On the Griffiths-Yukawa coupling length of some Calabi-Yau families

TL;DR

This paper investigates the Griffiths-Yukawa coupling length in certain Calabi-Yau families derived from hyperplane arrangements, providing insights into their geometric and Hodge-theoretic properties.

Contribution

It precisely determines the Griffiths-Yukawa coupling length for specific Calabi-Yau families associated with hyperplane arrangements, a novel calculation in this context.

Findings

Calculated the Griffiths-Yukawa coupling length for these families.

Enhanced understanding of the geometric structure of Calabi-Yau moduli.

Connected hyperplane arrangements with Hodge-theoretic invariants.

Abstract

We determine the Griffiths-Yukawa coupling length of the Calabi-Yau universal families coming from hyperplane arrangements.

Equations

$$r|m,n=m-\frac{m}{r}-1.$$

$$F^n\mathcal{V}\subset F^{n-1}\mathcal{V}\subset \cdots \subset F^1\mathcal{V}\subset F^0\mathcal{V}=\mathcal{V}:=\mathbb{V}{\otimes }_{\mathbb{C}}{\mathcal{O}}_M$$

$$D{F}^p\mathcal{V}\subset F^{p-1}\mathcal{V}{\otimes }_{{\mathcal{O}}_M}{\Omega }_M,\forall 1\le p\le n,$$

$${\theta }^{p,n-p}:E^{p,n-p}\to E^{p-1,n-p+1}{\otimes }_{{\mathcal{O}}_M}{\Omega }_M$$

$${\theta }_x^{p,n-p}(v):E_x^{p,n-p}\to E_x^{p-1,n-p+1},$$

$$E^{n,0}\stackrel{{\theta }^{n,0}}{\to }{E}^{n-1,1}\otimes {\Omega }_M\stackrel{{\theta }^{n-1,1}}{\to }\cdots \stackrel{{\theta }^{n-q+1,q-1}}{\to }{E}^{n-q,q}\otimes S^q{\Omega }_M$$

$${\theta }^q:Sy{m}^q(TM)\to Hom(E^{n,0},E^{n-q,q})$$

$$\varsigma (\mathbb{V})=min\{q\ge 1\mid {\theta }^q=0\}-1.$$

$$\{e_1\wedge \cdots \wedge e_p\wedge e_{p+1}\wedge \cdots \wedge e_n|e_1,\cdots ,e_p\in V_1,e_{p+1},\cdots ,e_n\in V_2\}.$$

$$\{e_1\wedge \cdots \wedge e_n|e_i\in W_i,1\le i\le n\}.$$

$$\sum _{i_1+\cdots +i_n\ge p}{F}^{i_1}{\mathcal{V}}_x\wedge F^{i_2}{\mathcal{V}}_x\wedge \cdots \wedge F^{i_n}{\mathcal{V}}_x\subset {\wedge }^n{\mathcal{V}}_x.$$

$${\theta }_x^{k-1}(v^{k-1}):E_x^{n,0}\to E_x^{n-k+1,k-1}$$

$$\begin{array}[]{ccc}E_{x}^{n,0}&\xrightarrow{\theta_{x}^{k-1}(v^{k-1})}&E_{x}^{n-k+1,k-1}\\ \downarrow{\wr}&&\downarrow{\wr}\\ \wedge^{n}F_{x}^{1,0}&\xrightarrow{\wedge^{n}\eta_{x}(v)}&\wedge^{n-k+1}F_{x}^{1,0}\wedge^{k-1}F_{x}^{0,1}\end{array}$$

$${\wedge }^n{\eta }_x(v)(e_1\wedge \cdots e_n):=(k-1)!\sum _{1\le i_1<\cdots <i_{k-1}\le n}{e}_1\wedge \cdots \wedge {\eta }_x(v)e_{i_1}\wedge \cdots \wedge {\eta }_x(v)e_{i_{k-1}}\wedge \cdots \wedge e_n.$$

$$\mathbb{V}:=R^n{f}_{\ast }\mathbb{C},\tilde{\mathbb{V}}:=R^n{\tilde{f}}_{\ast }\mathbb{C},{\tilde{\mathbb{V}}}_{pr}:=(R^n{\tilde{f}}_{\ast }\mathbb{C}{)}_{pr},\mathbb{W}:=R^1{g}_{\ast }\mathbb{C},$$

$$\mathbb{V}={\oplus }_{i=0}^{r-1}{\mathbb{V}}_{(i)},\mathbb{W}={\oplus }_{i=0}^{r-1}{\mathbb{W}}_{(i)}.$$

$$h^{n-q,q}=\left\{\begin{array}[]{ll}{n\choose q}{\frac{m}{r}-1\choose q},&\hbox{$\ 0\leq q\leq\frac{m}{r}-1$;}\\ 0,&\hbox{$\frac{m}{r}\leq q\leq n$.}\end{array}\right.$$

$$\begin{array}{rl}U& \stackrel{\sim }{\to }{\mathfrak{M}}_C\\ (a_1,\cdots ,a_{m-3})& \mapsto ([1:0],[1;1],[0:1],[1:a_1],\cdots ,[1:a_{m-3}])\end{array}$$

$$\begin{array}{rl}& y_2^r-(y_0^r+y_1^r)=0;\\ & y_{2+i}^r-(y_0^r+a_i{y}_1^r)=0,1\le i\le m-3.\end{array}$$

$$\begin{array}{rl}{N}_1:=Ker(N& \to \mathbb{Z}/r\mathbb{Z})\\ (a_j)& \mapsto \sum _{j=0}^{m-1}{a}_j\end{array}$$

$$\alpha \cdot y_j:={\zeta }^{a_j}{y}_j,\forall 0\le j\le m-1.$$

$$\begin{array}[]{ccc}F^{1,0}_{a}\otimes T_{a}\mathfrak{M}_{C}&\xrightarrow{\eta_{a}}&F^{0,1}_{a}\\ \downarrow{\wr}&&\downarrow{\wr}\\ H^{1,0}(C,{\mathbb{C}})_{(1)}\otimes T_{a}\mathfrak{M}_{C}&\xrightarrow{\psi_{a}}&H^{0,1}(C,{\mathbb{C}})_{(1)}\end{array}$$

$$\begin{array}[]{ccc}H^{1,0}(C,{\mathbb{C}})_{(1)}\otimes T_{a}\mathfrak{M}_{C}&\xrightarrow{\psi_{a}}&H^{0,1}(C,{\mathbb{C}})_{(1)}\\ \downarrow{\wr}&&\downarrow{\wr}\\ H^{1,0}(Y,{\mathbb{C}})^{N_{1}}_{(1)}\otimes T_{a}\mathfrak{M}_{C}&\xrightarrow{\tilde{\psi}_{a}}&H^{0,1}(Y,{\mathbb{C}})^{N_{1}}_{(1)}\end{array}$$

$$F={\mu }_0{F}_0+\cdots +{\mu }_{m-3}{F}_{m-3}$$

$$\begin{array}{rl}& F_0:=y_2^r-(y_0^r+y_1^r),\\ & F_i:=y_{i+2}^r-(y_0^r+a_i{y}_1^r),1\le i\le m-3.\end{array}$$

$$R:=\mathbb{C}[{\mu }_0,\cdots ,{\mu }_{m-3},y_0,\cdots ,y_{m-1}]/J$$

$$\begin{array}{rl}\alpha \cdot y_j& :={\zeta }^{{\alpha }_j}{y}_j,\forall 0\le j\le m-1.\\ \alpha \cdot {\mu }_i& :={\mu }_i,\forall 0\le i\le m-3.\end{array}$$

$$H^{1,0}(Y,\mathbb{C}{)}_{(1)}^{N_1}\simeq R_{(0,mr-m-2r)}^N,H^{0,1}(Y,\mathbb{C}{)}_{(1)}^{N_1}\simeq R_{(1,mr-m-r)}^N.$$

$$\begin{array}{rl}{T}_a{\mathfrak{M}}_C& \stackrel{\varphi }{\to }{R}_{(1,r)}\\ \frac{\partial }{\partial x_i}& \mapsto -{\mu }_i{y}_1^r\end{array}$$

$$\begin{array}[]{ccc}H^{1,0}(Y,{\mathbb{C}})^{N_{1}}_{(1)}\otimes T_{a}\mathfrak{M}_{C}&\xrightarrow{\tilde{\psi}_{a}}&H^{0,1}(Y,{\mathbb{C}})^{N_{1}}_{(1)}\\ \downarrow{id\otimes\phi}&&\downarrow{\simeq}\\ R^{N}_{(0,mr-m-2r)}\otimes R_{(1,r)}&\xrightarrow{}&R^{N}_{(1,mr-m-r)}\end{array}$$

Full text

1. Introduction

In the study of moduli spaces of Calabi-Yau (CY) manifolds, an effective method is to investigate the associated variation of Hodge structure. To every variation of Hodge structure ${\mathbb{V}}$, we can associate an interesting numerical invariant $\varsigma({\mathbb{V}})$, called the Griffiths-Yukawa coupling length, which is introduced in [10]. The connection of Griffiths-Yukawa coupling length with Shararevich s conjecture for CY manifolds has been intensively studied (see e.g., [4, 12]). It has been shown that, for example, a CY family with maximal Griffiths-Yukawa coupling length is rigid.

On the other hand, among the various Calabi-Yau moduli spaces, the ones coming from hyperplane arrangements are particular interesting, due to their analogue of elliptic curves and their relations to Gross’ geometric realization problem (see e.g., [2, 3, 8]).

The main purpose of this note is the determination of the Griffiths-Yukawa coupling length for the CY families coming from hyperplane arrangements. More precisely, let $m,n,r$ be positive integers satisfying the condition:

[TABLE]

Let $\mathfrak{M}_{AR}$ be the coarse moduli space of ordered $m$ hyperplane arrangements in ${\mathbb{P}}^{n}$ in general position and $\tilde{\mathcal{X}}_{AR}\xrightarrow{\tilde{f}}\mathfrak{M}_{AR}$ be the family of CY $n$-folds which is obtained by a resolution of $r$-fold covers of ${\mathbb{P}}^{n}$ branched along $m$ hyperplanes in general position. This family gives a weight $n$ complex variation of Hodge structure (${\mathbb{C}}$-VHS ) $\tilde{{\mathbb{V}}}_{pr}:=(R^{n}\tilde{f}_{*}{\mathbb{C}})_{pr}$ over $\mathfrak{M}_{AR}$. Our main result is:

Theorem 1.1**.**

The Griffiths-Yukawa coupling length of $\tilde{{\mathbb{V}}}_{pr}$ is $\varsigma(\tilde{{\mathbb{V}}}_{pr})=\frac{m}{r}-1$.

Remark 1.2**.**

By the theorem above, among the Calabi-Yau families coming from hyperplane arrangements, the ones with minimal Griffiths-Yukawa coupling length (i.e. $\varsigma(\tilde{{\mathbb{V}}}_{pr})=1$) are those with $(m,n,r)=(n+3,n,\frac{n+3}{2})$, and these are exactly the families considered in [7]. In particular, we get a positive answer to one of the questions listed at the end of [7]. Besides, the ones with maximal Griffiths-Yukawa coupling length (i.e. $\varsigma(\tilde{{\mathbb{V}}}_{pr})=n$) are those with $(m,n,r)=(2n+2,n,2)$, and these are exactly the Calabi-Yau families coming from double covers of ${\mathbb{P}}^{n}$ branched along $2n+2$ hyperplanes in general position.

In section 2, we recall the notion of ${\mathbb{C}}$-VHS, which is more general than the usual notion of ${\mathbb{Q}}$-VHS or ${\mathbb{R}}$-VHS. Then we recall the definition of the Griffiths-Yukawa coupling length in the context of ${\mathbb{C}}$-VHS. In section 3, we introduce the various ${\mathbb{C}}$-VHS coming from hyperplane arrangements. The key observation leading to the main result is a special locus $\mathfrak{M}_{C}$ in $\mathfrak{M}_{AR}$ which parameterizes distinct points on ${\mathbb{P}}^{1}$. We use the weight one ${\mathbb{C}}$-VHS associated to $\mathfrak{M}_{C}$ to compute some of the Hodge numbers of $\tilde{{\mathbb{V}}}_{pr}$, hence obtain an upper bound of $\varsigma(\tilde{{\mathbb{V}}}_{pr})$. In section 4, we use the tool of Jacobian rings to compute the Higgs map associated to the weight one ${\mathbb{C}}$-VHS on $\mathfrak{M}_{C}$. In this way, we obtain the lower bound of $\varsigma(\tilde{{\mathbb{V}}}_{pr})$ and finish the proof of the main result.

2. Definitions

Throughout this section, we let $M$ be a complex manifold and ${\mathcal{O}}_{M}$ be the sheaf of holomorphic functions on $M$. We identify holomorphic vector bundles of finite rank on $M$ and sheaves of locally free ${\mathcal{O}}_{M}$-modules of finite rank on $M$ by the well known way. We first review the definition of complex variation of Hodge structure, which seem to vary slightly according to the source.

Definition 2.1**.**

(c.f. [1]) A complex variation of Hodge structure (${\mathbb{C}}-$VHS) of weight $n$ on $M$ is a local system ${\mathbb{V}}$ of finite dimensional ${\mathbb{C}}$-vector spaces on $M$ together with a filtration of holomorphic vector bundles:

[TABLE]

satisfying the following Griffiths transversality condition:

[TABLE]

where $D:{\mathcal{V}}\rightarrow{\mathcal{V}}\otimes_{{\mathcal{O}}_{M}}\Omega_{M}$ is the Gauss-Manin connection induced by the local system ${\mathbb{V}}$. The filtration $F^{\cdot}{\mathcal{V}}$ is called the Hodge filtration and the integers $h^{p,n-p}({\mathbb{V}}):=rank\frac{F^{p}{\mathcal{V}}}{F^{p+1}{\mathcal{V}}}$ $(0\leq p\leq n)$ are called the Hodge numbers. Here by convention, $F^{n+1}{\mathcal{V}}=0$.

Remark 2.2**.**

If ${\mathbb{V}}$ is a weight $n$ rational variation of Hodge strucutre (${\mathbb{Q}}$-VHS), then it is easy to see ${\mathbb{V}}\otimes_{{\mathbb{Q}}}{\mathbb{C}}$ is admits a structure of ${\mathbb{C}}$-VHS of weight $n$ .

Obviously, a morphism $\phi:{\mathbb{W}}\rightarrow{\mathbb{V}}$ between two ${\mathbb{C}}$-VHS of weight $n$ on $M$ is a morphism of local systems preserving the Hodge filtration.

If $({\mathbb{V}},F^{n}{\mathcal{V}}\subset F^{n-1}{\mathcal{V}}\subset\cdots\subset F^{1}{\mathcal{V}}\subset F^{0}{\mathcal{V}}={\mathcal{V}}:={\mathbb{V}}\otimes_{{\mathbb{C}}}{\mathcal{O}}_{M})$ is a ${\mathbb{C}}$-VHS of weight $n$ on $M$, then by the Griffiths transversality, $\forall\ 0\leq p\leq n$, the Gauss-Manin connection $D$ induces a ${\mathcal{O}}_{M}$-linear homomorphism:

[TABLE]

where $E^{p,n-p}:=\frac{F^{p}{\mathcal{V}}}{F^{p+1}{\mathcal{V}}}$. For $x\in M$ and $v\in T_{x}M$, we denote the contraction of $\theta^{p,n-p}$ with $v$ by

[TABLE]

which is a ${\mathbb{C}}$-linear map between the fibers $E_{x}^{p,n-p}$ and $E_{x}^{p-1,n-p+1}$. We call $(E:=\oplus_{p=0}^{n}E^{p,n-p},\theta:=\oplus_{p=0}^{n}\theta^{p,n-p})$ the Higgs bundle associated to the ${\mathbb{C}}$-VHS ${\mathbb{V}}$. A morphism between two ${\mathbb{C}}$-VHS of weight $n$ induces a morphism between the associated Higgs bundles in an obviously way.

Let ${\mathbb{V}}$ be a ${\mathbb{C}}$-VHS of weight $n$ over $M$ and $(E,\theta)$ the associated Higgs bundle. For every $q$ with $1\leq q\leq n$, the $q$th iterated Higgs field

[TABLE]

defines a morphism

[TABLE]

where $TM$ is the holomorphic tangent bundle of $M$. The Griffiths-Yukawa coupling length $\varsigma({\mathbb{V}})$ of ${\mathbb{V}}$ is defined by

[TABLE]

Lemma 2.3**.**

Let $({\mathbb{V}},F^{\cdot}{\mathcal{V}})$ and $(\tilde{{\mathbb{V}}},F^{\cdot}\tilde{{\mathcal{V}}})$ be ${\mathbb{C}}$-VHS of weight $n$ on $M$ with Hodge numbers $h^{n,0}({\mathbb{V}})=h^{n,0}(\tilde{{\mathbb{V}}})=1$. Suppose $\phi:{\mathbb{V}}\rightarrow\tilde{{\mathbb{V}}}$ is a morphism of ${\mathbb{C}}$-VHS satisfying: $\forall\ 0\leq p\leq n$, $\forall\ x\in M$, the induced linear map between the fibers of the associated Higgs bundles $\phi_{x}^{p,n-p}:E^{p,n-p}_{x}\rightarrow\tilde{E}^{p,n-p}_{x}$ is injective. Then $\varsigma({\mathbb{V}})=\varsigma(\tilde{{\mathbb{V}}})$.

Proof.

The assumptions give us the following commutative diagram, $\forall\ x\in M$, $\forall\ 1\leq q\leq n$ : {diagram} where $\phi_{x}^{n,0}$ is an isomorphism, and $\phi_{x}^{n-q,q}$ is injective, $\forall\ 1\leq q\leq n$. It is easy to see that $\varsigma({\mathbb{V}})=\varsigma(\tilde{{\mathbb{V}}})$ follows from this commutative diagram and the definitions of $\varsigma({\mathbb{V}})$ and $\varsigma(\tilde{{\mathbb{V}}})$. ∎

If $V_{1}$, $V_{2}$ are linear subspaces of a ${\mathbb{C}}$-linear space $V$, for $0\leq p\leq n$, we let $\wedge^{p}V_{1}\wedge^{n-p}V_{2}$ denote the linear subspace of $\wedge^{n}V$ spanned by elements in the set

[TABLE]

Similarly, if $W_{i}$ ($1\leq i\leq n$) are linear subspaces of a ${\mathbb{C}}$-linear space $W$, we let $W_{1}\wedge\cdots\wedge W_{n}$ denote the linear subspace of $\wedge^{n}W$ spanned by elements in the set

[TABLE]

Now suppose $({\mathbb{V}},F^{\cdot}{\mathcal{V}})$ is a ${\mathbb{C}}$-VHS of weight $n$ on $M$, for any $m\geq 1$, we endow the local system $\wedge^{m}{\mathbb{V}}$ a Hodge filtration $F^{\cdot}\wedge^{m}{\mathbb{V}}$ such that $\wedge^{m}{\mathbb{V}}$ becomes a ${\mathbb{C}}$-VHS of weight $mn$ with this Hodge filtration. The Hodge filtration is defined as follows:

$\forall\ 0\leq p\leq mn$, $\forall\ x\in M$, the fiber of the holomorphic bundle $F^{p}\wedge^{m}{\mathbb{V}}$ at $x$ is

[TABLE]

With this definition of Hodge filtration, the Griffiths transversality is easy to verify.

We give the following lemma for the purpose of later use:

Lemma 2.4**.**

Let ${\mathbb{W}}$ be a weight one ${\mathbb{C}}$-VHS on $M$ with associated Higgs bundle $(F=F^{1,0}\oplus F^{0,1},\eta:F^{1,0}\rightarrow F^{0,1}\otimes\Omega_{M})$. The Hodge numbers are $h^{1,0}({\mathbb{W}})=n$, $h^{0,1}({\mathbb{W}})=k-1\leq n$. Suppose there exist $x\in M$ and $v\in T_{x}M$ satisfying that the Higgs map $\eta_{x}(v):F^{1,0}_{x}\rightarrow F^{0,1}_{x}$ is surjective. Then the Griffiths-Yukawa coupling length of $\wedge^{n}{\mathbb{W}}$ is $\varsigma(\wedge^{n}{\mathbb{W}})=k-1$.

Proof.

Let ${\mathbb{V}}=\wedge^{n}{\mathbb{W}}$ be the ${\mathbb{C}}$-VHS of weight $n$ on $M$. Denote the Higgs bundle associated to ${\mathbb{V}}$ by $(E=\oplus E^{p,q},\theta=\oplus\theta^{p,q})$. A direct computation shows that $h^{n-q,q}({\mathbb{V}})=0$, $\forall\ q\geq k$. From this we get $\varsigma({\mathbb{V}})\leq k-1$. In order to prove $\varsigma({\mathbb{V}})\geq k-1$, it suffices to show that the iterated Higgs map

[TABLE]

is nonzero.

By definitions, for any $0\leq q\leq n$, we can identify $E^{n-q,q}_{x}$ with $\wedge^{n-q}F^{1,0}_{x}\wedge^{q}F^{0,1}_{x}\subset\wedge^{n}F_{x}$. With these identifications, we have the following commutative diagram:

[TABLE]

where for $e_{1}\wedge\cdots e_{n}\in\wedge^{n}F_{x}^{1,0}$,

[TABLE]

From this diagram and the explicit expression of $\wedge^{n}\eta_{x}(v)$, it is easy to deduce the non-vanishing of the map $\theta^{k-1}_{x}(v^{k-1})$ from the surjectivity of the map $\eta_{x}(v):F^{1,0}_{x}\rightarrow F^{0,1}_{x}$.

∎

3. ${\mathbb{C}}$-VHS from hyperplane arrangements

Now the meaning of letters in the tuple $(m,n,r,\zeta)$ will be fixed to the end of the paper: $m$, $n$, $r$ are positive integers satisfying the condition (1.0.1), and $\zeta$ is a fixed primitive $r$-th root of unity. If the cyclic group ${\mathbb{Z}}/r{\mathbb{Z}}=<\sigma>$ acts on a ${\mathbb{C}}$-VHS ${\mathbb{V}}$, we denote ${\mathbb{V}}_{(i)}$ as the $i$-th eigen-sub ${\mathbb{C}}$-VHS of ${\mathbb{V}}$, i.e. the sections of ${\mathbb{V}}_{(i)}$ consist sections $s$ of ${\mathbb{V}}$ satisfying $\sigma\cdot s=\zeta^{i}s$.

An ordered arrangement $\mathfrak{A}=(H_{1},\cdots,H_{m})$ of $m$ hyperplanes in ${\mathbb{P}}^{n}$ is in general position if no $n+1$ of the hyperplanes intersect in a point. Let $\mathfrak{M}_{AR}$ denote the coarse moduli space of ordered arrangements of $m$ hyperplanes in ${\mathbb{P}}^{n}$ in general position. As shown in [8], $\mathfrak{M}_{AR}$ can be realized as an open subvariety of the affine space ${\mathbb{C}}^{n(\frac{m}{r}-1)}$ and it admits a natural family $f:\mathcal{X}_{AR}\rightarrow\mathfrak{M}_{AR}$, where each fiber $f^{-1}(\mathfrak{A})$ is the $r$-fold cyclic cover of ${\mathbb{P}}^{n}$ branched along the hyperplane arrangement $\mathfrak{A}$. It is easy to see the crepant resolution process in [7] gives a simultaneous crepant resolution $\pi:\tilde{\mathcal{X}}_{AR}\rightarrow\mathcal{X}_{AR}$ for the family $f$. We denote this smooth projective family of CY manifolds by $\tilde{f}:\tilde{\mathcal{X}}_{AR}\rightarrow\mathfrak{M}_{AR}$.

Let $\mathfrak{M}_{C}$ be the moduli space of ordered distinct $m$ points on ${\mathbb{P}}^{1}$ and $g:\mathcal{C}\rightarrow\mathfrak{M}_{C}$ be the universal family of $r$-fold cyclic covers of ${\mathbb{P}}^{1}$ branched at $m$ distinct points. There is a natural embedding $\mathfrak{M}_{C}\hookrightarrow\mathfrak{M}_{AR}$ (for details, see [8], section 2.3).

We consider the various ${\mathbb{C}}$-VHS attached to the three families $f$, $\tilde{f}$, $g$:

[TABLE]

where $(R^{n}\tilde{f}_{*}{\mathbb{C}})_{pr}$ means the local system on $\mathfrak{M}_{AR}$ whose fiber over $\mathfrak{A}\in\mathfrak{M}_{AR}$ is the primitive $n$-th cohomology of $\tilde{f}^{-1}(\mathfrak{A})$. Note that ${\mathbb{V}}$ is indeed a ${\mathbb{C}}$-VHS, although the family $f$ is not smooth (see [8], section 6). Note also the weights of ${\mathbb{V}}$, $\tilde{{\mathbb{V}}}$ and $\tilde{{\mathbb{V}}}_{pr}$ are $n$, while the weight of ${\mathbb{W}}$ is one.

Since ${\mathbb{Z}}/r{\mathbb{Z}}$ acts naturally on the families $f:\mathcal{X}_{AR}\rightarrow\mathfrak{M}_{AR}$ and $g:\mathcal{C}\rightarrow\mathfrak{M}_{C}$, we have a decomposition of the ${\mathbb{C}}$-VHS into eigen-sub ${\mathbb{C}}$-VHS:

[TABLE]

Proposition 3.1**.**

Notations as above, then

• (1)

as ${\mathbb{C}}$-VHS of weight $n$ on $\mathfrak{M}_{C}$, we have ${\mathbb{V}}_{(1)}\mid_{\mathfrak{M}_{C}}\simeq\wedge^{n}{\mathbb{W}}_{(1)}$;

• (2)

the Hodge numbers of ${\mathbb{W}}_{(1)}$ are: $h^{1,0}({\mathbb{W}}_{(1)})=n$,   $h^{0,1}({\mathbb{W}}_{(1)})=\frac{m}{r}-1$;

• (3)

the Hodge numbers of ${\mathbb{V}}_{(1)}$ are:

[TABLE]

• (4)

$\varsigma(\tilde{{\mathbb{V}}}_{pr})=\varsigma({\mathbb{V}}_{(1)})$;

• (5)

$\varsigma(\tilde{{\mathbb{V}}}_{pr})\leq\frac{m}{r}-1$.

Proof.

For $(1)$, one can see [8], Proposition 6.3.

$(2)$ follows from a standard computation of the Hodge numbers of cyclic covers of ${\mathbb{P}}^{1}$. One can see for example [5], $(2.7)$.

$(3)$ follows from $(1)$ and $(2)$.

$(4)$ follows from Lemma 2.3. Indeed, one can verify directly that the embeddings ${\mathbb{V}}_{(1)}\hookrightarrow{\mathbb{V}}$ and $\tilde{{\mathbb{V}}}_{pr}\hookrightarrow\tilde{{\mathbb{V}}}$ satisfy the hypothesis of Lemma 2.3, so we get $\varsigma({\mathbb{V}}_{(1)})=\varsigma({\mathbb{V}})$, $\varsigma(\tilde{{\mathbb{V}}}_{pr})=\varsigma(\tilde{{\mathbb{V}}})$. One can use Theorem 5.41 in [6] to show the natural morphism ${\mathbb{V}}\rightarrow\tilde{{\mathbb{V}}}$ satisfies the hypothesis of Lemma 2.3, which gives $\varsigma({\mathbb{V}})=\varsigma(\tilde{{\mathbb{V}}})$. Combining these equalities, we get $\varsigma(\tilde{{\mathbb{V}}}_{pr})=\varsigma({\mathbb{V}}_{(1)})$.

$(5)$ follows from $(3)$ and $(4)$.

∎

4. Computations in Jacobian ring

In this section, we keep the notations in section 3. We want to analyse the Higgs maps associated to the universal family $g:{\mathcal{C}}\rightarrow\mathfrak{M}_{C}$ in some detail. Recall $\mathfrak{M}_{C}$ is the coarse moduli space of ordered pairwise distinct $m$ points in ${\mathbb{P}}^{1}$. It is well known that $\mathfrak{M}_{C}$ can be identified with a Zariski open subset $U$ of ${\mathbb{C}}^{m-3}$ via the map:

[TABLE]

where $[z_{0}:z_{1}]$ are the homogeneous coordinates on ${\mathbb{P}}^{1}$. We fix this identification and view $\mathfrak{M}_{C}$ as a Zariski open subset of ${\mathbb{C}}^{m-3}$.

For $a=(a_{1},\cdots,a_{m-3})\in\mathfrak{M}_{C}\subset{\mathbb{C}}^{m-3}$, let $C=g^{-1}(a)$ be the fiber over $a$ of the universal family ${\mathcal{C}}$, then $C$ is the $r$-fold cyclic cover of ${\mathbb{P}}^{1}$ branched at the $m$ points $[1:0],[1;1],[0:1],[1:a_{1}],\cdots,[1:a_{m-3}]$. We let $Y$ denote the smooth curve which is the complete intersection of the $m-2$ hypersurfaces in ${\mathbb{P}}^{m-1}$ defined by the equations:

[TABLE]

Here $[y_{0}:\cdots:y_{m-1}]$ are the homogeneous coordinates on ${\mathbb{P}}^{m-1}$. $Y$ is called the Kummer cover of $C$, and when $a$ varies in $\mathfrak{M}_{C}$, the Kummer covers of $g^{-1}(a)$ form a family of curves over $\mathfrak{M}_{C}$.

Let $N=\oplus_{j=0}^{m-1}{\mathbb{Z}}/r{\mathbb{Z}}$. Consider the following group

[TABLE]

We define a natural action of $N$ on $Y$. $\forall\alpha=(a_{0},\cdots,a_{m-1})\in N$, the action of $\alpha$ on $Y$ is induced by

[TABLE]

Recall $\zeta$ is a fixed $r$-th primitive root of unity.

Proposition 4.1**.**

The following statements hold:

• (1)

The map $\pi_{1}:Y\rightarrow{\mathbb{P}}^{1}$, $[y_{0}:\cdots:y_{m-1}]\mapsto[y_{0}^{r}:y_{1}^{r}]$ defines a cover of degree $r^{m-1}$.

• (2)

$C\simeq Y/N_{1}$.

• (3)

There exists a natural isomorphism of rational Hodge structures $H^{1}(C,{\mathbb{Q}})\simeq H^{1}(Y,{\mathbb{Q}})^{N_{1}}$, where $H^{1}(Y,{\mathbb{Q}})^{N_{1}}$ denotes the subspace of invariants under $N_{1}$.

Proof.

$(1)$ can be verified directly.

$(2)$: By $(1)$, one can verify $Y/N_{1}$ is a $r$-fold cyclic cover of ${\mathbb{P}}^{1}$ branched at the $m$ points $[1:0],[1;1],[0:1],[1:a_{1}],\cdots,[1:a_{m-3}]$. Then $C\simeq Y/N_{1}$ follows from the uniqueness of this kind of covers.

$(3)$ follows from $(2)$ directly. ∎

Recall ${\mathbb{W}}=R^{1}g_{*}{\mathbb{C}}$ is the weight one ${\mathbb{C}}$-VHS coming from the universal family $g:{\mathcal{C}}\rightarrow\mathfrak{M}_{C}$, and under the natural ${\mathbb{Z}}/r{\mathbb{Z}}$-action, ${\mathbb{W}}_{(1)}$ is the first eigen-sub ${\mathbb{C}}$-VHS of ${\mathbb{W}}$. Let $(F=F^{1,0}\oplus F^{0,1},\eta:F^{1,0}\rightarrow F^{0,1}\otimes\Omega_{\mathfrak{M}_{C}})$ be the Higgs bundle associated to ${\mathbb{W}}_{(1)}$. At the point $a=(a_{1},\cdots,a_{m-3})\in\mathfrak{M}_{C}\subset{\mathbb{C}}^{m-3}$, we can identify the fiber $F^{1,0}_{a}$ with $H^{1,0}(C,{\mathbb{C}})_{(1)}$, where $H^{1,0}(C,{\mathbb{C}})_{(1)}:=\{\alpha\in H^{1,0}(C,{\mathbb{C}})|\sigma^{*}\alpha=\zeta\alpha\}$ is the first eigen subspace of $H^{1,0}(C,{\mathbb{C}})$ under the natural ${\mathbb{Z}}/r{\mathbb{Z}}=<\sigma>$-action. Similarly, we can identify $F^{0,1}_{a}$ with $H^{0,1}(C,{\mathbb{C}})_{(1)}$. Under these identifications, it is a standard fact that we have the following commutative diagram (see e.g., [11], Theorem 10.21):

[TABLE]

where $\psi_{a}$ means the composition of the Kodaira-Spencer map $T_{a}\mathfrak{M}_{C}\rightarrow H^{1}(C,TC)$ and the cup product $H^{1,0}(C,{\mathbb{C}})_{(1)}\otimes H^{1}(C,TC)\rightarrow H^{0,1}(C,{\mathbb{C}})_{(1)}$.

Since the isomorphism in Proposition 4.1, $(3)$ is equivariant with respect to the ${\mathbb{Z}}/r{\mathbb{Z}}$-action (note ${\mathbb{Z}}/r{\mathbb{Z}}=N/N_{1}$ acts naturally on $H^{1}(Y,{\mathbb{Q}})^{N_{1}}$), we can identify the corresponding eigen subspaces: $H^{1,0}(C,{\mathbb{C}})_{(1)}=H^{1,0}(Y,{\mathbb{C}})^{N_{1}}_{(1)}$, $H^{0,1}(C,{\mathbb{C}})_{(1)}=H^{0,1}(Y,{\mathbb{C}})^{N_{1}}_{(1)}$. Under these identifications, we have the commutative diagram:

[TABLE]

where $\tilde{\psi}_{a}$ is defined similarly as $\psi_{a}$.

In order to represent $\tilde{\psi}_{a}$ more explicitly, we use the tool of Jacobian ring. It is constructed as follows. In the polynomial ring of $2m-2$ variables ${\mathbb{C}}[\mu_{0},\cdots,\mu_{m-3},y_{0},\cdots,y_{m-1}]$, consider the polynomial

[TABLE]

where

[TABLE]

Let $J=<\frac{\partial F}{\partial\mu_{i}},\frac{\partial F}{\partial y_{j}}\mid 0\leq i\leq m-3,0\leq j\leq m-1>$ be the ideal of ${\mathbb{C}}[\mu_{0},\cdots,\mu_{m-3},y_{0},\cdots,y_{m-1}]$ generated by the partial derivatives of $F$. Define the Jacobian ring to be

[TABLE]

There is a natural bigrading on the polynomial ring ${\mathbb{C}}[\mu_{0},\cdots,\mu_{m-3},y_{0},\cdots,y_{m-1}]$, that is: the $(p,q)-$part ${\mathbb{C}}[\mu_{0},\cdots,\mu_{m-3},y_{0},\cdots,y_{m-1}]_{(p,q)}$ is linearly spanned by the monomials $\Pi_{i=0}^{m-3}\mu_{i}^{\alpha_{i}}\Pi_{j=0}^{m-1}y_{j}^{\beta_{j}}$ with $\sum_{i=0}^{m-3}\alpha_{i}=p$, $\sum_{j=0}^{m-1}\beta_{j}=q$. Since the ideal $J$ is a homogeneous ideal, there is a naturally induced bigrading on $R$, written as $R=\oplus_{p,q\geq 0}R_{(p,q)}$.

The group $N=\oplus_{j=0}^{m-1}{\mathbb{Z}}/r{\mathbb{Z}}$ acts on $R$ through $y_{0},\cdots,y_{m-1}$. Explicitly, recall $\zeta$ is a fixed primitive $r-$th root of unit, $\forall\alpha=(\alpha_{j})\in N$, we define the action of $\alpha$ on $R$ by

[TABLE]

It is obviously that the action of $N$ on $R$ preserves the bigrading. Let $R_{(p,q)}^{N}$ be the $N-$invariant part of $R_{(p,q)}$, then we have the decomposition of the $N-$invariant subring: $R^{N}=\oplus_{p,q\geq 0}R_{(p,q)}^{N}$. Recall $N_{1}=Ker(\oplus_{j=0}^{m-1}{\mathbb{Z}}/r{\mathbb{Z}}\xrightarrow{\sum}{\mathbb{Z}}/r{\mathbb{Z}})$ is the kernel subgroup of $N=\oplus_{j=0}^{m-1}{\mathbb{Z}}/r{\mathbb{Z}}$ under the summation homomorphism.

Proposition 4.2**.**

The following statements hold:

• (1)

There are isomorphisms

[TABLE]

• (2)

Let $(x_{1},\cdots,x_{m-3})$ be the coordinates on $\mathfrak{M}_{C}\subset{\mathbb{C}}^{m-3}$, define a map

[TABLE]

Under this map and the isomorphisms in $(1)$, we have a commutative diagram

[TABLE]

where the lower horizontal map is the natural multiplication in $R$.

Proof.

$(1)$ follows from [9], Corollary 2.5 and its proof.

$(2)$ follows from $(1)$ and [9], Proposition 2.6. ∎

For $\beta=\sum_{i=1}^{m-3}\lambda_{i}\mu_{i}y_{1}^{r}\in R_{(1,r)}$, let $\psi_{\beta}$ be the following multiplication homomorphism

[TABLE]

Now our key computations are included in the following

Proposition 4.3**.**

For a generic $\lambda=(\lambda_{1},\cdots,\lambda_{m-3})\in{\mathbb{C}}^{m-3}$, the homomorphism $\psi_{\beta}$ is surjective.

Proof.

We first analyse the relations in $R$, in order to obtain bases of the ${\mathbb{C}}$-linear spaces $R^{N}_{(0,mr-m-2r)}$ and $R^{N}_{(1,mr-m-r)}$.

By the definition, the following relations hold in $R$:

[TABLE]

From these relations, it is easy to see the ${\mathbb{C}}$-linear space $R^{N}_{(0,mr-m-2r)}$ is linearly spanned by elements in the set

[TABLE]

By Proposition 3.1, (2), Proposition 4.1, (2) amd Proposition 4.2, (1), we can deduce

[TABLE]

This implies that $\{y_{0}^{br}y_{1}^{cr}\mid b,c\in{\mathbb{Z}}_{\geq 0},\ b+c=m-\frac{m}{r}-2\}$ is a ${\mathbb{C}}-$basis of $R^{N}_{(0,mr-m-2r)}$.

From the relations (4.3.1), we get the following relations in $R$:

[TABLE]

The relations above imply directly that the ${\mathbb{C}}$-linear space $R^{N}_{(1,mr-m-r)}$ is linearly spanned by elements in the set $\{\mu_{i}y_{1}^{mr-m-r}\mid i=1,2,\cdots,m-3.\}$. Next we want to get a ${\mathbb{C}}$-basis of $R^{N}_{(1,mr-m-r)}$ from this set.

By the relations (4.3.2), we know in $R$,

[TABLE]

Then we get $\forall e\geq 0$,

[TABLE]

From the relations (4.3.2) again, we get $\forall e\geq 0$,

[TABLE]

From the identities (4.3.3) and (4.3.4), we get $\forall e\geq 0$,

[TABLE]

From this we get the following identity

[TABLE]

Note that $a\in\mathfrak{M}_{C}$ implies that $\forall\ 1\leq i\leq m-3$, $a_{i}\neq 0,1$ and $\forall\ i\neq j$, $a_{i}\neq a_{j}$. So that from the matrix equality above, we know that any $m-3-(m-\frac{m}{r}-2)=\frac{m}{r}-1$ distinct elements in $\{\mu_{i}y_{1}^{mr-m-r}\mid i=1,2,\cdots,m-3.\}$ form a ${\mathbb{C}}-$basis of $R^{N}_{(1,mr-m-r)}$. We can represent the map $\psi_{\beta}$ as follows:

[TABLE]

From this matrix representation, we can easily see that for a generic $\lambda=(\lambda_{1},\cdots,\lambda_{m-3})\in{\mathbb{C}}^{m-3}$, the homomorphism $\psi_{\beta}$ is surjective. ∎

Now we can prove our main theorem. Recall the notations from section 3. We have:

Theorem 4.4**.**

$\varsigma(\tilde{{\mathbb{V}}}_{pr})=\frac{m}{r}-1$.

Proof.

In Proposition 3.1, $(5)$, we have established the inequality $\varsigma(\tilde{{\mathbb{V}}}_{pr})\leq\frac{m}{r}-1$. So it suffices to prove $\varsigma(\tilde{{\mathbb{V}}}_{pr})\geq\frac{m}{r}-1$.

Combining the commutative diagrams (4.1.1), (4.1.2), (4.2.1) and Proposition 4.3 together shows that for any $a\in\mathfrak{M}_{C}$, for a generic tangent vector $v\in T_{a}\mathfrak{M}_{C}$, the Higgs map $\eta_{a}(v):F_{a}^{1,0}\rightarrow F_{a}^{0,1}$ associated to the ${\mathbb{C}}$-VHS ${\mathbb{W}}_{(1)}$ is surjective. This and Proposition 3.1, $(2)$ imply ${\mathbb{W}}_{(1)}$ satisfies the hypothesis of Lemma 2.4. Applying this lemma to ${\mathbb{W}}_{(1)}$, we get $\varsigma(\wedge^{n}{\mathbb{W}}_{(1)})=\frac{m}{r}-1$. This gives $\varsigma({\mathbb{V}}_{(1)}|_{\mathfrak{M}_{C}})=\frac{m}{r}-1$ by Proposition 3.1, $(1)$. By the definition of the length of Griffith-Yukawa coupling, $\varsigma({\mathbb{V}}_{(1)})\geq\varsigma({\mathbb{V}}_{(1)}|_{\mathfrak{M}_{C}})$. So finally we get the desired inequality $\varsigma(\tilde{{\mathbb{V}}}_{pr})\geq\frac{m}{r}-1$ by combining the (in)equalities above and Proposition 3.1, $(4)$.

∎

Acknowledgements This work is supported by Wu Wen-Tsun Key Laboratory of Mathematics, Chinese Academy of Sciences. The first named author is supported by National Natural Science Foundation of China (Grant No. 11622109, No. 11721101).