Kodaira-type Vanishings via Non-abelian Hodge Theory

Let $X$ be a complex projective manifold. Given a reduced normal crossing divisor $D=D_{1}+...+D_{n}$ on $X$, and a vector $\mathbf{d}=(d_{1},...,d_{n})\in\mathbb{R}^{n},$ we set $\mathbf{d}D=d_{1}D_{1}+...+d_{n}D_{n}$. We say $\mathbf{d}\geq\mathbf{a}$, if $d_{i}\geq a_{i}$ for all $i$, similarly defined for other inequalities. We use $\mathbf{1}=(1,...,1)\in\mathbb{R}^{n}$, similarly for $\mathbf{0}$. We also use $aD$ to denote $aD_{1}+...+aD_{n}$, for $a\in\mathbb{R}$. A divisor denoted by a single letter, e.g. $D$, is an effective integral divisor, unless there is a coefficient in front of it, like $\mathbf{d}D,aD,$ as above, or explicitly stated otherwise. We mainly consider $\mathbb{Q}$-divisors, and the $\mathbb{R}$-divisor case can be reduced to such case by an argument appeared later in Remark 1.7.

The renowned Kodaira-Akizuki-Nakano Vanishing Theorem has the following generalized form, which is due to Esnault-Viehweg [EV86], [EV92], combined with the Kawamata-Viehweg type Vanishing formulation [AMPW]:

There is another direction to generalize Kodaira-Akizuki-Nakano Vanishing using Saito’s mixed Hodge Module. In this paper, we say a pair $(\mathcal{M},F_{\bullet})$ admits a mixed Hodge Module, where $\mathcal{M}$ is a right coherent $\mathscr{D}$-module, and $F_{\bullet}$ is its increasing Hodge filtration, if we can add a weight filtration to make it an algebraic graded polarizable mixed Hodge module in the sense of [Sai90, §2]. We remark that, among other things, the existence of the weight compatible polarization on $(\mathcal{M},F_{\bullet})$ is essential while defining mixed Hodge Module. However, for the vanishing theory we studying in this paper, we only use its formal property: the strictness of certain functors with respect to $F_{\bullet}$. Hence we mute those information for the ease of notation.

In the case with the absence of $D$, it is just Saito’s vanishing [Sai90, (2.g)]. Please refer [Wei20b, Theorem 20] for this general form, with the proof follows Saito’s proof of Saito’s vanishing. Please note that, in loc. cit., we use Sp to denote the de Rham functor on right modules. We use the notation DR here, to follow the convention in [Sab05] and [Moc15].

In 80s’, the non-abelian Hodge Theory is mainly developed by C. Simpson. As the starting point, he studies the harmonic bundles and uses that to give extra structures on cohomologies of a compact Kählar manifold, with the coefficient being a semisimple local system, instead of the constant local system as studied in the classical Hodge theory. Simpson also proposes the Meta theorem which states that what works for classical Hodge theory shall still work for non-abelian Hodge theory. Later on, following Saito’s idea of constructing mixed Hodge Modules, the theory of mixed twistor $\mathscr{D}$-modules is initiated by C. Sabbah. Then, T. Mochizuki completes this spectacular theory in the past decade, which can also be viewed as a satisfying answer towards Simpson’s Meta Theorem. In the same vein, heuristically, the vanishing theorem above can be generalized to the corresponding mixed twistor $\mathscr{D}$-modules setting. Before we state the generalization, let’s set up the notations for mixed twistor $\mathscr{D}$-modules.

For any complex manifold $X$, we denote $\mathcal{X}=X\times\mathbb{C}_{\lambda}$, where $\mathbb{C}_{\lambda}$ is the affine line with $\lambda$ as its coordinate, and $p:\mathcal{X}\to X,$ $q:\mathcal{X}\to\mathbb{C}_{\lambda}$ the natural projections. Following the notations in [Sab05] and [Moc15], in this paper, we say that a right algebraic $\mathscr{R}_{X}$-module $\mathcal{M}$ on $\mathcal{X}$ admits a mixed twistor $\mathscr{D}$-module, if there exists an algebraic graded polarizable mixed twistor $\mathscr{D}$-module represented by a filtered $\mathscr{R}$-triple $(\mathcal{T},W)$, with the triple $\mathcal{T}=(\mathcal{M}^{\prime},\mathcal{M}^{\prime\prime},C),$ and $\mathcal{M}=\mathcal{M}^{\prime\prime}$. We also use $\Xi_{\textup{Dol}}:=\mathcal{M}/\lambda\mathcal{M}$ and $\Xi_{\textup{DR}}:=\mathcal{M}/(\lambda-1)\mathcal{M}$, which are exact functors from strict coherent $\mathscr{R}_{X}$-modules to coherent $\mathscr{D}_{X}$-modules and coherent $\mathscr{A}_{X}:=\textup{Gr}^{F}(\mathscr{D}_{X})$-modules, respectively [Sab05, 1.1.a, Definition 1.2.1]. As we remarked in the mixed Hodge Module case, the other data while defining a graded polarizable mixed twistor $\mathscr{D}$-module is important. However, for the vanishing theorem, we only use the holomorphic picture of a mixed twistor $\mathscr{D}$-module, and its formal property like the strictness of certain natural functors, so we will only keep track the $\mathscr{R}_{X}$-module $\mathcal{M}=\mathcal{M}^{\prime\prime}$.

The following vanishing result is a direct non-abelian Hodge theoretic generalization of Theorem 1.2.

In the previous theorem, $\mathcal{M}[*D]$ and $\mathcal{M}[!D]$ are prolongations of $\mathcal{M}$ as coherent $\mathscr{R}_{X}$-modules as in [Moc15, 3.1.2, 3.1.3]. We will use $\mathcal{M}(*D)$ to denote the canonical prolongation as a coherent $\mathscr{R}_{X(*D)}$-module [Moc15, 3.1.1].

Naturally, we want to give the previous vanishing a Kawamata-Viehweg Vanishing type formulation, as in Theorem 1.1. The next vanishing theorem can be viewed as a such generalization, and it seems to be new even in the setting of mixed Hodge Modules. The formulation is quite technical, and we will define the notations in §3. At this stage, for reader’s convenience, we introduce the notation

where $V^{D_{i}}_{\bullet}\mathcal{M}^{(\alpha)}(*D)$ is the KM-filtration of $\mathcal{M}(*D)$ locally around $\lambda=\alpha$. Please note that the multi-indexed filtration $\mathbf{V}^{D}_{\bullet}$ does not behave well in general. We need the $V$-compatibility on $\mathcal{M}(*D)$, Definition 3.3.

We will also establish the relative versions of the previous two vanishing theorems in §2. Actually, it is not straight-forward to get such generalization as e.g. [KMM87, Theorem 1-2-3], using a log-smooth compactification and Serre vanishing on coherent $\mathcal{O}$-modules. This is because, in our case, we are not working in the derived category of coherent sheaves. However, due to our proof of Theorem 1.3 being functorial, we can also get a functorial proof of its relative version, Theorem 2.1, which does not use a log-smooth compactification. We then show a Nadel-type vanishing with multiplier-ideals from $\mathbb{Q}$-divisors, and state an effective global generalization theorem, as an application.

In §3, we recall the notion of multi-indexed Kashiwara-Malgrange (KM) filtration, its compatibility, and using it to get the logarithmic comparison in the setting of $\mathscr{R}$-modules as in [Wei20a]. In §4, we recall the geometric construction in [EV92], which is constructing a cyclic cover from a normal crossing $\mathbb{Q}$-divisor, and study the induced local systems and their multi-indexed KM filtration. We give the proofs of our main vanishing results in §5.

It is also natural to generalize the injectivity/surjectivity results in [EV92, §5], [Wu17], [Wu21], to the Non-abelian Hodge setting. However, the author finds such type of results have a different nature of approach, and they cannot directly imply those vanishing results above using the method summerized in [EV92, §1, 2. proof], with the help of Serre Vanishing, for the same reason in the relative vanishing case. Hence, the author decides to leave it for another occasion.

Let’s explain how the previous two vanishing theorems cover essentially all Kodaira-type vanishings in algebraic geometry we know so far. Nevertheless, the author has no intention to say that we have new proofs to those brilliant results. On the contrary, the vanishing theorems above shall be taken as a natural summary of currently known Kodaira-type vanishings, and the proofs are based on the proofs of those results. The proofs mainly adopt Esnault and Viehweg’s geometric construction [EV92], Saito’s proof of Saito Vanishing, and the Theory of mixed twistor $\mathscr{D}$-modules, due to Sabbah and Mochizuki. However, we still cannot recover Nadel vanishing in the analytic setting, that is proved using $L^{2}$ method.

We first restrict ourselves to the case that $\mathcal{M}$ being the Rees algebra associated to a mixed Hodge Module $(\mathcal{N},F_{\bullet})$, with a right $\mathscr{D}$-module $\mathcal{N}$, and an increasing Hodge filtration $F_{\bullet}$. We have that $\Xi_{\textup{Dol}}\mathcal{M}:=\mathcal{M}/\lambda\mathcal{M}$ is just taking the associated graded pieces, $\textup{Gr}^{F}_{\bullet}\mathcal{N}$. Then, Theorem 1.3 in this case is just Theorem 1.2.

For Theorem 1.4, if we only consider $\mathcal{M}$ admits a graded polarizable variation of mixed Hodge structures on $X$, $C=0$, and $\mathbf{0}\leq\mathbf{b}<\mathbf{1},$ we have $V^{B}_{-\mathbf{b}}\mathcal{M}(*D)\simeq\mathcal{M}$. In particular, if we take $\mathcal{M}\simeq\omega_{\mathcal{X}}$, i.e. it corresponds to the trivial variation, we get the Kawamata-Viehweg vanishing for klt pairs. Hence, via a standard argument, e.g. Corollary 2.3, it implies the big and nef vanishing for the lowest graded piece of a mixed Hodge Module, which has been proved by Suh [Suh18] and Wu [Wu17] independently. Let’s also remark that the $C$ part can be useful in case we want to deal with log-canonical singularities, e.g. the application on showing the zero locus of holomorphic log-one forms, [Wei20b].

If we have a reduced, possibly singular Cartier divisor $D$, by taking a log resolution $f:(X^{\prime},D^{\prime})\to(X,D)$, with $f^{*}dD=\mathbf{d}^{\prime}D^{\prime}$. We have the multiplier ideal

Hence we get the Nadel vanishing with the multiplier ideal of $\mathbb{Q}$-divisors. Such relation between multiplier ideals and Hodge modules has also been studied in [BS05] and [MP20]. See also Corollary 2.3 in the next section.

In the case that $\mathcal{M}$ is tame and underlies a variation of twistor structures on $X\setminus D$, this is equivalent to that $\Xi_{\textup{Dol}}\mathcal{M}$ gives us a slope parabolic polystable higgs bundle on $X\setminus D$, due to [Moc06, Theorem 1.4]. Since a parabolic semistable higgs bundle can be realized as a sequentially extension of stable higgs bundles, as argued in [AHL19, Lemma 7.1], then the two vanishings above generalize those vanishing results proved by Arapura-Hao-Li [AHL19], and by Deng-Hao [DH22]. In [DH22], their vanishing is more refined in the sense that they build a relation between the range of the vanishing degrees and the number of positive eigenvalues of the curvature form of $\mathcal{L}$. In the case that $\mathcal{L}$ being $k$-ample as in [Som78, §1], (which is a stronger condition, see also [Tot13],) we have a projective dominate morphism $f:X\to Y$, with $\mathcal{L}^{n}=f^{*}\mathcal{A}$ for some ample line bundle $\mathcal{A}$ on $Y$, and $n\in\mathbb{Z}^{+}$. Hence we can achieve the vanishing with the expected range as in [DH22], due to the compatibility of the direct image, DR and $\Xi_{\textup{Dol}}$ functor. We leave the details to the interested readers.

In this paper, we use $\otimes$ (without sub-index) to denote the tensor product over the corresponding structure sheaf $\mathcal{O}$, unless specified otherwise. We use $f_{+}$ and $f_{\dagger}$ (resp. $f_{*}$ and $f_{!}$) to denote the derived direct image and direct image with proper support in the derived category of $\mathscr{R}$-modules (resp. quasi-coherent sheaves or constructable sheaves). We use $\mathbb{H}^{\bullet}$ (resp. $\mathbb{H}^{\bullet}_{c}$) to denote taking the hypercohomoloy (resp. hypercohomoloy with proper support), which is the same as the (derived) functor of taking derived direct image $a_{*}$ (resp. $a_{!}$), where $a$ is the canonical map to $\text{Spec}(\mathbb{C})$. We consider right $\mathscr{D}$-modules or $\mathscr{R}$-modules by default, unless specified otherwise. Although we only consider the vanishing theory for algebraic mixed twistor $\mathscr{D}$-modules on a quasi-projective variety, the constructions in §3 and §4 work for the analytic setting.

In this section, we state the relative vanishings, a generalized nef and big vanishing, and an effective global generalization theorem, as an application. Most notations shall be standard, following [Sab05] and [Moc15]. Some notations will be carefully defined in the next section.

Let’s first state a relative version of Theorem 1.3.

We can also show a relative version of Theorem 1.4:

Recall that, due to Hitchin-Kobayashi correspondence, we have that, for a stable vector bundle $\mathcal{E}$ on a smooth projective variety $X$ with vanishing Chern classes, we can view it as a Higgs bundle with the trivial Higgs map, that can be lift as a harmonic bundle corresponding to a unitary representation. A semistable vector bundle $\mathcal{N}$ with vanishing Chern classes can be realized as extensions of stable vector bundle $\mathcal{E}$ with vanishing Chern classes, [Sim91, Theorem 2]. Hence $\mathcal{N}$, equipped with the trivial higgs map, can be lift as a smooth variation of mixed twistor structures.

Then, we can show the following Nadel vanishing on a semistable vector bundle with vanishing Chern classes.

$\mathcal{J}(bB)$ can also be replaced by $\mathcal{J}(b|B|)$ the multiplier ideal of linear series, or $\mathcal{J}(||bB||)$ the asymptotic multiplier ideal. Let’s refer [Laz04b] for more details.

It is natural to apply the previous Nadel vanishing to get the following effective global generation result, and we follow the exposition in [Laz04b, 10.4]. Please also refer [dC98b], [dC98a] for a much more comprehensive study on this topic.

In this section, we would love to generalize some results in [Wei20a] about logarithmic comparison in mixed Hodge modules setting, to the mixed twistor $\mathscr{D}$-modules case.

Let’s first recall the definition of the Kashiwara-Malgrange (KM-)filtration on a coherent $\mathscr{R}_{X(*D)}$-module, with respect to a smooth component $H$ of the reduced normal crossing divisor $D$ on a complex manifold $X$. Since in this paper, given a $\mathscr{R}$-module $\mathcal{M}$ on $X$, we will find ourselves only use its information on the open part $X\setminus D$, so it is more natural and easier to consider the KM-filtration on the coherent $\mathscr{R}_{X(*D)}$-module $\mathcal{M}(*D):=\mathcal{M}\otimes\mathcal{O}_{\mathcal{X}}(*(p^{*}D))$ than to consider the KM-filtration on the coherent $\mathscr{R}_{X}$-modules like $\mathcal{M}[*D]$ or $\mathcal{M}[!D]$, see also Remark 1.5. Actually, $\mathcal{M}[*D]$ and $\mathcal{M}[!D]$ themselves are built from the KM-filtration on $\mathcal{M}(*D)$, as in [Moc15, 3.1].

There is a canonical $\mathbb{Z}$-indexed increasing filtration on $\mathscr{R}_{X}$, stalk-wise defined by, for any $(x,\alpha)\in\mathcal{X}$,

We note that $V^{H}_{0}\mathscr{R}_{X(*D)}$ is a coherent ring. It can be directly checked that $V^{H}_{k}\mathscr{R}_{X(*D)}=V^{H}_{0}\mathscr{R}_{X(*D)}\otimes\mathcal{O}_{X}(kH)$, and $V^{H}_{\bullet}\mathscr{R}_{X(*D)}$ is a good filtration, [Bjo93, Appendix III], as a $V^{H}_{0}\mathscr{R}_{X(*D)}$-module.

Now, we use $\mathcal{N}$ to denote a strict coherent $\mathscr{R}_{X(*D)}$-module. For $\forall\alpha\in\mathbb{C}$, we use $\mathcal{N}^{(\alpha)}$ to denote restricting $\mathcal{N}$ onto an open subset $X\times\Delta(\alpha,\epsilon)\subset\mathcal{X}$, for some small $\epsilon>0$, where $\Delta(\alpha,\epsilon)$ is the open disk in $\mathbb{C}_{\lambda}$, centered at $\lambda=\alpha$ and with radius $\epsilon$.

Then, we consider the multi-indexed KM-filtration. Let $X$ be a complex manifold, with a reduced normal crossing divisor $D=D_{1}+...+D_{n}$, with irreducible components $D_{i}$. For any

we denote

For $\mathbf{a}=\mathbf{0}:=(0,...,0)$, $\mathbf{V}_{\mathbf{0}}^{D}\mathscr{R}_{X(*D)}$ is a coherent sub-ring of $\mathscr{R}_{X}$, denoted by $\mathscr{R}_{(X,D)}$.

Let $\mathcal{N}$ be a coherent right $\mathscr{R}_{X(*D)}$-module, which is strictly specializable with respect to all $D_{i}$. In particular, it is the case when $\mathcal{M}$, a coherent $\mathscr{R}_{X}$-module, admits a graded polarizable mixed twistor $\mathscr{D}$-module, and $\mathcal{N}=\mathcal{M}(*D)$.

Define a multi-indexed Kashiwara-Malgrange filtration with respect to $D$ by

for any $\mathbf{a}=(a_{1},...,a_{n})\in\mathbb{R}^{n}.$ It is not hard to see that $\mathbf{V}^{D}_{\bullet}\mathcal{N}^{(\alpha)}$ is a multi-indexed filtered module over the filtered ring $\mathbf{V}_{\bullet}^{D}\mathscr{R}_{X}$, and their filtrations are compatible in the sense that

for any $\mathbf{a},\mathbf{b}\in\mathbb{R}^{n}.$

If we separate $D$ into two groups of components $D=B+C$, we will also use $\mathbf{V}^{B}_{\mathbf{b}}\mathbf{V}^{C}_{\mathbf{c}}\mathcal{N}^{(\alpha)}:=\mathbf{V}^{B}_{\mathbf{b}}\mathcal{N}^{(\alpha)}\cap\mathbf{V}^{C}_{\mathbf{c}}\mathcal{N}^{(\alpha)}.$

Recall that we say $\mathbf{b}<\mathbf{a}$, if $b_{i}<a_{i}$ for all $i$. We denote

In general, the $n$ filtrations $V^{D_{i}}_{\bullet}$ do not behave well between each other. This motivates us to make the following definition, which will be essentially used in the logarithmic comparison, Proposition 3.7.

As in the filtered $\mathscr{D}$-module case in [Wei20a, Lemma 12], we have the following