Minimal extension property of direct images

Let $f:X\rightarrow Y$ be a projective surjective morphism from a complex space $X$¹¹1All complex space is assumed to be reduced and irreducible. to a complex manifold $Y$. Let $X^{o}\subset X$ be a dense Zariski open subset and $(E,h)$ be a holomorphic vector bundle endowed with a singular Hermitian metric (Definition 2.2). Let $S_{X}(E,h)$ be the sheaf on $X$ defined as follows. Let $U\subset X$ be an open subset. The space $S_{X}(E,h)(U)$ consists of holomorphic $E$-valued $(n,0)$-forms $\alpha$ on $U\cap X^{o}$ such that $\{\alpha,\alpha\}$ is locally integrable near every point of $U$. We define $S_{X/Y}(E,h)$ as $S_{X}(E,h)\otimes f^{\ast}(\omega_{Y}^{-1})$.

The first example is the case when the bundle $E$ do not degenerate. When $X=X^{o}$ ($X$ is smooth in particular), $E$ is a holomorphic line bundle with a singular Hermitian metric of semi-positive curvature, the aforementioned theorem implies the positivity result of the direct image sheaf $f_{\ast}(\omega_{X/Y}\otimes E\otimes\mathscr{I}(h))$, as proven by Paun-Takayama [PT2018] and Hacon-Popa-Schnell [HPS2018, Theorem 21.1]. If $E$ is of higher rank and $h$ is a metric satisfying conditions in Theorem 1.1, then using Caltaldo’s notation $E(h)$ (consisting of locally $L^{2}$ holomorphic sections in $E$), $f_{\ast}(\omega_{X/Y}\otimes E(h))$ has a canonical singular Hermitian metric which is Griffiths semi-positive and satisfies the minimal extension property.

Let $\mathbb{V}$ be a variation of Hodge structure on a regular Zariski open subset of a projective variety $X$. Kollár [Kollar1986_2] introduced a coherent sheaf $S(IC_{X}(\mathbb{V}))$ that generalizes the dualizing sheaf. He conjectured that $S(IC_{X}(\mathbb{V}))$ satisfies Kollár’s package (including the torsion-freeness, the injectivity theorem, Kollár’s vanishing theorem and the decomposition theorem). This conjecture was subsequently proven by Saito [MSaito1991] using the theory of mixed Hodge modules. Saito’s proof is based on the observation that $S(IC_{X}(\mathbb{V}))$ represents the highest index Hodge component of the intermediate extension $IC_{X}(\mathbb{V})$ as a Hodge module. In [SZ2022], the authors provide a new proof of Kollár’s conjecture using the $L^{2}$-method. This is based on their observation that $S(IC_{X}(\mathbb{V}))$ is isomorphic to some $S_{X}(E,h)$ for certain Hermitian bundle $(E,h)$ arising from the variation of Hodge structure (see below). The $S$-sheaf has played a crucial role in the application of Hodge module theory to complex algebraic geometry (see [Popa2018] for a comprehensive survey).

Let $X$ be a complex space and $X^{o}\subset X_{\rm reg}$ a Zariski open subset. Let $\mathbb{V}:=(\mathcal{V},\nabla,\mathcal{F}^{\bullet},Q)$ be a polarized complex variation of Hodge structure on $X^{o}$. To establish the Nadel vanishing theorem for $S(IC_{X}(\mathbb{V}))$, the authors of [SC2021] introduce a multiplier $S$-sheaf, denoted as $S(IC_{X}(\mathbb{V}),\varphi)$, a combination of the $S$-sheaf $S(IC_{X}(\mathbb{V}))$ and the multiplier ideal sheaf associated with a quasi-plurisubharmonic (quasi-psh) function $\varphi:X\to[-\infty,\infty)$. Let $h_{Q}$ be the Hodge metric defined as $(u,v)_{h_{Q}}:=Q(Cu,\overline{v})$, where $Q$ is the polarization of $\mathbb{V}$ and $C$ is the Weil operator. The multiplier $S$-sheaf is then defined as

where $S(\mathbb{V}):=\mathcal{F}^{\max\{k|\mathcal{F}^{k}\neq 0\}}$ is the top indexed nonzero piece of the Hodge filtration $\mathcal{F}^{\bullet}$. This type of sheaf possesses several good properties, such as the strong openness property and the Ohsawa-Takegoshi extension property. Additionally, $S(IC_{X}(\mathbb{V}),0)=S(IC_{X}(\mathbb{V}))$. For more details on the multiplier $S$-sheaf and its relation to $S(IC_{X}(\mathbb{V}))$ and $\mathscr{I}(\varphi)$, readers may refer to [SC2021].

Let $S_{X/Y}(IC_{X}(\mathbb{V}),\varphi):=S(IC_{X}(\mathbb{V}),\varphi)\otimes f^{\ast}(\omega_{Y}^{-1})$. As a consequence of Theorem 1.6, we obtain the following.

In particular, if $E$ is a holomorphic vector bundle on $X$ endowed with a smooth Hermitian metric with Nakano semi-positive curvature (i.e., $\varphi=0$ in Theorem 1.5), then $f_{\ast}\left(S_{X/Y}(IC_{X}(\mathbb{V}))\otimes E\right)$ has a singular Hermitian metric that is Griffiths semi-positive and satisfies the minimal extension property. This generalizes the result of Schnell-Yang [SY2023] to the relative case.

The concept of the multiplier $S$-sheaf, as shown in the previous example, can be extended to the framework of non-abelian Hodge theory. This extension of the multiplier $S$-sheaf is elaborated on in [SZ2022].

Let $X$ be a smooth, projective variety, and let $D$ be a reduced simple normal crossing divisor on $X$. Let’s consider a locally abelian parabolic Higgs bundle $(H,\{{{}_{E}}H\}_{E\in{\rm Div}_{D}(X)},\theta)$ on $(X,D)$. This bundle consists of the following data:

• •

  A locally abelian parabolic vector bundle $(H,\{{{}_{E}}H\}_{E\in{\rm Div}_{D}(X)})$ with parabolic structures on $D$. Here, the filtration $\{{{}_{E}}H\}$ is indexed by the set ${\rm Div}_{D}(X)$, which consists of $\mathbb{R}$-divisors whose support lies in $D$.

• •

  A Higgs field $\theta:H|_{X\backslash D}\to H|_{X\backslash D}\otimes\Omega_{X\backslash D}$, which has regular singularity along $D$.

This parabolic Higgs bundle is required to have vanishing parabolic Chern classes and to be polystable with respect to an ample line bundle $A$ on $X$.

The main focus of this study is to examine a specific extension, denoted as $P_{E,(2)}(H)$, of $H|_{X\backslash D}$. To define this extension, let $E$ be an $\mathbb{R}$-divisor supported on $D$. We denote ${{}_{<E}}H$ as $\cup_{E^{\prime}<E}{{}_{E^{\prime}}}H$. The coherent sheaf $P_{E,(2)}(H)$ is determined by the following conditions.

1. (1)

   ${{}_{<E}}H\subset P_{E,(2)}(H)\subset{{}_{E}}H$.

2. (2)

   Let $x$ be a point in $D$ and $(U;z_{1},\dots,z_{n})$ be holomorphic local coordinates on some open neighborhood $U$ of $x$ in $X$, such that $D=\{z_{1}\cdots z_{r}=0\}$. We denote $D_{i}=\{z_{i}=0\}$ for $i=1,\dots,r$. Now, let $\{W_{m,i}({{}_{E}}H)\}_{m\in\mathbb{Z}}$ be the monodromy weight filtration on ${{}_{E}}H|_{U}$ at $x$, with respect to the nilpotent part of the residue map ${\rm Res}_{D_{i}}(\theta)$ of the Higgs field along $D_{i}$. Then, we have:

   (1.1)

   $\displaystyle P_{E,(2)}(H)|_{U}={{}_{<E}}H+\bigcap_{i=1}^{r}W_{-2,i}({{}_{E}}H).$

When $E=D$, $P_{D,(2)}(H)$ represents the sheaf of $L^{2}$-holomorphic sections with coefficients in $H$. This construction was originally introduced by S. Zucker [Zucker1979] for algebraic curves, and it involves $H$ arising from a variation of Hodge structure. Consequently, it is a significant subject of study in the context of $L^{2}$-cohomology of a variation of Hodge structure.

To extend Zucker’s construction [Zucker1979] to higher-dimensional bases and non-canonical indexed extensions, we introduce $P_{E,(2)}(H)$. Specifically, when $E\geq 0$, $P_{D-E,(2)}(H)$ combines elements from both $P_{D,(2)}(H)$ and the multiplier ideal sheaf associated with $E$. This aspect makes $P_{E,(2)}(H)$ more convenient in applications where $E\neq D$. It can be proven that $P_{E,(2)}(H)$ is always locally free.

According to the non-abelian Hodge theory of Simpson [Simpson1988, Simpson1990] and Mochizuki [Mochizuki2006, Mochizuki20071], a $\mu_{A}$-polystable regular parabolic flat bundle $(V,\{{{}_{E}}V\}_{E\in{\rm Div}_{D}(X)},\nabla)$ is associated with the parabolic Higgs bundle $(H,\{{{}_{E}}H\}_{E\in{\rm Div}_{D}(X)},\theta)$. Moreover, there exists an isomorphism between the $C^{\infty}$ complex bundles:

In particular, the $C^{\infty}$ complex bundle associated with $H|_{X\backslash D}$ has two complex structures: $\bar{\partial}$, the complex structure of the Higgs bundle $H|_{X\backslash D}$, and $\nabla^{0,1}$, the $(0,1)$-part of $\nabla$ in the flat bundle $V|_{X\backslash D}$.

In this setting, Theorem 1.1 implies the following result.

In [SZtwisted], the authors demonstrate that $\omega_{X}\otimes(P_{D-N,(2)}(H)\cap j_{\ast}K)\otimes F\otimes L$ satisfies Kollár’s package. In particular, they establish that $f_{\ast}\left(\omega_{X/Y}\otimes(P_{D-N,(2)}(H)\cap j_{\ast}K)\otimes F\otimes L\right)$ is weakly positive in the sense of Viehweg.

Let $X$ be a smooth projective variety and $D\subset X$ be a simple normal crossing divisor on $X$. Let $(H,\{{{}_{E}}H\}_{E\in{\rm Div}_{D}(X)})$ be a locally abelian parabolic bundle on $(X,D)$ with vanishing parabolic Chern classes. This bundle is also polystable with respect to an ample line bundle $A$ on $X$. In this case, we can consider $(H,\{{{}_{E}}H\}_{E\in{\rm Div}_{D}(X)})$ as a parabolic Higgs bundle with a vanishing Higgs field. Consequently, $P_{E,(2)}(H)={{}_{<E}}H$. By selecting $K=H|_{X\backslash D}$ in Theorem 1.1, the conditions of holomorphicity and weak transversality are satisfied for $K$. Therefore, Theorem 1.6 implies the following.

This article is structured as follows. In Section 2, we provide a review of basic concepts such as a singular metric on a torsion-free sheaf, Caltaldo’s notion of Nakano semi-positivity, Hacon-Popa-Schnell’s notion of minimal extension property, and Guan-Mi-Yuan’s optimal Ohsawa-Takegoshi extension theorem. Section 3 introduces and examines $S_{X}(E,h)$, while also explaining its connection to significant transcendental and Hodge theoretic objects. The main result is demonstrated in Section 4.

In this subsection, we review the notion of Nakano/Griffiths positivity for singular Hermitian metrics.

Throughout this subsection, let $X$ be a complex manifold of dimension $n$ and $E$ be a vector bundle of rank $r$ on $X$.

We review the singular version of the Griffiths positivity and Nakano positivity of a singular hermitian metric on a vector bundle. First, the concept of a singular Hermitian metric, as introduced by Berndtsson-Paun [BP2008], Paun-Takayama [PT2018], and Hacon-Popa-Schnell [HPS2018], is defined as follows.

A singular Hermitian metric $h$ on $E$ induces a dual singular Hermitian metric $h^{\ast}$ on $E^{\ast}$.

When $h$ is a smooth Hermitian metric on $E$, the above definition coincides with the classical Griffiths positivity.

The following definition, which can be regarded as the singular version of Nakano positivity, was introduced by Cataldo [CataldoAndrea1998] and Guan-Mi-Yuan [GMY2023].

Let $X$ be a complex manifold of dimension $n$ and let $X^{o}\subset X$ be a Zariski open subset. Let $(E,h)$ be a holomorphic vector bundle on $X^{o}$ with a singular Hermitian metric such that $\Theta_{h}(E)\geq_{\rm Nak}^{s}0$ (Definition 2.5). Let $\alpha=\alpha^{\prime}\otimes\omega$ be an $E$-valued $(n,0)$-form, where $\alpha^{\prime}$ is a section of $E$ and $\omega$ is an $(n,0)$-form. We use the notation

The $L^{2}$-norm of $\alpha\in H^{0}(X^{o},\omega_{X^{o}}\otimes E)$ is defined as

We follow [HPS2018] to use the scaling $c_{n}$ in the $L^{2}$-norm.

Suppose $f:X\rightarrow B$ is a projective map to the open unit ball $B\subset\mathbb{C}^{r}$, where $0\in B$ is a regular value of $f$. Then the central fiber $X_{0}=f^{-1}(0)$ is a projective manifold of dimension $n-r$. We assume that $X_{0}\cap X^{o}\neq\emptyset$ and denote by $(E_{0},h_{0})$ the restriction of $(E,h)$ to $X_{0}\cap X^{o}$. The $L^{2}$ extension theorem for vector bundles equipped with a singular Hermitian metric, originally developed by Ohsawa-Takegoshi [OT1987], has been further elaborated by Guan-Zhou [GZ2015] and Guan-Mi-Yuan [GMY2023]. This theorem plays a crucial role in the proof of the main theorem of this article. For more results on this direction, we refer the readers to [Blocki2013, CPB2024, BP2008, Berndtsson1996, Demailly2000, DHP2013, OT1988] and the references therein.

The minimal extension property for singular Hermitian metrics, which is closely related to the Ohsawa-Takegoshi extension theorem, was introduced by Hacon, Popa, and Schnell [HPS2018]. This property allows for the extension of sections across a bad locus while maintaining control over the norm of the section.

Let us still denote by $B\subset\mathbb{C}^{n}$ the open unit ball.

According to [DNWZ2023], the minimal extension property of a $C^{2}$ metric is equivalent to the Nakano semi-positivity of its curvature form.

We use the notations in §1.4. Let $X$ be a smooth projective variety and $D$ a reduced simple normal crossing divisor on $X$. Consider $(H,\{{{}_{E}}H\}_{E\in{\rm Div}_{D}(X)},\theta)$ as a locally abelian parabolic Higgs bundle on $(X,D)$, which is polystable with respect to an ample line bundle $A$ on $X$. Let $h$ be a tame harmonic metric on $H|_{X\backslash D}$, which is compatible with the parabolic structure. The existence of such a metric is ensured by Simpson [Simpson1990] for algebraic curves and by Mochizuki [Mochizuki2006] in higher dimensions. Let $\overline{\theta}$ be the adjoint of $\theta$, and $\partial$ be the unique $(1,0)$-connection such that $\partial+\bar{\partial}$ is compatible with $h$. Consequently, $(H|_{X\backslash D}\otimes_{\mathscr{O}_{X\backslash D}}\mathscr{A}^{0}_{X\backslash D},\nabla:=\partial+\bar{\partial}+\theta+\overline{\theta})$ specifies a meromorphic flat connection that remains regular along the divisor $D$. Let $\nabla=\nabla^{1,0}+\nabla^{0,1}$ be the decomposition with respect to the bi-degree. Notice that $\nabla^{1,0}=\partial+\theta$ and $\nabla^{0,1}=\bar{\partial}+\overline{\theta}$.