{\eta}_{kjl} dz_l\wedge\widehat{dz_l}\wedge d\overline{z}_l \wedge \widehat{d\overline{z}_l}\\ &=-\sum \eta_{jkl}\overline{\eta}_{kjl} dz\wedge d\overline{z}, \end{align*} $$

where $c_ndz\wedge d\overline {z}=dV_z$ .

Hence, we obtain the following calculation:

$$ \begin{align*} i\partial\overline{\partial}\widetilde{T}^H_u&=-(-1)^nc_Nf_{*}(\tilde{u}\wedge\overline{\tilde{u}}\wedge i\partial\overline{\partial}\varphi e^{-\varphi})+c_n\Bigl(\int_{X_0}\sum\eta^j_k\wedge\overline{\eta}^k_j e^{-\varphi}\Bigr)dV_t \end{align*} $$

at $t=0$ . Let $u_j=U_jdz$ and $\varphi _{jk}:=\partial _{t_j}\overline {\partial }_{t_k}\varphi $ . Here, if $\varphi $ is plurisubharmonic, then $c_N\hat {u}\wedge \overline {\hat {u}}\wedge i\partial \overline {\partial }\varphi =\sum \varphi _{jk}U_j\overline {U}_kdV_z \wedge dV_t\geq 0$ . By $\tilde {u}=\sum \overline {u}_j\wedge \widehat {dt_j}=\sum \overline {U}_j d\overline {z} \wedge \widehat {dt_j}$ and $d\overline {z}\wedge dz=(-1)^{n^2}dz\wedge d\overline {z}=(-1)^ndz\wedge d\overline {z}$ , we have that

$$ \begin{align*} c_N\tilde{u}\wedge\overline{\tilde{u}}\wedge i\partial\overline{\partial}\varphi=(-1)^n\sum \varphi_{jk}\overline{U}_j U_kdV_z \wedge dV_t =(-1)^nc_N\hat{v}\wedge\overline{\hat{v}}\wedge i\partial\overline{\partial}\varphi, \end{align*} $$

where $\hat {v}=\sum \overline {U}_j\wedge dz \wedge \widehat {dt_j}$ and that

$$ \begin{align*} (-1)^nc_Nf_{*}(\tilde{u}\wedge\overline{\tilde{u}}\wedge i\partial\overline{\partial}\varphi e^{-\varphi})&=c_Nf_{*}(\hat{v}\wedge\overline{\hat{v}}\wedge i\partial\overline{\partial}\varphi e^{-\varphi})\\ &=f_{*}(\sum \varphi_{jk}\overline{U}_j U_k e^{-\varphi}dV_z \wedge dV_t)\\ &=\Bigl(\int_{X_0}\sum \varphi_{jk}\overline{U}_j U_k e^{-\varphi}dV_z\Bigr)dV_t\\ &\geq0, \end{align*} $$

if $\varphi $ is plurisubharmonic.

3.4 Proof of Theorem 1.1 and projectivized bundles

Let V be a holomorphic vector bundle of finite rank r over a compact complex manifold Y. Let $\pi :\mathbb {P}(V)\to Y$ be a projectivized bundle whose fiber at $t\in Y$ is the projective space of lines in $V^*_t$ , that is, $\mathbb {P}(V^*_t)$ . For any point $t\in Y$ , we get $\pi ^{-1}(t)=\mathbb {P}(V^*_t)\cong \mathbb {P}^{r-1}$ , then $\mathbb {P}(V)$ is a holomorphically locally trivial fibration. This projectivized bundle carries the tautological line bundle $\mathcal {O}_{\mathbb {P}(V)}(1)$ over $\mathbb {P}(V)$ whose restriction to any fiber $\mathbb {P}(V^*_t)$ is identical to $\mathcal {O}_{\mathbb {P}^{r-1}}(1)$ .

We shall apply Proposition 3.3 to the line bundles $\mathcal {O}_{\mathbb {P}(V)}(k)\to \mathbb {P}(V)$ where $k\in \mathbb {Z}$ . Let $E(k)$ be the vector bundle whose fiber over a point $t\in Y$ is the space of global holomorphic sections of $K_{\mathbb {P}(V^*_t)}\otimes \mathcal {O}_{\mathbb {P}(V)}(k)$ , that is,

$$ \begin{align*} E(k)&:=\bigcup_{t\in Y}H^0(\mathbb{P}(V^*_t),K_{\mathbb{P}(V^*_t)}\otimes\mathcal{O}_{\mathbb{P}(V)}(k)|_{\mathbb{P}(V^*_t)})\\ &=\pi_{*}(K_{\mathbb{P}(V)/Y}\otimes\mathcal{O}_{\mathbb{P}(V)}(k)), \end{align*} $$

where $E(k)_t=H^0(\mathbb {P}(V^*_t),K_{\mathbb {P}(V^*_t)}\otimes \mathcal {O}_{\mathbb {P}(V)}(k)|_{\mathbb {P}(V^*_t)})\cong H^0(\mathbb {P}^{r-1},\mathcal {O}_{\mathbb {P}^{r-1}}(k-r))$ . If $k<r$ , then each fiber $E(k)_t$ is zero. Berndtsson asserted the following fact:

$$ \begin{align*} E(r+m)=S^m(V)\otimes\mathrm{det}\,V, \end{align*} $$

where $S^m(V)$ is the mth symmetric power of V, and showed the following theorem using Theorem 3.2.

Here, the vector bundle V is called ample in the sense of Hartshorne (see [Reference Hartshorne15]) if the tautological line bundle $\mathcal {O}_{\mathbb {P}(V)}(1)$ is ample. Replacing $\mathcal {O}_{\mathbb {P}(V)}(r+1)$ by $\mathcal {O}_{\mathbb {P}(V)}(r+m)$ , we also get that $S^m(V)\otimes \mathrm {det}\,V$ is Nakano (semi)-positivity for any $m\in \mathbb {N}$ .

It is a well-known Griffiths conjecture that an ample vector bundle is Griffiths-positive, that is, has a smooth Griffiths-positive Hermitian metric. From Demailly–Skoda’s theorem (see [Reference Demailly and Skoda10]) that if V is Griffiths (semi)-positive, then $V\otimes \mathrm {det}\,V$ is (dual) Nakano (semi)-positive, this theorem may be regarded as indirect evidence of Griffiths conjecture. After that, it was shown that $S^m(V)\otimes \mathrm {det}\,V$ has Nakano-positive metric and dual Nakano-positive metric (see [Reference Liu, Sun and Yang18, Cor. 4.12]). Griffiths conjecture is known when Y is a compact curve (cf. [Reference Umemura27]), and it was recently shown to hold under a certain condition for the $L^2$ metric (see [Reference Naumann21]). Since the Kodaira–Spencer forms vanish under certain condition in [Reference Naumann21], we obtain the following theorem for dual Nakano positivity of the canonical Hermitian metric which is a different metric in [Reference Liu, Sun and Yang18].

Theorem 3.6. Let V be an ample holomorphic vector bundle of rank r over a complex manifold Y. If the canonical isomorphism

$$ \begin{align*} K^{-1}_{\mathbb{P}(V)/Y}\cong \mathcal{O}_{\mathbb{P}(V)}(r)\otimes\pi^*\mathrm{det}\,V^* \end{align*} $$

becomes an isometry for an positive metric on $\mathcal {O}_{\mathbb {P}(V)}(1)$ and some Hermitian metric on $\mathrm {det}\,V$ , then for any $m\in \mathbb {N}$ and for a smooth (semi)-positive Hermitian metric h on $\mathcal {O}_{\mathbb {P}(V)}(r+m)$ , the smooth canonical Hermitian metric H induced by h on $S^m(V)\otimes \mathrm {det}\,V$ is dual Nakano (semi)-positive.

We prove this below. Let $(t_1,\ldots ,t_m)$ and $(z_1,\ldots ,z_n)$ be the local coordinates on Y and the fibers, respectively. By the ampleness of V, there is a smooth positive Hermitian metric $h_{O(1)}$ on $\mathcal {O}_{\mathbb {P}(V)}(1)$ . We write locally for the curvature of the positively curved metric

$$ \begin{align*} \omega_{\mathbb{P}(V)}:&=-i\partial\overline{\partial}\log h_{O(1)}\\ &=i\Bigl(g_{\alpha\overline{\beta}}dz_\alpha\wedge d\overline{z}_\beta +h^{O(1)}_{k\overline{\beta}}dt_k\wedge d\overline{z}_\beta +h^{O(1)}_{\alpha\overline{l}}dz_\alpha\wedge d\overline{z}_l +h^{O(1)}_{k\overline{l}}dz_k\wedge d\overline{z}_l\Bigr). \end{align*} $$

Thus, the Kähler forms on each fibers are given by $\omega _t:=i\sum g_{\alpha \overline {\beta }}dz_\alpha \wedge d\overline {z}_\beta $ and the induced metric on $K^{-1}_{\mathbb {P}(V)/Y}$ can be written as $\mathrm {det}\,(g_{\alpha \overline {\beta }})$ . Here, this positive metric $h_{O(1)}$ induces the above canonical isomorphism.

According to [Reference Schumacher25], we denote the horizontal lift of a tangent vector $\partial /\partial t_j$ on the base Y by $v_j$ . It is given by

$$ \begin{align*} v_j=\frac{\partial}{\partial t_j}+\sum a^\alpha_j\frac{\partial}{\partial z_\alpha} \quad \mathrm{and} \quad a^\alpha_j=-\sum g^{\overline{\beta}\alpha}h^{O(1)}_{j\overline{\beta}}. \end{align*} $$

For a fibration $\pi :\mathbb {P}(V)\to Y$ , we obtain the Kodaira–Springer forms by

$$ \begin{align*} \theta_j:=\overline{\partial}(v_j)|_{X_t}, \end{align*} $$

where $\theta _j\in \rho _t(\partial /\partial t_j)$ .

Since it is a projectivized bundle, we get $\mathscr {H}^{0,1}(\mathbb {P}(V^*_t),T^{1,0}_{\mathbb {P}(V^*_t)})\cong H^{0,1}(\mathbb {P}(V^*_t),T^{1,0}_{\mathbb {P}(V^*_t)})\cong H^{0,1}(\mathbb {P}^{r-1},T^{1,0}_{\mathbb {P}^{r-1}})=0$ . Then the value of Kodaira–Spencer map is zero. Here $\{\theta _j\}=\rho _t(\partial /\partial t_j)=0$ . By the forms $\theta _j$ is harmonic, $\theta _j$ is zero as differential forms.

Proof of Theorem 3.6

From the Kodaira–Spencer forms, $\theta _j$ are zero and the definition of the complex structure in $E(r+m)$ , for any local holomorphic section $u\in \mathcal {O}(E(r+m))_t$ , the restriction of

$$ \begin{align*} \overline{\partial}u=\sum \eta^j\wedge dt_j \end{align*} $$

to each fiber is zero. In fact, the smooth $(n-1,1)$ -forms $\eta ^j$ equal $\theta _j\rfloor u$ in each fiber. In particular, we get $\eta ^j=\theta _j\rfloor u=0$ in each fiber.

By Proposition 3.3, for any local holomorphic section $u_j\in \mathcal {O}(E(r+m))$ such that $D^{\prime H}u_j=0$ at $t=0$ , we have that

$$ \begin{align*} i\partial\overline{\partial}\widetilde{T}^H_u=-c_N\pi_{*}(\hat{v}\wedge\overline{\hat{v}}\wedge i\partial\overline{\partial}\varphi e^{-\varphi}) \end{align*} $$

at $t=0$ , where $u_j=U_jdz,\, \hat {v}=\sum \overline {U}_j\wedge dz \wedge \widehat {dt_j}$ , and $\varphi =-\log h$ on locally. Here $c_N\pi _{*}(\hat {v}\wedge \overline {\hat {v}}\wedge i\partial \overline {\partial }\varphi e^{-\varphi })\geq 0$ (resp. $\,>0$ ) if $\varphi $ is (strictly) plurisubharmonic.

Hence, this theorem follows from Proposition 2.4.

Similar to this proof, Theorem 1.1 can be shown from Proposition 3.3, since if Kodaira–Spencer forms $\theta _j$ can be taken to be zero, then $\eta ^j$ vanishes where $\theta _j\in \rho _t(\partial /\partial t_j)$ . Furthermore, the following corollary is obtained.

4.1 Singular Hermitian metric on vector bundle and positivity

For any holomorphic vector bundle E, we introduce the definition of singular Hermitian metrics h on E, its various notions of positivity, and the $L^2$ -subsheaf $\mathscr {E}(h)$ of $\mathcal {O}(E)$ analogous to the multiplier ideal sheaf.

Definition 4.2 (cf. [Reference de Cataldo5, Def. 2.3.1]).

Let h be a singular Hermitian metric on E. We define the $L^2$ -subsheaf $\mathscr {E}(h)$ of germs of local holomorphic sections of E by

$$ \begin{align*} \mathscr{E}(h)_x:=\{s_x\in\mathcal{O}(E)_x\mid|s_x|^2_h~ \mathrm{is ~locally ~integrable ~around~} x\}. \end{align*} $$

If E is a holomorphic line bundle, then we get $\mathscr {E}(h)=\mathcal {O}(E)\otimes \mathscr {I}(h)$ . Moreover, we define positivity and negativity such as Griffiths and dual Nakano.

For a singular Hermitian metric h on E, the following is already known (see [Reference Berndtsson and Păun2], [Reference Raufi24]): h being Griffiths semi-negative is equivalent to $T^h_{\xi u}$ being plurisubharmonic, that is, $i\partial \overline {\partial } T^h_{\xi u}\geq 0$ in the sense of currents, for any local section $u\in \mathcal {O}(E)$ and any $\xi =(\xi _1,\ldots ,\xi _n)\in \mathbb {C}^n$ satisfying $u_j=\xi _j u$ and written $\xi u=(u_1,\ldots ,u_n)$ .

For singular Hermitian metrics, we cannot always define the curvature currents with measure coefficients (see [Reference Raufi24]). However, the above definitions can be defined by not using the curvature currents. In general, the dual of a Nakano negative bundle is not Nakano-positive, then we cannot define Nakano semi-positivity as in the case of Griffiths, but this definition of dual Nakano semi-positivity is natural. The characterization of Nakano semi-positivity using $L^2$ -estimate by the following definition is already known by Deng–Ning–Wang–Zhou’s work (see [Reference Deng, Ning, Wang and Zhou11, Th. 1.1]).

Definition 4.6 (cf. [Reference Deng, Ning, Wang and Zhou11, Def. 1.1]).

Let X be a complex manifold of dimension n, and let U be an open subset of X with a Kähler metric $\omega $ on U which admits a positive Hermitian holomorphic line bundle. Let $(E,h)$ be a (singular) Hermitian vector bundle over X. We call $(E,h)$ satisfies the optimal $L^2$ -estimate on U if for any positive Hermitian holomorphic line bundle $(A,h_A)$ on U, for any $f\in \mathscr {D}^{n,1}(U,E\otimes A)$ satisfying $\overline {\partial }f=0$ on U and $\int _U\langle B^{-1}_{A,h_A}f,f\rangle _{h\otimes h_A,\omega }dV_\omega <+\infty $ , there is $u\in L^2_{n,0}(U,E\otimes A)$ satisfying $\overline {\partial }u=f$ on U and

$$ \begin{align*} \int_U|u|^2_{h\otimes h_A,\omega}dV_\omega\leq\int_U\langle B^{-1}_{A,h_A}f,f\rangle_{h\otimes h_A,\omega}dV_\omega, \end{align*} $$

where $B_{A,h_A}=[i\Theta _{A,h_A}\otimes \mathrm {id}_E,\Lambda _\omega ]$ and $\mathscr {D}$ denotes the space of $C^\infty $ sections with compact support, i.e. $\mathscr {D}=\mathcal {C}^\infty _c$ .

In other words, when $ h $ is a smooth metric, satisfying the optimal $ L^2 $ -estimate is equivalent to being Nakano semi-positive. Therefore, since the above optimal $ L^2 $ -estimate does not depend on curvature $ \Theta _{E,h} $ , this definition itself can be extended to singular Hermitian metrics, allowing us to define singular Nakano semi-positivity. In the next subsection, we will define singular semi-positivity for torsion-free coherent sheaves.

It is already known that multiplier ideal sheaves are coherent in [Reference Nadel20]. After that, Hosono and Inayama proved that the $L^2$ -subsheaf $\mathscr {E}(h)$ is coherent if h is Nakano semi-positive in the singular sense as in Definition 4.6 in [Reference Hosono and Inayama16], [Reference Inayama17].

4.2 Singular Hermitian metrics on torsion-free sheaves and positivity

Let X be a complex manifold, and let $\mathscr {F}$ be a torsion-free coherent sheaf on X. Let $X(\mathscr {F})\subseteq X$ denote the maximal open subset where $\mathscr {F}$ is locally free, then $Z_{\mathscr {F}}:=X\setminus X(\mathscr {F})$ is a closed analytic subset of codimension $\geq 2$ . If $\mathscr {F}\ne 0$ , then the restriction of $\mathscr {F}$ to the open subset $X(\mathscr {F})$ is a holomorphic vector bundle F of some rank $r\geq 1$ .

Using the natural inclusion $j:X(\mathscr {F})=X\setminus Z_{\mathscr {F}}\hookrightarrow X$ , we define a natural extension of the $L^2$ -subsheaf $\mathscr {E}(h)$ as follows. Here, $j_{*}\mathcal {O}_{X\setminus Z_{\mathscr {F}}}\cong \mathcal {O}_X$ is already known.

The definition of the minimal extension property for singular Hermitian metrics on torsion-free coherent sheaves is already known.

Based on this definition and Definition 4.6, we define Nakano positivity for singular Hermitian metrics on torsion-free coherent sheaves.

Here, the positivity of holomorphic line bundles $(A, h_A)$ in Definition 4.6 can be replaced by $e^{-\psi }$ using smooth strictly plurisubharmonic function $\psi $ in the local case.

Nakano semi-positivity for singular Hermitian metrics is defined by $ L^2 $ -estimates, so there is a drawback in that it cannot be derived from local positivity to global positivity. The Nakano semi-positivity of [Reference Inayama17, Def. 1.1] establishes the vanishing theorem, making it a stronger definition than Definition 4.10, as it is globally defined.

5.1 Canonical singular Hermitian metric on direct image sheaves

In this subsection, we define the canonical singular Hermitian metric on the direct image sheaf $\mathcal {E}:=f_{*}(K_{X/Y}\otimes L\otimes \mathscr {I}(h))$ in the same way as in [Reference Hacon, Popa and Schnell14].

Construct a Hermitian metric of $\mathcal {E}$ over a Zariski-open subset $Y\setminus \mathscr {Z}_{\mathcal {E}}$ where everything is nice, and then to extend it over the bad locus $\mathscr {Z}_{\mathcal {E}}$ . First, we choose a nowhere dense closed analytic subset $Z\subseteq \mathscr {Z}_{\mathcal {E}}$ with the following three properties:

1. (1) The morphism f is submersion over $Y\setminus \mathscr {Z}_{\mathcal {E}}$ .

2. (2) Both $\mathcal {E}$ and the quotient sheaf $f_{*}(K_{X/Y}\otimes L)/\mathcal {E}$ are locally free on $Y\setminus \mathscr {Z}_{\mathcal {E}}$ .

3. (3) On $Y\setminus \mathscr {Z}_{\mathcal {E}}$ , the locally free sheaf $f_{*}(K_{X/Y}\otimes L)$ has the base change property.

By the base change theorem, the third condition will hold as long as the coherent sheaves $R^if_{*}(K_{X/Y}\otimes L)$ are locally free on $Y\setminus \mathscr {Z}_{\mathcal {E}}$ . The restriction of $\mathcal {E}$ to the open subset $Y\setminus \mathscr {Z}_{\mathcal {E}}$ is a holomorphic vector bundle E of some rank $r\geq 1$ . The second and third conditions together guarantee that

$$ \begin{align*} E_t:=\mathcal{E}|_t\subseteq f_{*}(K_{X/Y}\otimes L)|_t=H^0(X_t,K_{X_t}\otimes L|_{X_t}) \end{align*} $$

whenever $t\in Y\setminus \mathscr {Z}_{\mathcal {E}}$ . Here, note that $Z_{\mathcal {E}}=X\setminus X(\mathcal {E})\subseteq \mathscr {Z}_{\mathcal {E}}$ .

Lemma 5.1 (cf. [Reference Hacon, Popa and Schnell14, Lem. 22.1]).

For any $t\in Y\setminus \mathscr {Z}_{\mathcal {E}}$ , we have inclusions

$$ \begin{align*} H^0(X_t,K_{X_t}\otimes L|_{X_t}\otimes\mathscr{I}(h|_{X_t}))\subseteq E_t\subseteq H^0(X_t,K_{X_t}\otimes L|_{X_t}). \end{align*} $$

Here, we can immediately see that the two subspaces

$$ \begin{align*} H^0(X_t,K_{X_t}\otimes L|_{X_t}\otimes\mathscr{I}(h|_{X_t}))\subseteq E_t \end{align*} $$

are equal for almost everywhere $t\in Y\setminus \mathscr {Z}_{\mathcal {E}}$ . But unless $\mathcal {E}=0$ , the two subspaces are different, for example, at points where $h|_{X_t}$ is identically equal to $+\infty $ .

On each $E_t$ with $t\in Y\setminus \mathscr {Z}_{\mathcal {E}}$ , we can define a singular Hermitian metric H as follows. For any element $\alpha \in E_t\subseteq H^0(X_t,K_{X_t}\otimes L|_{X_t})$ , we can integrate over the compact complex manifold $X_t$ and define the inner product of $\alpha $ with respect to H by

$$ \begin{align*} |\alpha|^2_H(t):=\int_{X_t}|\alpha|^2_h\in[0,+\infty]. \end{align*} $$

Clearly, $|\alpha |_H(t)<+\infty $ if and only if $\alpha \in H^0(X_t,K_{X_t}\otimes L|_{X_t}\otimes \mathscr {I}(h|_{X_t}))$ . By Ehresmann’s fibration theorem and Fubini’s theorem, the function $t\mapsto |s|_H(t)$ is measurable for any local holomorphic section s of E.

From the discussion in [Reference Hacon, Popa and Schnell14], the singular Hermitian metric H over $Y\setminus \mathscr {Z}_{\mathcal {E}}$ is well defined on the entire open set $Y(\mathcal {E})$ . Then we say that this extended metric H on E over $Y(\mathcal {E})$ is a canonical singular Hermitian metric of $\mathcal {E}$ .

Definition 5.2. We define the set $\Sigma _H$ on Y related to the unboundedness of H by

$$ \begin{align*} \Sigma_H:=\{t\in Y\mid \mathcal{E}_t\subsetneq H^0(X_t,K_{X_t}\otimes L|_{X_t})\}. \end{align*} $$

Here, for any $t\in Y\setminus \mathscr {Z}_{\mathcal {E}}$ , if $\mathscr {I}(h|_{X_t})=\mathcal {O}_{X_t}$ , then $t\notin \Sigma _H$ and $H(t)$ is bounded by $\int _{X_t}e^{-\varphi }<+\infty $ , where $h=e^{-\varphi }$ on local. Let $\Sigma _h:=\{t\in Y\mid \int _{X_t}e^{-\varphi }=+\infty \}$ be a set related to the unboundedness of h, then we have that $\Sigma _H\setminus \mathscr {Z}_{\mathcal {E}}\subseteq \Sigma _h\setminus \mathscr {Z}_{\mathcal {E}}$ .

In fact, from the analytical characterization of nef and big line bundles (see [Reference Demailly8, Chap. 6]), there exists a singular Hermitian metric h on L such that $\mathscr {I}(h)=\mathcal {O}_X$ and $i\Theta _{L,h}\geq \varepsilon \omega $ in the sense of currents for some $\varepsilon>0$ , where $\omega $ is a Kähler metric on X. Therefore, from Lemma 5.1 and $\mathcal {E}:=f_{*}(K_{X/Y}\otimes L\otimes \mathscr {I}(h))=f_{*}(K_{X/Y}\otimes L)$ , we have $\mathcal {E}|_t=E_t=H^0(X_t,K_{X_t}\otimes L|_{X_t})$ for any $t\in Y\setminus \mathscr {Z}_{\mathcal {E}}$ .

5.2 Locally $L^2$ -type Nakano (semi)-positivity of H

Recall that $f:X\to Y$ is a projective surjective morphism between two connected complex manifolds. For the canonical singular Hermitian metric H on the direct image sheaf $\mathcal {E}=f_{*}(K_{X/Y}\otimes L\otimes \mathscr {I}(h))$ induced by the singular Hermitian metric h which is pseudo-effective, the following theorem is known with respect to the positivity property.

In this subsection, we show that this metric H on $\mathcal {E}$ has locally Nakano (semi)-positivity. This proof is inspired by the proof of the smooth case using $L^2$ -estimates in [Reference Deng, Ning, Wang and Zhou11, Th. 1.6].

Proof. First, we prove that H is locally $L^2$ -type Nakano semi-positive, that is, for any $t\in Y\setminus \mathscr {Z}_{\mathcal {E}}$ , there exists a Stein open neighborhood $U\subset Y\setminus \mathscr {Z}_{\mathcal {E}}$ of t such that H has the optimal $L^2$ -estimate on U. Here, $E:=\mathcal {E}|_{Y\setminus Z_{\mathcal {E}}}$ is a holomorphic vector bundle, and f is proper submersion over U from the construction of $\mathscr {Z}_{\mathcal {E}}$ .

By projectivity of X, there exists an analytic subset D such that $S:=X\setminus D$ is Stein and that $L|_S$ is trivial. Let $\varphi :=-\log h|_S$ , then $\varphi $ is plurisubharmonic function on S and $h=e^{-\varphi }$ on S. By [Reference Fornaess and Narasimhan12, Th. 5.5], there exist a sequence of smooth plurisubharmonic functions $(\varphi _\nu )_{\nu \in \mathbb {N}}$ on S decreasing to $\varphi $ almost everywhere pointwise. Here, there is a smooth exhaustive strictly plurisubharmonic function $\Psi $ on S such that $\textrm{sup}_S \psi =+\infty $ .

Let $X_U=f^{-1}(U)$ , then $X_U\setminus D$ is also Stein by f is holomorphic. In fact, there is a strictly plurisubharmonic function $\varPhi $ on U which is exhaustive and smooth by Stein-ness of U. Thus, the function $\Psi +f^*\varPhi $ on $X_U\setminus D$ is strictly plurisubharmonic, smooth, and exhaustive. We take a local coordinate $(t_1,\ldots ,t_m,z_1,\ldots ,z_n)$ on $X_U$ near $f^{-1}(t)=X_t$ , where $t_1,\ldots ,t_m$ is the standard coordinate on $U\subset \mathbb {C}^m$ . Let $\widetilde {\omega }=i\sum ^m_{j=1}dt_j\wedge d\overline {t}_j+i\sum ^n_{j=1}dz_j\wedge d\overline {z}_j$ and $\omega =i\sum ^m_{j=1}dt_j\wedge d\overline {t}_j$ .

For any smooth strictly plurisubharmonic function $\psi $ on U and any section $g\in \mathscr {D}^{m,1}(U,E,he^{-\psi },\omega )$ satisfying $\overline {\partial }g=0$ on U and $\int _U\langle [i\partial \overline {\partial }\psi \otimes \mathrm {id}_E,\Lambda _\omega ]^{-1}g,g\rangle _{H,\omega }e^{-\psi }dV_\omega <+\infty $ , we can write $g(t)=\sum ^m_{j=1} g_j(t)d\overline {t}_j\wedge dt$ , with $g_j(t)\in E_t\subseteq H^0(X_t,K_{X_t}\otimes L|_{X_t})$ . We can identify g as a smooth compact supported $(n+m,1)$ -form $\widetilde {g}(t,z):=\sum ^m_{j=1}g_j(t,z)d\overline {t}_j\wedge dt$ on X, with $g_j(t,z)$ begin holomorphic section $K_{X_t}\otimes L|_{X_t}$ . We have the following observations:

• • $\overline {\partial }_zg_j(t,z)=0$ for any fixed $t\in U$ , since $g_j(t,z)$ are holomorphic sections of $K_{X_t}\otimes L|_{X_t}$ ,

• • $\overline {\partial }_tg_j=0$ , since g is a $\overline {\partial }$ -closed form on U.

Here, we obtain that

$$ \begin{align*} i\partial\overline{\partial} f^*\psi&=\sum^m_{j,k=1}\frac{\partial^2\psi}{\partial t_j\partial\overline{t}_k}dt_j\wedge d\overline{t}_k, \\ [i\partial\overline{\partial} f^*\psi,\Lambda_{\widetilde{\omega}}]\widetilde{g}&=\sum_{j,k}\frac{\partial^2\psi}{\partial t_j\partial\overline{t}_k}g_j(t,z)dt \wedge d\overline{t}_k,\\ [i\partial\overline{\partial} f^*\psi,\Lambda_{\widetilde{\omega}}]^{-1}\widetilde{g}&=\sum_{j,k}\psi^{jk}g_j(t,z)dt \wedge d\overline{t}_k,\\ \langle[i\partial\overline{\partial} f^*\psi,\Lambda_{\widetilde{\omega}}]^{-1}\widetilde{g},\widetilde{g}\rangle_{\widetilde{\omega}} dV_{\widetilde{\omega}}&=\sum_{j,k}\psi^{jk}c_ng_j\wedge\overline{g}_kc_mdt\wedge d\overline{t}. \end{align*} $$

at any $t\in U$ , where $(\psi ^{jk})=(\frac {\partial ^2\psi }{\partial t_j\partial \overline {t}_k})^{-1}$ . By Fubini’s theorem, we have

$$ \begin{align*} \int_{X_U\setminus D}\langle[i\partial\overline{\partial} f^*\psi,\Lambda_{\widetilde{\omega}}]^{-1}\widetilde{g},\widetilde{g}\rangle_{\widetilde{\omega}} e^{-\varphi_\nu-f^*\psi}dV_{\widetilde{\omega}} &=\int_{X_U\setminus D}\sum_{j,k}\psi^{jk}c_ng_j\wedge\overline{g}_ke^{-\varphi_\nu-f^*\psi}c_mdt\wedge d\overline{t}\\ &\leq\int_{X_U}\sum_{j,k}\psi^{jk}c_ng_j\wedge\overline{g}_ke^{-\varphi-f^*\psi}c_mdt\wedge d\overline{t}\\ &=\int_U(g_j,g_k)_H(t)\psi^{jk}e^{-\psi}c_mdt\wedge d\overline{t}\\ &=\int_U\langle[i\partial\overline{\partial}\psi\otimes\mathrm{id}_E,\Lambda_{\omega}]^{-1}g,g\rangle_{H,\omega}e^{-\psi}dV_{\omega}\\ &<+\infty, \end{align*} $$

where by $(\bullet ,\bullet )_H(t)$ , we mean that pointwise inner product with respect to H.

Note that, acting on $\bigwedge ^{n+m,1}T^*_X\otimes L|_{X_U\setminus D}=\bigwedge ^{n+m,1}T^*_X|_{X_U\setminus D}$ , we have

(*) $$ \begin{align} [i\partial\overline{\partial}\varphi_\nu+i\partial\overline{\partial} f^*\psi,\Lambda_{\widetilde{\omega}}]\geq[i\partial\overline{\partial} f^*\psi,\Lambda_{\widetilde{\omega}}]\geq0, \end{align} $$

over $X_U\setminus D$ for any $\nu \in \mathbb {N}$ .

From Hörmander’s $L^2$ -estimate, that is, Theorem 2.5, there exists a solution $\widetilde {v}_\nu \in L^2_{n+m,0}(X_U\setminus D,e^{-\varphi _\nu },\widetilde {\omega })$ such that $\overline {\partial }\widetilde {v}_\nu =\widetilde {g}$ on $X_U\setminus D$ and satisfies the following estimate:

$$ \begin{align*} \int_{X_U\setminus D}|\widetilde{v}_\nu|e^{-\varphi_\nu-f^*\psi}dV_{\widetilde{\omega}}&=\int_{X_U\setminus D}c_{n+m}\widetilde{v}_\nu\wedge\overline{\widetilde{v}}_\nu e^{-\varphi_\nu-f^*\psi}\\ &\leq\int_{X_U\setminus D}\langle[i\partial\overline{\partial}\varphi_\nu+i\partial\overline{\partial} f^*\psi,\Lambda_\omega]^{-1}\widetilde{g},\widetilde{g}\rangle_{\widetilde{\omega}}e^{-\varphi_\nu-f^*\psi}dV_{\widetilde{\omega}}\\ &\leq\int_{X_U\setminus D}\langle[i\partial\overline{\partial} f^*\psi,\Lambda_{\widetilde{\omega}}]^{-1}\widetilde{g},\widetilde{g}\rangle_{\widetilde{\omega}} e^{-\varphi_\nu-f^*\psi}dV_{\widetilde{\omega}}\\ &\leq\int_U\langle[i\partial\overline{\partial}\psi\otimes\mathrm{id}_E,\Lambda_{\omega}]^{-1}g,g\rangle_{H,\omega}e^{-\psi}dV_{\omega}<+\infty. \end{align*} $$

Letting $\widetilde {v}_\nu =0$ on D, then we have $\overline {\partial }\widetilde {v}_\nu =\widetilde {g}$ on $X_U$ by Lemma 5.6. Since $e^{-\varphi _\nu }$ increases and converges to $h=e^{-\varphi }$ on S as $ \nu $ tends to $ +\infty $ , the sequence $(\widetilde {v}_\nu )_{\nu _1\leq \nu \in \mathbb {N}}$ forms a bounded sequence in $L^2_{n+m,0}(X_U,e^{-\varphi _{\nu _1}},\widetilde {\omega })$ . Therefore, we can obtain a weakly convergence subsequence in $L^2_{n+m,0}(X_U,e^{-\varphi _{\nu _1}},\widetilde {\omega })$ . By using a diagonal argument, we get a subsequence $(\widetilde {v}_{\nu _k})_{k\in \mathbb {N}}$ of $(\widetilde {v}_\nu )_{\nu _1\leq \nu \in \mathbb {N}}$ converging weakly in $L^2_{n+m,0}(X_U,e^{-\varphi _{\nu _1}},\widetilde {\omega })$ for any $\nu _1$ , where $\widetilde {v}_{\nu _k}\in L^2_{n+m,0}(X_U,e^{-\varphi _{\nu _k}},\widetilde {\omega })\subset L^2_{n+m,0}(X_U,e^{-\varphi _{\nu _1}},\widetilde {\omega })$ .

We denote by $\widetilde {v}$ the weakly limit of $(\widetilde {v}_{\nu _k})_{k\in \mathbb {N}}$ , then $\widetilde {v}$ satisfies $\overline {\partial }\widetilde {v}=\widetilde {g}$ on $X_U$ and

$$ \begin{align*} \int_{X_U\setminus D}|\widetilde{v}|^2e^{-\varphi_{\nu_k}-f^*\psi}dV_\omega\leq\int_U\langle[i\partial\overline{\partial}\psi,\Lambda_{\omega_0}]^{-1}g,g\rangle_{H,\omega_0}e^{-\varphi}dV_{\omega_0}<+\infty \end{align*} $$

for any $k\in \mathbb {N}$ . Taking weakly limit $k\to +\infty $ and using the monotone convergence theorem, we have the following $L^2$ -estimate:

$$ \begin{align*} \int_{X_U}|\widetilde{v}|^2_he^{-f^*\psi}dV_\omega&=\int_{X_U\setminus D}|\widetilde{v}|^2_he^{-f^*\psi}dV_\omega\\ &\leq\int_U\langle[i\partial\overline{\partial}\psi,\Lambda_{\omega_0}]^{-1}g,g\rangle_{H,\omega_0}e^{-\psi}dV_{\omega_0}<+\infty, \end{align*} $$

that is, $\widetilde {v}\in L^2_{n+m,0}(X_U,L,h,\omega )$ .

Here we write $\widetilde {v}(t,z)=\widetilde {V}(t,z)dz\wedge dt$ , then $\frac {\partial \widetilde {V}}{\partial \overline {z}_j}=0$ , that is, $\overline {\partial }\widetilde {v}|_{X_t}=0$ for any fixed $t\in U$ , since $\overline {\partial }\widetilde {v}=\widetilde {g}$ on $X_U$ . This means that $\widetilde {V}(t,\cdot )dz\in H^0(X_t,\omega _{X_t}\otimes L|_{X_t})$ . We can identify $\widetilde {v}$ as an $(m,0)$ -form $v(t):=V(t)dt$ on U, with $V(t)=\widetilde {V}(t,\cdot )dz\in H^0(X_t,\omega _{X_t}\otimes L|_{X_t})$ .

Fubini’s theorem implies the following:

$$ \begin{align*} \int_{X_U}|\widetilde{v}|^2_he^{-f^*\psi}dV_\omega=\int_{X_U}c_{n+m}\widetilde{v}\wedge\overline{\widetilde{v}}e^{-\varphi-f^*\psi}=\int_U||v||^2_He^{-\psi}dV_{\omega_0}. \end{align*} $$

Therefore, we obtain that

$$ \begin{align*} \int_U||v||^2_{H,\omega_0}e^{-\psi}dV_{\omega_0}\leq\int_U\langle[i\partial\overline{\partial}\psi,\Lambda_{\omega_0}]^{-1}g,g\rangle_{H,\omega_0}e^{-\psi}dV_{\omega_0}<+\infty. \end{align*} $$

Here, by the boundedness of the integral of $||v||^2_H$ , for any almost everywhere $t\in U$ , we have that $||v||^2_H(t)<+\infty $ , that is, $V(t)\in H^0(X_t,K_{X_t}\otimes L|_{X_t}\otimes \mathscr {I}(h|_{X_t}))\subseteq \mathcal {E}_t$ .

Form the assumption $\Sigma _H\subseteq A$ , replacing $v=0$ , that is, $V=0$ , on A then for any $t\in U$ we get $V(t)\in H^0(X_t,K_{X_t}\otimes L|_{X_t})=\mathcal {E}_t$ . By the Lebesgue measure of A is zero, this means that $v\in L^2_{m,0}(U,E,H,\omega _0)$ and $\overline {\partial }v=g$ on $U\setminus A$ . From Lemma 5.6, we get $\overline {\partial }v=g$ on U. Hence, we showed that H satisfies the optimal $L^2$ -estimate on U.

Finally, we prove that H is full locally $L^2$ -type Nakano semi-positive on $Y(\mathcal {E})$ . By $Z_{\mathcal {E}}:=Y\setminus Y(\mathcal {E})$ and $Z_{\mathcal {E}}\subseteq \mathscr {Z}_{\mathcal {E}}$ , there exists an analytic subset B such that $\mathscr {Z}_{\mathcal {E}}=Z_{\mathcal {E}}\bigcup B$ . Therefore, it is sufficient to show that for any $t\in B\setminus Z_{\mathcal {E}}$ , there exists a open neighborhood $U\subset Y(\mathcal {E})$ of t such that H has the optimal $L^2$ -estimate on U. This can be shown in the same way as above by using Lemma 5.6.

By using Lemma 5.7 and Demailly’s approximation theorem (see [Reference Demailly7]), the following can be shown similarly as above. Here, we do not use Demailly’s approximation theorem in the proof of Theorem 5.5 because the left term of $(\ast )$ is not necessarily semi-positive and Hörmander’s $L^2$ -estimate cannot be used.

Here, the $L^2$ -subsheaf $\mathscr {E}(H)$ of H is a subsheaf of $E=\mathcal {E}|_{Y(\mathcal {E})}$ over $Y(\mathcal {E})$ . For the natural inclusion $j:Y\setminus Z_{\mathcal {E}}=Y(\mathcal {E})\hookrightarrow Y$ , the natural extended $L^2$ -subsheaf with respect to H over Y is defined by $\mathscr {E}_Y(H):=j_{*}\mathscr {E}(H)$ as in Definition 4.8.

Here, when L is nef and big, the following is known from [Reference Zou and Watanabe30].

Remark 5.13 (cf. [Reference Zou and Watanabe30, Th. 1.3 and Cor. 1.4]).

Let $f:X\to Y$ be a smooth fibration of smooth projective varieties with connected fibers. If a holomorphic line bundle L on X is nef and big, then the holomorphic vector bundle $f_{*}(K_{X/Y}\otimes L)$ is also nef and Viehweg-big and has a canonical singular Hermitian metric H induced by a nef and big singular metric h as in Example 5.3 and there exists a proper analytic subset Z such that H is smooth and Nakano positive on $Y\setminus Z$ .

Moreover, we have the following cohomology vanishing

$$ \begin{align*} H^q(Y,f_{*}(K_X\otimes L))=0 \end{align*} $$

for any integers $q\leq 1$ . Here, we have $\Sigma _H=\emptyset $ by Example 5.3.

6.1 Exact sequences of torsion-free coherent sheaves and positivity

Consider the inheritance of positivity in exact sequences. The following is already known for the minimal extension property.

Proposition 6.2 (cf. [Reference Lombardi and Schnell19, Props. 6 and 7]).

Let

$$ \begin{align*} 0\longrightarrow \mathscr{S} \stackrel{j}{\hookrightarrow} \mathscr{F} \stackrel{g}{\twoheadrightarrow} \mathscr{Q} \longrightarrow 0 \end{align*} $$

be an exact sequence of torsion-free coherent sheaves. Then we have the following.

1. (a) Let h be a singular Hermitian metric on $\mathscr {S}$ which satisfies the minimal extension property. If j is generically an isomorphism, then h extends to a singular Hermitian metric $h_{\mathscr {F}}$ on $\mathscr {F}$ satisfying the minimal extension property,

2. (b) If $\mathscr {F}$ has a singular Hermitian metric satisfying the minimal extension property, then the induced metric $h_{\mathscr {Q}}$ has also the minimal extension property.

For Griffiths and Nakano positivity of smooth metrics, the following is known.

Here, for the inheritance of semi-positivity from E to Q, Nakano semi-positivity has a counterexample (see Proposition 6.7), but by rephrasing condition $(c)$ , we find the following with respect to dual Nakano positivity.

We consider the positivity of singular Hermitian metrics. For Griffiths positivity, [Reference Hacon, Popa and Schnell14, Prop. 19.3] is already known, and we obtain the following proposition for (dual) Nakano positivity.

In particular, Proposition 6.3 and Corollary 6.4 follow from this proposition.

Proof. $(a)$ We define the natural singular Hermitian metric $h_S$ of S induced from h by $|u|_{h_S}:=|ju|_h$ for any section u of S. By the assumption, for any local holomorphic section $s_j\in \mathcal {O}(E)$ , the $(n-1,n-1)$ -form $T^h_u=\sum (s_j,s_k)_h\widehat {dz_j\wedge d\overline {z}_k}$ is plurisubharmonic, that is, $i\partial \overline {\partial } T^h_u\geq 0$ . For any local holomorphic section $u_k\in \mathcal {O}(S)$ , image $ju_k$ is also a local holomorphic section of E, that is, $ju_k\in \mathcal {O}(E)$ . Then, from the equality

$$ \begin{align*} T^{h_S}_u=\sum(u_j,u_k)_{h_S}\widehat{dz_j\wedge d\overline{z}_k}=\sum(ju_j,ju_k)_h\widehat{dz_j\wedge d\overline{z}_k}=T^h_{ju}, \end{align*} $$

we have that $T^{h_S}_u$ is also plurisubharmonic, that is, $h_S$ is Nakano semi-negative.

$(b)$ Here, the sequence $0\to Q^*\to E^*\to S^*\to 0$ is also exact. Similarly to the proof of $(a)$ , $Q^*$ has a Nakano semi-negative singular Hermitian metric.

6.2 A concrete example

We consider the following exact sequence of holomorphic vector bundles:

$$ \begin{align*} 0\longrightarrow \mathcal{O}_{\mathbb{P}^n}(-1) \stackrel{j}{\hookrightarrow} \underline{V}:=\mathbb{P}^n\times\mathbb{C}^{n+1} \stackrel{g}{\twoheadrightarrow} Q:=\underline{V}/\mathcal{O}_{\mathbb{P}^n}(-1) \longrightarrow 0. \end{align*} $$

From this sequence, we get $\mathrm {det}\,\underline {V}=\mathrm {det}\,Q\otimes \mathcal {O}_{\mathbb {P}^n}(-1)$ and get isomorphisms

$$ \begin{align*} \mathrm{det}\,Q\cong \mathcal{O}_{\mathbb{P}^n}(1), \quad T_{\mathbb{P}^n}=Q\otimes \mathcal{O}_{\mathbb{P}^n}(1)\cong Q\otimes \mathrm{det}\,Q, \end{align*} $$

where $\mathrm {det}\,\underline {V}$ is also trivial. By Griffiths semi-positivity of $\underline {V}$ and Corollary 6.4, the bundle Q is dual Nakano semi-positive and then Griffiths semi-positive. Therefore, $T_{\mathbb {P}^n}$ is Nakano semi-positive from Demailly–Skoda’s theorem (see [Reference Demailly and Skoda10]), and is Griffiths positive from $Q\geq _{Grif}0$ and $\mathrm {det}\,Q\cong \mathcal {O}_{\mathbb {P}^n}(1)>0$ . But the tangent bundle $T_{\mathbb {P}^n}$ has no smooth Nakano positive metric. In fact, if $T_{\mathbb {P}^n}>_{Nak}0$ , then $H^q(\mathbb {P}^n,K_{\mathbb {P}^n}\otimes T_{\mathbb {P}^n})=0$ for any $q\geq 1$ by the Nakano vanishing theorem. However, this contradicts the following:

$$ \begin{align*} H^{n-1}(\mathbb{P}^n,K_{\mathbb{P}^n}\otimes T_{\mathbb{P}^n})\cong H^1(\mathbb{P}^n,T^*_{\mathbb{P}^n})=H^{1,1}(\mathbb{P}^n,\mathbb{C})=\mathbb{C}. \end{align*} $$

Proof. First, if Q has a smooth Griffiths-positive Hermitian metric, then $T_{\mathbb {P}^n}\cong Q\otimes \mathrm {det}\,Q$ has a smooth Nakano-positive Hermitian metric by Demailly–Skoda’s theorem. Second, if Q has a smooth Nakano semi-positive Hermitian metric, then $T_{\mathbb {P}^n}\cong Q\otimes \mathrm {det}\,Q$ has a smooth Nakano-positive Hermitian metric by $\mathrm {det}\,Q\cong \mathcal {O}_{\mathbb {P}^n}(1)$ is positive line bundle. But these contradict that $T_{\mathbb {P}^n}$ is not Nakano-positive.

Finally, if Q has a singular Hermitian metric h which is globally Nakano semi-positive and satisfying $\nu (-\log \mathrm {det}\,h,x)<2$ for any point $x\in \mathbb {P}^n$ , then from the vanishing theorem (see [Reference Watanabe28, Th. 6.1]) for singular Nakano semi-positivity, we have

$$ \begin{align*} H^q(\mathbb{P}^n,K_{\mathbb{P}^n}\otimes\mathcal{O}_{\mathbb{P}^n}(1)\otimes\mathscr{E}(h))=0 \end{align*} $$

for $q\geq 1$ . By the fact that if $\nu (-\log \mathrm {det}\,h,x)<2$ , then $\mathscr {E}(h)=\mathcal {O}_{\mathbb {P}^n}(Q)$ (see the proof of [Reference Watanabe28, Th. 6.2]), we get

$$ \begin{align*} 0=H^q(\mathbb{P}^n,K_{\mathbb{P}^n}\otimes\mathcal{O}_{\mathbb{P}^n}(1)\otimes\mathscr{E}(h))\cong H^q(\mathbb{P}^n,K_{\mathbb{P}^n}\otimes \mathrm{det}\,Q\otimes Q) \cong H^q(\mathbb{P}^n,K_{\mathbb{P}^n}\otimes T_{\mathbb{P}^n}). \end{align*} $$

But this vanishing contradicts that $H^{n-1}(\mathbb {P}^n,K_{\mathbb {P}^n}\otimes T_{\mathbb {P}^n})\cong \mathbb {C}$ .

Finally, we ascertain by concrete calculations that the naturally induced smooth metric $h_Q$ of Q has indeed the minimal extension property. Here, this metric $h_Q$ induced from $I_{\underline {V}}$ and g defined by $|u|_{h_Q}:=|g^*u|_{I_V}$ for any section u of Q.

Let $a\in \mathbb {P}^n$ be fixed. Choose an orthonormal basis $(e_0,e_1,\ldots ,e_n)$ of $\mathbb {C}^{n+1}$ such that $a=[e_0]$ . Consider the natural embedding $\mathbb {C}^n \hookrightarrow \mathbb {P}^n:0\mapsto a$ which sends $z=(z_1,\ldots ,z_n)\mapsto [e_0+z_1e_1+\cdots +z_ne_n]$ . Then $\varepsilon (z)=e_0+z_1e_1+\cdots +z_ne_n$ defines a nonzero hol section of $\mathcal {O}_{\mathbb {P}^n}(-1)|_{\mathbb {C}^n}$ . The adjoint homomorphism $g^*:Q\to \underline {V}$ is $C^\infty $ and can be described as the orthogonal splitting of the above exact sequence. The images $(\tilde {e}_1,\ldots ,\tilde {e}_n)$ of $(e_1,\ldots ,e_n)$ in Q define a local holomorphic frame of $Q|_{\mathbb {C}^n}$ , and we already know that $gg^*=\mathrm {id}_{\underline {V}}$ and

$$ \begin{align*} g^*\cdot\tilde{e}_j=e_j-\frac{\langle e_j,\varepsilon\rangle}{|\varepsilon|^2}\varepsilon=e_j-\frac{\overline{z}_j}{1+|z|^2}\varepsilon=e_j-\zeta_j\varepsilon, \end{align*} $$

where put $\zeta _j=\frac {\overline {z}_j}{1+|z|^2}$ (see [Reference Demailly9, Chap. V]). By $gg^*=\mathrm {id}_{\underline {V}}$ and $\varepsilon \in \mathrm {ker}g$ , we get $\tilde {e}_j=gg^*\tilde {e}_j=g(e_j-\zeta _j\varepsilon )=ge_j$ . From these, the matrix representations of g and $g^*$ with respect to frames $(\tilde {e}_1,\ldots ,\tilde {e}_n)$ and $(e_1,\ldots ,e_n)$ are as follows.

$$ \begin{align*} g=\begin{pmatrix} -z_1 & & \\ \vdots & I_n & \\ -z_n & & \\ \end{pmatrix}, \quad g^*=\begin{pmatrix} 0 \\ I_n \\ \end{pmatrix} +G^*, \quad G^*= \begin{pmatrix} -\zeta_1 & \cdots & \cdots & -\zeta_n \\ -\zeta_1z_1 & & & -\zeta_nz_1 \\ \vdots & & & \vdots \\ -\zeta_1z_n & \cdots & \cdots & -\zeta_nz_n \\ \end{pmatrix}, \end{align*} $$

where we can write $G^*=(-\zeta _1\varepsilon ,\cdots ,-\zeta _n\varepsilon )$ . In this setting, we prove the following.

Proof. Let $I_V$ be a trivial Hermitian metric on $\underline {V}$ , then $I_V$ has the minimal extension property by the optimal Ohsawa–Takegoshi theorem. We define the natural smooth Hermitian metric $h_Q$ of Q induced from $I_V$ by $|u|_{h_Q}:=|g^*u|_{I_V}$ for any section u of Q. We show that $h_Q$ has the minimal extension property. By the minimal extension property of $I_V$ , for any $a\in \mathbb {P}^n$ and any $v\in Q_a$ with $|v|_{h_Q}=|g^*v|_{I_V}=1$ , there is a holomorphic section $s\in H^0(B,\underline {V})$ such that

$$ \begin{align*} s(0)=g^*v \quad \mathrm{and} \quad \frac{1}{\mathrm{Vol}(B)}\int_B|s|^2_{I_V}dV_B\leq1, \end{align*} $$

where $g^*v\in \underline {V}_a$ . From $gg^*=\mathrm {id}_{\underline {V}}$ , then the composition $gs$ is a holomorphic section, that is, $gs\in H^0(B,Q)$ , and $gs(0)=gg^*v=v$ . Hence, if $|gs|^2_{h_Q}=|g^*gs|^2_{I_V}\leq |s|^2_{I_V}$ on B, then $h_Q$ has the minimal extension property.

We can write $s=\sum ^n_{j=0}s_je_j=\sigma _0\varepsilon +\sum ^n_{j=0}\sigma _je_j\in H^0(B,\underline {V})$ , where $\sigma _0=s_0, \,\sigma _j=s_j-s_0z_j$ , and $s_j\in \mathcal {O}(B)$ . Then we have that $gs=\sum ^n_{j=1}\sigma _jge_j=\sum ^n_{j=1}\sigma _j\widetilde {e}_j$ and

where $\sum ^n_{j=1}\overline {z}_j\sigma _j=\sum ^n_{j=1}(s_j\overline {z}_j-s_0|z_j|^2)=\sum ^n_{j=1}s_j\overline {z}_j-s_0|z|^2$ . Therefore, if

$$ \begin{align*} \frac{1+2|z|^2}{1+|z|^2}\Bigl(\sum^n_{j=1}|s_j|^2+|s_0|^2|z|^2\Bigr)\leq|s|^2_{I_V}=\sum^n_{j=0}|s_j|^2, \end{align*} $$

that is, $|z|^2\sum ^n_{j=1}|s_j|^2\leq |s_0|^2(1-2|z|^4)$ , then we obtain $|g^*gs|^2_{I_V}\leq |s|^2_{I_V}$ .

Here, $s_j$ is expressed as a scalar multiple of $s_0$ for any j. In fact, by the optimal Ohsawa–Takegoshi extension theorem (see [Reference Błocki4], [Reference Guan and Zhou13]) for trivial line bundle, there is a holomorphic function $f\in \mathcal {O}(B)$ such that

$$ \begin{align*} f(0)=1 \quad \mathrm{and} \quad \frac{1}{\mathrm{Vol}(B)}\int_B|f|^2dV_B\leq1. \end{align*} $$

We write $g^*v=\sum ^n_{j=0}w_je_j\in \underline {V_a}$ where $1=|g^*v|^2_{I_V}=\sum ^n_{j=0}|w_j|^2$ and $w_j\in \mathbb {C}$ . By changing the subscript of the local trivial frame $(e_j)$ , $w_0\ne 0$ can be assumed. Therefore, we can take $s_j:=w_jf=\frac {w_j}{w_0}f\in \mathcal {O}(B)$ . Indeed, it is $s(0)=\sum ^n_{j=0}w_jf(0)e_j=\sum ^n_{j=0}w_je_j=g^*v$ and $|s|^2_{I_V}=(\sum ^n_{j=0}|w_j|^2)|f|^2=|f|^2$ .

Thus, the condition $|z|^2\sum ^n_{j=1}|s_j|^2\leq |s_0|^2(1-2|z|^4)$ is sufficient for $2|z|^2+|z|^2(1-1/|w_0|^2)-1\leq 0$ . Since $w_0$ is taken as one of the nonzeros in $\{w_0,\ldots ,w_n\}$ that satisfy $\sum ^n_{j=0}|w_j|^2=1$ , we get $|w_0|^2\geq \frac {1}{1+n}$ . Hence, if the radius of B is taken to be smaller than $(-n+\sqrt {n^2+8})/4>0$ , which is a solution of $2r^2+nr-1=0$ , then we have that $|gs|^2_{h_Q}=|g^*gs|^2_{I_V}\leq |s|^2_{I_V}$ on B for any solution s of the optimal Ohsawa–Takegoshi extension theorem for any $g^*v\in \underline {V}_a$ .