Remarks on the Griffiths infinitesimal invariant of algebraic curves

Let $C$ be a non-hyperelliptic smooth algebraic curve of genus $g\geq 3$. In [Cer83] Ceresa showed there is a canonically defined homologous algebraic cycle in $\mathrm{CH}^{g-1}(\mathrm{Jac}(C))$ which is not algebraically trivial, verifies the non-triviality of the corresponding Griffiths group. This cycle is known as Ceresa cycle.

Properties of Ceresa cycle is a board and active research topic. In this article we focus on the complex geometric and Hodge theoretic aspect. In particular, we focus on the Griffiths’ Abel-Jacobi map (3.18) associated to the Ceresa cycle. When the curve varies in a family, a natural tool in Hodge theory to study deformation of the Ceresa cycle is the associated admissible normal function, which we denoted as $\nu_{c}$.

Given a family of curves over a quasi-projective base, we are particularly interested in the positive-dimensional locus over which the Ceresa cycle is torsion. Such locus must be contained in the positive-dimensional torsion locus of $\nu_{c}$ which is shown to be algebraic by [KT24]. To obtain a better picture of this locus, One way is to study the Griffiths infinitesimal invariant $\delta\nu_{c}$ associated to the normal function $\nu_{c}$. $\delta\nu$ is intuitively the ”derivative” of normal functions and could be study in various ways. For example, in [CP95] Collino-Pirola studied $\delta\nu_{c}$ over the moduli of genus $3$ curves by using the adjunction morphism and produced several interesting results.

In this paper we are going to apply Collino-Pirola’s adjunction formula to $\delta\nu_{c}$ for the genus $4$ case. Firstly, over all rank $1$ deformations we search vanishing criteria for $\delta\nu_{c}$. Secondly, we study a specific family $\mathcal{C}\rightarrow B$ of genus $4$ curves. We will show $\delta\nu_{c}$ has maximal rank (See Section 3 for definitions) over this family, and moreover give nice control on the locus on which $\nu_{c}$ drops rank.

The first main result of this paper is to generalize part of Collino-Pirola’s results to the genus $4$ case. Besides the Ceresa normal function $\nu_{c}$, there is another canonical normal function associated to family of genus $4$ curves defined by Griffiths in [Gri83, Sec. 6], which we denote as $\nu_{0}$¹¹1Strictly speaking, $\nu_{0}$ is only well-defined over $\mathcal{M}_{4}[2]$, the moduli space with an additional level-$2$ structure, but $\delta\nu_{0}$ is well-defined for any $C\in\mathcal{M}_{4}$..

The main strategy for the proof is to find relations between the Griffiths infinitesimal invariants and the canonical embedding. In particular, we are able to realize $\delta\nu_{c}$ as a certain projective hyperplane cutting off the canonical curve, generalize a similar result of [CP95, Thm 4.2.4] for the genus $3$ case.

While coming to general first-order deformations, the two normal functions $\nu_{0}$ and $\nu_{c}$ behave very differently. In the second part of the paper we study a certain family of genus $4$ curves which can be realized as a triple cover of the projective line branched over $6$ double points. We denote this family of curves as $\mathcal{C}\rightarrow B$ (See Section 5). Our second main theorem is about the behaviors of $\nu_{0}$ and $\nu_{c}$ over this family:

The equation defining the subbundle of $TB$ controlling $TX$ is found by explicitly calculating the Kodaira-Spencer classes of the family, see Theorem 5.5.

To show $\nu_{c}$ has maximal rank over this family, we need an alternative description on the rank of normal function which is the third main result of this paper:

The proof of algebraicity of these $r_{c}$-dimensional submanifolds requires o-minimal geometry and its application in Hodge theory. We give a brief survey in Appendix 6. Next we will consider the algebraic monodromy group of the variation over each of these subvarieties. Theorem 1.2 will be concluded from a Zariski density argument.

Normal functions over the moduli space of (stable) curves including the Ceresa normal function has been extensively studied, we refer readers to [Hai13] for a comprehensive overview.

Rank of the Ceresa normal function $\nu_{c}$ over $\mathcal{M}_{g}$ for $g\geq 3$ was proven to be maximal, independently by Gao-Zhang [GZ24] and Hain [Hai24]. As a generalization of [CP95], Pirola-Zucconi showed in [PZ03] that over a dimension-$\geq 2g-1$ subvariety of $\mathcal{M}_{g}$, the Griffiths infinitesimal invariant of $\nu_{c}$ must not vanish unless the subvariety is the hyperelliptic locus.

One may also interested in finding specific families of curves on which $\nu_{c}$ drops rank, including those (positive-dimensional) families on which the Ceresa cycle itself is torsion. Some known examples include [Lat23], [QZ24]. Relation between the Ceresa cycle themselves and their Abel-Jacobi image is also an interesting topic, see for example [LS24].

The author is aware of the preprint [GZ24] which contains some results similar to the Appendix 6. In particular, the maximal rank part of Theorem 1.2 can also be concluded from [GZ24, Cor. 1.8].

Let $\mathcal{V}\rightarrow S$ be an integral polarized variation of Hodge structures ($\mathbb{Z}$-PVHS) of weight $-1$ with type $(V_{\mathbb{Z}},Q,h^{p,q})$ defined over a quasi-projective base $S$. The intermediate Jacobian associated to $\mathcal{V}\rightarrow S$ is $\mathcal{J}(\mathcal{V}):=\frac{\mathcal{V}_{\mathbb{C}}}{\mathcal{F}^{0}\mathcal{V}_{\mathbb{C}}+\mathcal{V}_{\mathbb{Z}}}$. We also define

(3.1)

$$\mathcal{J}_{h}(\mathcal{V}):=\mathrm{ker}\{\overline{\nabla}:\mathcal{J}(\mathcal{V})\rightarrow\frac{\mathcal{V}}{\mathcal{F}^{0}}\otimes\Omega_{S}^{1}\}$$

as the horizontal part of $\mathcal{J}(\mathcal{V})$.

In general, we only care about normal functions which are admissible. The precise definition can be found at for example [Sai96]. Briefly speaking, admissibility means the existence of relative monodromy weight filtrations and nilpotent orbit theorem ([Sch73]) along the boundary strata.

Let $\mathcal{X}\rightarrow S$ be a family of smooth projective varieties of dimension $n$ and $\mathcal{Z}\subset\mathcal{X}$ be a subvariety of codimension $r$ such that for each $s\in S$, $\mathcal{Z}_{s}:=\mathcal{Z}\cap X_{s}\in\mathrm{CH}^{r}_{\mathrm{hom}}(X_{s})$. For each $s\in S$ the Griffiths Abel-Jacobi map associates $Z_{s}$ an element in $J(H_{s}):=\frac{H_{s}(\mathbb{C})}{F^{0}H_{s}+H_{s}(\mathbb{Z})}$ where $H_{s}:=H^{2n-2r+1}(X_{s})[n-r+1]$ defined as follows.

Use the identification:

(3.2)

$$J(H_{s}):=\frac{H_{s}(\mathbb{C})}{F^{0}H_{s}+H_{s}(\mathbb{Z})}\cong\frac{(F^{0}H_{s})^{\vee}}{H_{s}(\mathbb{Z})},$$

there is the defining formula

(3.3)

$$\nu(\omega(s))=\int_{\Gamma_{s}}\omega_{s}$$

where $\omega(s)\in F^{0}H_{s}$ and $\Gamma_{s}\subset X_{s}$ is a $(2n-2r+1)$-chain satisfies $\partial\Gamma_{s}=\mathcal{Z}_{s}$, both vary smoothly locally around $s$.

By patching together all intermediate Jacobians $J(H_{s})$ and the Abel-Jacobi image given by $Z_{s}$, we obtain a normal function $\nu$ as a section of $\mathcal{J}\rightarrow S$. Such a normal function is said to be of geometric origin.

In this subsection we fix a $\mathbb{Z}$-PVHS $\mathcal{V}\rightarrow S$ of weight $-1$ and an admissible normal function $\nu\in H^{0}(S,\mathcal{J}(\mathcal{V}))$. For any $s\in S$, we have the (split) exact sequence:

(3.4)

$$0\rightarrow V_{s}/F^{0}V_{s}\rightarrow T_{\nu(s)}\mathcal{J}(\mathcal{V})\rightarrow T_{s}S\rightarrow 0$$

Let $\pi_{v}$ be the natural projection $T_{\nu(s)}\mathcal{J}(\mathcal{V})\rightarrow V_{s}/F^{0}V_{s}$.

The following lemma is an immediate consequence from the definition:

There will be more discussions about the normal function rank in Section 6.

Associated to a normal function $\nu$ is a differential invariant of $\nu$ studied by Griffiths [Gri83], denoted as $\tilde{\delta}\nu$ which is described as follows.

For $s\in S$, choose any local lift $v$ of $\nu$ around $s$, we have

(3.8)

$$\nabla v\in\mathrm{Ker}\{\mathcal{F}^{-1}\otimes\Omega^{1}_{S}\xrightarrow{\nabla}\Omega^{2}_{S}\otimes\mathcal{F}^{-2}\}.$$

Consider the Koszul complex:

(3.9)

$$\mathcal{F}^{0}\xrightarrow{\nabla}\Omega^{1}_{s}\otimes\mathcal{F}^{-1}\xrightarrow{\nabla}\Omega^{2}_{s}\otimes\mathcal{F}^{-2},$$

since different choices of $v$ are differed by elements in $\mathcal{F}^{0}$ and $\nabla$-flat sections, $\tilde{\delta}\nu_{s}$ gives a well-defined element in the Koszul cohomology group:

(3.10)

$$\tilde{\delta}\nu_{s}\in\tilde{\mathbb{H}}_{s}:=\frac{\mathrm{Ker}\{\Omega^{1}_{s}\otimes F_{s}^{-1}\xrightarrow{\nabla}\Omega^{2}_{s}\otimes F_{s}^{-2}\}}{\mathrm{Img}\{F_{s}^{0}\xrightarrow{\nabla}\Omega^{1}_{s}\otimes F_{s}^{-1}\}}$$

This is usually called the (full) Griffiths infinitesimal invariant of $\nu$ at $s\in S$.

In practice a weaker invariant called the first Griffiths infinitesimal invariant $\delta\nu$ is more suitable for calculation. Consider the graded part of (3.9):

(3.11)

$$H_{s}^{0,-1}\xrightarrow{\overline{\nabla}}\Omega^{1}_{s}\otimes H_{s}^{-1,0}\xrightarrow{\overline{\nabla}}\Omega^{2}_{s}\otimes H_{s}^{-2,1},$$

The first Griffiths infinitesimal invariant $\delta\nu_{s}$ gives a well-defined element in:

(3.12)

$$\delta\nu_{s}\in\mathbb{H}_{s}:=\frac{\mathrm{Ker}\{\Omega^{1}_{s}\otimes H_{s}^{-1,0}\xrightarrow{\overline{\nabla}}\Omega^{2}_{s}\otimes H_{s}^{-2,1}\}}{\mathrm{Img}\{H_{s}^{0,-1}\xrightarrow{\overline{\nabla}}\Omega^{1}_{s}\otimes H_{s}^{-1,0}\}}.$$

By considering the dual complex of (3.11):

(3.13)

$$\wedge^{2}T_{s}\otimes H_{s}^{1,-2}\rightarrow T_{s}\otimes H_{s}^{0,-1}\xrightarrow{}H_{s}^{-1,0},$$

$\overline{\delta}\nu_{s}$ can also be seen as a linear function on:

(3.14)

$$\mathbb{H}_{s}^{\vee}:=\frac{\mathrm{Ker}\{T_{s}\otimes H_{s}^{0,-1}\xrightarrow{}H_{s}^{-1,0}\}}{\mathrm{Img}\{\wedge^{2}T_{s}\otimes H_{s}^{1,-2}\xrightarrow{}T_{s}\otimes H_{s}^{0,-1}\}}$$

given by:

(3.15)

$$\delta\nu_{s}(\sum\xi_{i}\otimes w_{i})=\sum\langle\nabla_{\xi_{i}}v,w_{i}\rangle$$

where $v$ is any local lift of $\nu$ around $s$.

The Griffiths infinitesimal invariant is a powerful tool to study the local behavior of a normal function. In particular, we have:

For the rest of this paper, by the Griffiths infinitesimal invariant we mean the first Griffiths infinitesimal invariant.

Fix a smooth genus $g$ algebraic curve $C$ and a marked point $p_{0}\in C$. The image of Abel-Jacobi map $C\rightarrow J(C),\ p\rightarrow\int_{p_{0}}^{p}$ in $J(C)$ gives an algebraic cycle $[C]-[C^{-}]\in Z^{g-1}(J(C))$. In [Cer83] Ceresa showed for a general curve $C$,

(3.16)

$$0\neq[C]-[C^{-}]\in\mathrm{Griff}^{g-1}(J(C)),$$

where

(3.17)

$$\mathrm{Griff}^{g-1}(J(C)):=\frac{\mathrm{Ch}^{g-1}_{\mathrm{hom}}(J(C))}{\mathrm{Ch}^{g-1}_{\mathrm{alg}}(J(C))}$$

is the Griffiths group. The Griffiths Abel-Jacobi map (3.3) gives:

(3.18)

$$\mathrm{AJ}([C]-[C^{-}])\in J^{g-1}(J(C))\simeq\frac{F^{2}H^{3}(J(C))^{\vee}}{H_{3}(J(C),\mathbb{Z})}.$$

The image depends on the base point $p_{0}\in C$, but its projection onto the primitive factor

(3.19)

$$PJ^{g-1}(J(C)):=\frac{F^{2}PH^{3}(J(C),C)^{\vee}}{H_{3}(J(C),\mathbb{Z})}$$

does not ([CP95], Prop. 2.2.1). Therefore, associated to the universal curve of genus $g$: $\mathcal{M}_{g,1}\rightarrow\mathcal{M}_{g}$ is a $\mathbb{Z}$-PVHS $\wedge^{3}_{0}\mathcal{V}\rightarrow\mathcal{M}_{g}$ with fibers being the primitive cohomology $PH^{3}(J(C),C)(2)$ and associated admissible normal function $\nu_{c}$ given by the Ceresa cycle. By the existence of the Kuranishi family, $\delta\nu_{c,C}$ is well-defined and called the Griffiths infinitesimal invariant of $C$.

More precisely, the family of intermediate Jacobians $\mathcal{PJ}\rightarrow\mathcal{M}_{g}$ has fiber at $C$ given by (3.19). Therefore $\delta\nu_{c,C}$ gives a linear function on $\overline{\mathbb{H}}_{s}$ defined by equation (3.14).

We introduce a result given by Collino-Pirola which is useful on computing the infinitesimal invariant. Let $\xi\in H^{1}(C,T_{C})$ be a transformation with $\mathrm{dim}(W_{\xi})\geq 2$, and $\sigma_{1},\sigma_{2}\in H^{1,0}(C)$ are $2$ independent forms annihilated by $\xi$.

The proof relies on the construction of the adjunction map defined in [CP95], we will not cover it in details here.

Since any quadric hypersurface $Q\subset\mathbb{P}^{3}$ is isomorphic to $\mathbb{P}^{1}\times\mathbb{P}^{1}$, the two rulings $l_{1},l_{2}$ of $Q$ cut off the canonical curve $C=Q\cap V$ at the divisor

(4.1)

$$\pm D_{0}(C):=\sum_{1\leq i\leq 3}[p_{i}]-\sum_{1\leq i\leq 3}[q_{i}]$$

which is homologous to zero and whose image in $\mathrm{CH}^{1}_{\mathrm{hom}}(C)$ is well-defined up to a sign. Therefore, $\nu_{0}$ is just the normal function associated to this family of cycles:

(4.2)

$$\nu_{0}:\mathcal{M}_{4}[2]\rightarrow\mathcal{J}(\mathcal{V})\simeq\frac{\mathcal{V}}{\mathcal{F}^{0}\mathcal{V}+\mathcal{V}_{\mathbb{Z}}}.$$

In [Gri83] Griffiths studied the infinitesimal invariant $\delta\nu_{0}$ over $\mathcal{M}_{4}$. Fix a non-hyperelliptic curve $C$, we have $T_{C}M_{4}\simeq H^{1}(C,T_{C})$ via the Kodaira-Spencer map. Consider the set of rank $1$ deformations of $C$:

$\mathcal{C}_{\Delta}\rightarrow\Delta$, $\Delta:=\mathrm{Spec}\{\frac{\mathbb{C}(\epsilon)}{\epsilon^{2}}\}$, $\mathcal{C}_{0}\simeq C$.

Such that the image of the Kodaira-Spencer class $\xi\in H^{1}(C,T_{C})$ in $\mathrm{Hom}(H^{1,0},H^{0,1})$ has rank $1$. Using the identification introduced in the beginning of Sec. 2, this set may be identified with the quadric hypersurface $Q\subset\mathbb{P}^{3}$.

Indeed, he showed that $\delta\nu_{0}$ defines an element in $H^{0}(Q,\mathcal{O}_{Q}(3))$ whose vanishing locus is exactly the canonical curve $C$.

In this subsection we consider the Ceresa normal function $\nu_{c}$ over $\mathcal{M}_{4}$ and its infinitesimal invariant $\delta\nu_{c}$. In particular, we show $\delta\nu_{c}$ and $\delta\nu_{0}$ has the same vanishing locus on rank $1$ deformations.

Let $\mathcal{M}_{4}\rightarrow\mathcal{A}_{4}$ be the period map where $\mathcal{A}_{4}$ is the moduli space of principally polarized abelian varieties of dimension $4$. The image of $\mathcal{M}_{4}$ is known as the Jacobian divisor of $\mathcal{A}_{4}$. Let $C\in\mathcal{M}_{4}$ and $T_{C}\mathcal{A}_{4}\simeq\mathrm{Hom}(H^{1,0}(C),H^{0,1}(C))$. Denote $\{P^{p,q},\ p+q=-1\}$ as the Hodge decomposition of $PH^{3}(J(C),C)[2]$. We have the commutative diagram:

(4.3)

Following the computation in [Nor93, Sec. 7], the bottom row of (4.3) is exact, while the top row has the cohomology group $\mathbb{H}_{C}^{\vee}$ defined in (3.14) (We regard the curve $C$ as a point in $\mathcal{M}_{4}$).

Let $\mathcal{K}:=\mathrm{Ker}\{T_{C}\mathcal{M}_{4}\otimes P^{1,-2}\rightarrow P^{0,-1}\}$. It follows that $\mathcal{K}\simeq\mathrm{Sym}^{3}(H^{0,1}(C))$. Choose an orthonormal basis $\{\omega_{i},\ 1\leq i\leq 4\}$ of $H^{1,0}(C)$. The equivalence is given on decomposible tensors by

(4.4)

$$\overline{d\omega_{1}}\cdot\overline{d\omega_{2}}\cdot\overline{d\omega_{3}}\in\mathrm{Sym}^{3}(H^{0,1}(C))\rightarrow\sum_{\sigma\in S_{3}}\overline{d\omega_{\sigma(1)}}\cdot\overline{d\omega_{\sigma(2)}}\otimes\star\omega_{\sigma(3)}\in\mathcal{K}$$

where $\star\omega$ is the Hodge-star operator.

Take any $0\neq\eta\in T_{C}\mathcal{A}_{4}-T_{C}\mathcal{M}_{4}$. There is a well-defined map $\rho_{\eta}:\mathcal{K}\rightarrow\mathbb{H}_{C}^{\vee}$ defined by:

(4.5)

$$\sum\xi_{i}\otimes\omega_{i}\in\mathcal{K}\xrightarrow{\rho_{\eta}}\sum\xi_{i}\otimes\nabla_{\eta}\omega_{i}\in\mathbb{H}_{C}^{\vee}.$$

Since different choice of $\eta$ are differed by a constant multiple or an element in $T_{C}M_{4}$, the map $\rho_{\eta}$ with image in $\mathbb{P}\mathbb{H}_{C}^{\vee}$ is canonically defined. By pulling back we may realize $\delta\nu_{c,C}$ as a linear functional on $\mathbb{P}\mathcal{K}\simeq\mathbb{P}\mathrm{Sym}^{3}(H^{0,1}(C))$.

Notice that for any rank $1$ transformation $\xi=\overline{d\omega}\cdot\overline{d\omega}$, take $\{\sigma_{1},\sigma_{2},\sigma_{3}\}$ as an orthonormal basis of $W_{\xi}$, the maps (4.4) and (4.5) are read as:

(4.6)

$$\overline{d\omega}\cdot\overline{d\omega}\cdot\overline{d\omega}\rightarrow\overline{d\omega}\cdot\overline{d\omega}\otimes\sigma_{1}\wedge\sigma_{2}\wedge\sigma_{3}\xrightarrow{\rho_{\eta}}\sum\overline{d\omega}\cdot\overline{d\omega}\otimes\nabla_{\eta}(\sigma_{1}\wedge\sigma_{2}\wedge\sigma_{3}).$$

The following proposition together with Theorm 4.1 implies the main Theorem 1.1.

For $u\in B$ there is a canonical basis of $H^{0}(C_{u},\Omega^{1}_{C_{u}})$ compatible with the eigenspace decomposition of the induced action from $\rho$:

(5.4)

$$H^{0}(C_{u},\Omega^{1}_{C_{u}})=\langle\frac{dx}{y}\rangle_{e^{\frac{4\pi i}{3}}}\oplus\langle\frac{dx}{y^{2}},\frac{xdx}{y^{2}},\frac{x^{2}dx}{y^{2}}\rangle_{e^{\frac{2\pi i}{3}}}=:\langle\omega_{0}\rangle\oplus\langle\omega_{1},\omega_{2},\omega_{3}\rangle.$$

Take this ordered basis and consider the corresponding canonical embedding $C_{u}\xrightarrow{\phi}\mathbb{P}(H^{0,1}(C_{u}))\simeq\mathbb{P}^{3}$, the canonical image $\phi(C_{u})$ lies on the quadric surface

(5.5)

$$Q:=\{[z_{0}:z_{1}:z_{2}:z_{3}]\in\mathbb{P}^{3}\ |\ z_{2}^{2}-z_{1}z_{3}=0\}.$$

The two rulings are given by the lines:

(5.6)		$\displaystyle L(t_{1}):=\{z_{2}+z_{1}=t_{1}(z_{3}-z_{1})\}\cap\{z_{1}=t_{1}(z_{2}-z_{1})\}$
(5.7)		$\displaystyle L(t_{2}):=\{z_{2}+z_{1}=t_{2}z_{1}\}\cap\{z_{3}-z_{1}=t_{2}(z_{2}-z_{1})\}$

for $t_{i}\in\mathbb{P}^{1}$. Take $t_{1},t_{2}$ such that

(5.8)

$$\frac{1}{t_{1}}+1=t_{2}-1,$$

it follows that $[L(t_{1})\cap C_{u}]=[L(t_{2})\cap C_{u}]$. Hence for any $u\in B$, $t_{1},t_{2}\in\mathbb{P}^{1}$, the cycle $D_{0}(C_{u})=[L(t_{1})\cap C_{u}]-[L(t_{2})\cap C_{u}]$ on $C_{u}$ is rationally trivial²²2This also implies $\nu_{0}$ can be defined over $\mathcal{C}\rightarrow B$ without adding any level structures..

To study the Ceresa normal function of the family, we need to compute the explicit Kodaira-Spencer image of $T_{u}B$. Clearly

(5.9)

$$T_{u}B=\{a_{1}\partial_{1}+a_{2}\partial_{2}+a_{3}\partial_{3},\ \partial_{j}:=\frac{\partial}{\partial u_{j}},\ a_{j}\in\mathbb{C}\}\simeq\mathbb{C}^{3}.$$

To find the Kodaira-Spencer image of $\partial_{j}$, we notice that

(5.10)		$\displaystyle\partial_{j}\omega_{0}=\frac{1}{3(x-u_{j})}\omega_{0},$
(5.11)		$\displaystyle\partial_{j}\omega_{k}=\frac{2}{3(x-u_{j})}\omega_{k},\ 1\leq k\leq 3.$

It follows that

(5.12)		$\displaystyle\partial_{j}(\omega_{2}-u_{j}\omega_{1})=\frac{2}{3(x-u_{j})}\omega_{1},$
(5.13)		$\displaystyle\partial_{j}(\omega_{3}-u_{j}\omega_{2})=\frac{2}{3(x-u_{j})}\omega_{2}$

are both holomorphic, therefore $\langle\omega_{2}-u_{j}\omega_{1},\omega_{3}-u_{j}\omega_{2}\rangle$ is annihilated by the Kodaira-Spencer class of $\partial_{j}$.

Moreover, it is clear for any $\xi\in T_{u}B$,

(5.14)

$$\nabla_{\xi}H^{0}(C_{u},\Omega^{1}_{C_{u}})_{e^{\frac{4\pi i}{3}}}\subset H^{1}(C_{u},\mathcal{O}_{C_{u}})_{e^{\frac{4\pi i}{3}}}=\langle\overline{\omega_{0}}\rangle$$

and similarly

(5.15)

$$\nabla_{\xi}H^{0}(C_{u},\Omega^{1}_{C_{u}})_{e^{\frac{2\pi i}{3}}}=\nabla_{\xi}\langle\omega_{0}\rangle\subset H^{1}(C_{u},\mathcal{O}_{C_{u}})_{e^{\frac{2\pi i}{3}}}.$$

As a consequence we have the following lemma.