1 Introduction
Given a proper smooth family of Kähler manifolds, we can associate the polarized Hodge structure of each fiber to the base point, and hence obtain a holomorphic map from the base to the moduli space of polarized Hodge structures of certain fixed type. This holomorphic map is called the period map, which is a central notion in Hodge theory, and is a powerful tool for studying moduli spaces of projective manifolds for which the period map is injective (we then say these manifolds satisfy the global Torelli theorem).
1.1 Occult period maps
In [Reference Kudla and RapoportKR12], Kudla and Rapoport discussed what they called the occult period maps. The key point is that, for some kinds of projective manifolds, by looking at the periods of certain canonically associated objects instead of the usual periods, we obtain better characterization of the moduli spaces. The examples addressed in [Reference Kudla and RapoportKR12] are cubic surfaces, cubic threefolds, and nonhyperelliptic curves of genus 3 and 4. We first sketch the constructions for those cases. More detailed treatments can be found in Sections 6 and 7.
(Cubic surface). For a smooth cubic surface 
$S$, we have 
$H^{2}(S,\mathbb{C})=H^{1,1}(S)$. Thus, the Hodge structures on smooth cubic surfaces are without moduli. A clever construction by Allcock, Carlson, and Toledo in [Reference Allcock, Carlson and ToledoACT02] is to consider the period of the cubic threefold 
$X$ which is a triple cover of 
$\mathbb{P}^{3}$ branched along 
$S$. The celebrated work [Reference Clemens and GriffithsCG72] by Clemens and Griffiths showed the global Torelli for cubic threefolds. Therefore, the period of 
$X$ should control the geometry of 
$S$ in a certain sense. The authors of [Reference Allcock, Carlson and ToledoACT02] associated the period of 
$X$ with 
$S$ and show that the resulting period map identifies the moduli space of smooth cubic surfaces with an open subset of an arithmetic ball quotient of dimension 4. This period map is called the occult period map for cubic surfaces.
(Cubic threefold). For cubic threefolds, the usual period map gives rise to an embedding from the moduli space of smooth cubic threefolds to the moduli space of five-dimensional principal polarized abelian varieties. For this usual period map, the source has dimension 
$10$, while the target has dimension 15. It turns out that an occult period map behaves better, in the sense that the source and target have the same dimension. To be concrete, let 
$T$ be a smooth cubic threefold. Denote by 
$X$ the triple cover of 
$\mathbb{P}^{4}$ branched along 
$T$. Then 
$X$ is a cubic fourfold with a natural action by the group 
$\unicode[STIX]{x1D707}_{3}$ of third roots of unity. The global Torelli theorem for cubic fourfolds is originally proved by Voisin [Reference VoisinVoi86, Reference VoisinVoi08]. A new and complete proof can also be found in [Reference LooijengaLoo09]. In [Reference Looijenga and SwierstraLS07] and [Reference Allcock, Carlson and ToledoACT11], the authors associated the period of 
$X$ with 
$T$, and show that the resulting period map identifies the moduli space of smooth cubic threefolds with an open subset of an arithmetic ball quotient of dimension 
$10$. This period map is called the occult period map for cubic threefolds.
(Genus 3 curve). For a smooth nonhyperelliptic curve 
$C$ with genus 
$3$, the linear system of the canonical bundle 
$K_{C}$ embeds 
$C$ as a smooth quartic curve in 
$\mathbb{P}^{2}$. Let 
$X$ be the fourth cover of 
$\mathbb{P}^{2}$ branched along 
$C$. Then 
$X$ is a smooth quartic surface with a natural action by 
$\unicode[STIX]{x1D707}_{4}=\{\pm 1,\pm \sqrt{-1}\}$. A smooth quartic surface is a 
$K3$ surface of degree 
$4$. The global Torelli theorem for polarized 
$K3$ surfaces is first proved in [Reference Pjateckiǐ-Šapiro and ŠafarevičPŠ71]. In [Reference KondōKon00], Kondō associated the period of 
$X$ with 
$C$ and showed that the resulting period map identifies the moduli space of smooth nonhyperelliptic curves of genus 
$3$ with an open subset of an arithmetic ball quotient of dimension 
$6$. This period map is called the occult period map for genus 
$3$ curves.
(Genus 4 curve). For a smooth nonhyperelliptic curve 
$C$ with genus 
$4$, the linear system of the canonical bundle 
$K_{C}$ embeds 
$C$ as a complete intersection of a quadric surface 
$Q$ (either smooth or with one node) and a smooth cubic surface in 
$\mathbb{P}^{3}$. Let 
$X$ be the triple cover of 
$Q$ branched along 
$C$. Then 
$X$ is a polarized 
$K3$ surface (either smooth or with one node) with a natural action by 
$\unicode[STIX]{x1D707}_{3}$. In [Reference KondōKon02], Kondō associated the period of 
$X$ with 
$C$ and showed that the resulting period map identifies the moduli space of smooth nonhyperelliptic curves of genus 
$4$ with an open subset of an arithmetic ball quotient of dimension 
$9$. This period map is called the occult period map for genus 
$4$ curves.
The sources and targets of those four occult period maps acquire natural orbifold structures. In [Reference Kudla and RapoportKR12], Kudla and Rapoport regarded those four ball quotients as the coarse moduli of the moduli stack of abelian varieties with certain additional structures. Moreover, they reinterpreted the occult period maps as morphisms between Deligne–Mumford stacks. This led them to raise and partially answer some natural descent problems, for example, whether the occult period maps can be defined over their natural fields of definition. See [Reference Kudla and RapoportKR12, Section 9].
The main result of this paper, Theorem 1.1, answers the conjectures made by Kudla and Rapoport about the orbifold aspects of the occult period maps; see [Reference Kudla and RapoportKR12, Remark 5.2, 6.2, 7.2, 8.2].
Theorem 1.1. (Main Theorem)
For smooth cubic surfaces, smooth cubic threefolds, and smooth nonhyperelliptic curves with genus 
$3$ or 
$4$, the occult period maps identify the orbifold structures on the moduli spaces and those on the ball quotients.
1.2 Structure of the proof
To prove Theorem 1.1, we need to characterize the actions of the automorphism groups of cubic threefolds, cubic fourfolds, and polarized 
$K3$ surfaces on the corresponding polarized Hodge structures. The following fact is useful in this paper (see [Reference Javanpeykar and LoughranJL17, Proposition 2.11] combining with [Reference Matsumura and MonskyMM64a]).
Proposition 1.2. When 
$d\geqslant 3$, 
$n\geqslant 2$, and 
$X$ is a smooth degree 
$d$ 
$n$-fold, the induced action of 
$\text{Aut}(X)$ on 
$H^{n}(X,\mathbb{Z})$ is faithful.
In order to prove Theorem 1.1 for cubic threefolds, we need the following.
Proposition 1.3. Let 
$X$ be a smooth cubic fourfold, then the group homomorphism
$$\begin{eqnarray}\text{Aut}(X)\longrightarrow \text{Aut}_{hs}(H^{4}(X,\mathbb{Z}),\unicode[STIX]{x1D702})\end{eqnarray}$$ is an isomorphism. Here 
$\unicode[STIX]{x1D702}$ is the square of the hyperplane class of 
$X$, and 
$\text{Aut}_{hs}(H^{4}(X,\mathbb{Z}),\unicode[STIX]{x1D702})$ is the group of automorphisms of the lattice 
$H^{4}(X,\mathbb{Z})$ preserving the Hodge decomposition and 
$\unicode[STIX]{x1D702}$.
The injectivity of the homomorphism (1) is a corollary of Proposition 1.2. The surjectivity of the homomorphism (1) is saying that any automorphism of the polarized Hodge structure on 
$H^{4}(X,\mathbb{Z})$ is induced by an automorphism of 
$X$. We recall the global Torelli theorem for cubic fourfolds.
Theorem 1.4. (Voisin)
Let 
$X_{1},X_{2}$ be two smooth cubic fourfolds. Suppose there exists an isomorphism 
$\unicode[STIX]{x1D711}:H^{4}(X_{2},\mathbb{Z})\cong H^{4}(X_{1},\mathbb{Z})$ respecting the Hodge decompositions and squares of hyperplane classes, then there exists a linear isomorphism 
$f:X_{1}\cong X_{2}$.
Actually, a stronger version of the global Torelli theorem for cubic fourfolds is claimed in [Reference VoisinVoi86]. Namely, with the conditions in Theorem 1.4, the linear isomorphism 
$f:X_{1}\cong X_{2}$ can be uniquely chosen such that 
$\unicode[STIX]{x1D711}$ is induced by 
$f$. Assuming the weak version (Theorem 1.4), the strong version of global Torelli is equivalent to Proposition 1.3. In Section 4, we show that Theorem 1.4, plus the injectivity of the group homomorphism (1) appearing in Proposition 1.3, implies the surjectivity of the same homomorphism.
Remark 1.5. By [Reference Beauville and DonagiBD85], the Fano scheme of lines on a smooth cubic fourfold is a hyper-Kähler fourfold of deformation type 
$K3^{[2]}$. Via this construction, the strong version of global Torelli for cubic fourfolds can be deduced from Verbitsky’s global Torelli theorem for hyper-Kähler manifolds. This is done by Charles [Reference CharlesCha12].
To show Theorem 1.1 for cubic surfaces, we need to characterize the action of the automorphism group of a smooth cubic threefold on its intermediate Jacobian. Recall that for a smooth cubic threefold 
$X$, we denote 
$J(X)=H^{3}(X,\mathbb{Z})\backslash H^{1,2}(X)$, which is a five-dimensional complex torus with a principal polarization given by the topological intersection on 
$H^{3}(X,\mathbb{Z})$. This principally polarized abelian variety 
$J(X)$ is called the intermediate Jacobian of 
$X$. See [Reference Clemens and GriffithsCG72]. By Proposition 1.2, we have an injective group homomorphism 
$\text{Aut}(X){\hookrightarrow}\text{Aut}(J(X))$. Note that we have naturally 
$\unicode[STIX]{x1D707}_{2}=\{\pm 1\}\subset \text{Aut}(J(X))$.
Proposition 1.6. Let 
$X$ be a smooth cubic threefold, then we have a natural group isomorphism 
$\text{Aut}(J(X))\cong \text{Aut}(X)\times \unicode[STIX]{x1D707}_{2}$.
One input of our proof for Propositions 1.3 and 1.6 is the existence of analytic slices for certain proper actions of complex Lie groups (see Proposition 2.2), which implies the existence of universal deformations for any smooth hypersurfaces of degree at least 
$3$. We discuss this in Section 2. As an application of the results in Section 2, we construct the moduli spaces of marked hypersurfaces in Section 3. In Sections 4 and 5, we present the proof of Propositions 1.3 and 1.6, respectively. In Section 6, we conclude Theorem 1.1 for cubic surfaces and cubic threefolds.
The action of the automorphism group of a polarized 
$K3$ surface on the corresponding Hodge structure is well-understood, thanks to the work by Rapoport and Burns [Reference Burns and RapoportBR75]. In Section 7, we prove Theorem 1.1 for smooth nonhyperelliptic curves with genus 
$3$ or 
$4$. Our proof relies on lattice theoretic analysis.
2 Universal deformation of smooth hypersurface
All algebraic varieties considered in this paper are over the complex field, and the topology we are using is the analytic topology. We use 
$\mathbb{P}^{n}$ to denote the complex projective space of dimension 
$n$. For a complex vector space 
$V$ of finite dimension, we denote by 
$\mathbb{P}V$ the projectivization of 
$V$. By a degree 
$d$
$n$-fold, we mean a hypersurface of degree 
$d$ in 
$\mathbb{P}^{n+1}$. In this section, we require 
$n\geqslant 2$, 
$d\geqslant 3$, and 
$(n,d)\neq (2,4)$.
Let 
$G$ be a complex Lie group acting on a complex manifold 
$M$. For 
$x\in M$, we denote by 
$Gx=\{gx\big|g\in G\}$ the orbit of 
$x$ and by 
$G_{x}=\{g\in G\big|gx=x\}$ the stabilizer group of 
$x$.
A subgroup 
$H$ of 
$G$ acts on 
$G\times M$ via 
$h(g,x)=(gh^{-1},hx)$ for 
$h\in H$ and 
$(g,x)\in G\times M$. We denote 
$G\times ^{H}M=H\backslash \!\backslash (G\times M)$ if 
$H$ is finite.
Let 
$X$ be a degree 
$d$
$n$-fold. We denote by 
$\text{Aut}(X)$ the group of automorphisms of 
$X$ induced from linear transformations of the ambient space. According to [Reference Matsumura and MonskyMM64a, Theorem 2], when 
$d\geqslant 3$, 
$n\geqslant 2$ and 
$(n,d)\neq (2,4)$, the group 
$\text{Aut}(X)$ is equal to the usual automorphism group of 
$X$ consisting of regular automorphisms. In particular, this is the case when 
$X$ is a smooth cubic of dimension 
$2,3$, or 
$4$.
The vector space 
$\text{Sym}^{d}((\mathbb{C}^{n+2})^{\ast })$ consists of degree 
$d$ polynomials with 
$n+2$ variables. We denote by 
${\mathcal{C}}^{n,d}\subset \text{Sym}^{d}((\mathbb{C}^{n+2})^{\ast })$ the subspace consisting of polynomials defining smooth degree 
$d$
$n$-folds. Recall that 
$\mathbb{P}{\mathcal{C}}^{n,d}$ is the projectivization of 
${\mathcal{C}}^{n,d}$.
For 
$F\in {\mathcal{C}}^{n,d}$ and 
$g\in \text{GL}(n+2,\mathbb{C})$, we define 
$g(F)=F\circ g^{-1}$. Thus, we have a left action of 
$\text{GL}(n+2,\mathbb{C})$ on 
${\mathcal{C}}^{n,d}$. This induces a left action of 
$\text{PGL}(n+2,\mathbb{C})$ on 
$\mathbb{P}{\mathcal{C}}^{n,d}$. Take a point 
$x$ in 
$\mathbb{P}{\mathcal{C}}^{n,d}$ and denote by 
$X$ the corresponding degree 
$d$
$n$-fold, we have 
$G_{x}=\text{Aut}(X)$. In our cases, 
$G_{x}$ is finite; see [Reference Matsumura and MonskyMM64a, Theorem 1].
For a complex submanifold 
$S$ of 
$\mathbb{P}{\mathcal{C}}^{n,d}$, we denote by 
$\mathscr{X}_{S}$ the tautological family of degree 
$d$
$n$-folds over 
$S$. The following result will be used in the proof of Propositions 1.3 and 1.6.
Proposition 2.1. For a smooth degree 
$d$
$n$-fold 
$X$ with corresponding point 
$x\in \mathbb{P}{\mathcal{C}}^{n,d}$, there exists a complex submanifold 
$S$ of 
$\mathbb{P}{\mathcal{C}}^{n,d}$ containing 
$x$, which satisfies the following properties.
(i) For any point
$x^{\prime }\in \mathbb{P}{\mathcal{C}}^{n,d}$ with the corresponding hypersurface $X^{\prime }$ linearly isomorphic to $X$ via $f:X^{\prime }\longrightarrow X$, we can find an open neighborhood $U$ of $x^{\prime }$ in $\mathbb{P}{\mathcal{C}}^{n,d}$, a map $U\longrightarrow S$, and a morphism $\widetilde{f}:\mathscr{X}_{U}\longrightarrow \mathscr{X}_{S}$ such that one has the following commutative diagram:with$\widetilde{f}|_{\mathscr{X}_{x^{\prime }}}=f:X^{\prime }\longrightarrow X$. The choice of $\widetilde{f}$ is unique.(ii) The submanifold
$S$ is $G_{x}$-invariant. In other words, any automorphism $a$ of $X$ induces an automorphism $a:S\longrightarrow S$ of S. We denote by $\widetilde{a}:\mathscr{X}_{S}\longrightarrow \mathscr{X}_{S}$ the pullback of $a$ on $\mathscr{X}_{S}$. We then have the following commutative diagram: (iii) Suppose there are
$x_{1},x_{2}\in S$ and $g\in G$ with $g:\mathscr{X}_{x_{1}}\cong \mathscr{X}_{x_{2}}$, then $g\in G_{x}$.
To prove this theorem, we need to understand the local structure of the action of 
$\text{PGL}(n+2,\mathbb{C})$ on 
$\mathbb{P}{\mathcal{C}}^{n,d}$ at 
$x$. The following proposition should be known to the experts. However, we did not find it in the literature; hence, we give a proof for completeness.
Proposition 2.2. Let 
$G$ be a complex Lie group acting holomorphically and properly on a complex manifold 
$M$. Suppose 
$x$ is a point in 
$M$ with the stabilizer group 
$G_{x}=\{g\in G\big|gx=x\}$ finite. Then there exists a smooth, locally closed, contractible, 
$G_{x}$-invariant submanifold 
$S$ of 
$M$ containing 
$x$ such that 
$GS$ is open and 
$G\times ^{G_{x}}S\longrightarrow GS$ is an isomorphism. In particular, 
$G\times S\longrightarrow GS$ is a covering map of degree 
$|G_{x}|$.
Proof. The orbit 
$Gx\cong G/G_{x}$ is a submanifold of 
$M$ containing 
$x$. There exists an open neighborhood 
$U$ of 
$x$ in 
$M$ with an open embedding 
$j:U{\hookrightarrow}T_{x}M$ such that 
$j(x)=0$ and the tangent map 
$j_{\ast }$ is equal to identity. For every 
$g\in G_{x}$, the tangent map 
$g_{\ast }:T_{x}M\longrightarrow T_{x}M$ of 
$g$ at 
$x$ is an invertible linear map. Consider a holomorphic map 
$F:U\longrightarrow T_{x}M$ sending 
$y\in U$ to
$$\begin{eqnarray}F(y)=\frac{1}{|G_{x}|}\mathop{\sum }_{g\in G_{x}}(g_{\ast }^{-1}j(g(y))).\end{eqnarray}$$ Then 
$F(x)=0$ and 
$F_{\ast }=\text{id}$. Moreover, for any 
$h\in G_{x}$, we have
$$\begin{eqnarray}F(h(y))=\frac{1}{|G_{x}|}\mathop{\sum }_{g\in G_{x}}(g_{\ast }^{-1}j(gh(y)))=\frac{1}{|G_{x}|}\mathop{\sum }_{g\in G_{x}}h_{\ast }((gh)_{\ast }^{-1}j(gh(y)))=h_{\ast }F(y).\end{eqnarray}$$ The representation of 
$G_{x}$ on 
$T_{x}M$ has an invariant subspace 
$T_{x}(Gx)$. By representation theory of finite groups, there exists an invariant subspace 
$T_{1}$ such that 
$T_{x}(Gx)\oplus T_{1}=T_{x}M$. By inverse function theorem, we can choose an open neighborhood 
$U_{1}$ of 
$x$ in 
$U$ such that the restriction of 
$F$ on 
$U_{1}$ is an open embedding into 
$T_{x}M$. We may shrink 
$U_{1}$ such that 
$F(U)$ is the product of an open subset of 
$T_{x}(Gx)$ and a 
$G_{x}$-invariant open subset 
$B$ of 
$U_{1}$. By Equation (2), the submanifold 
$S=F^{-1}(B)$ of 
$M$ is 
$G_{x}$-invariant.
Consider the natural map 
$p:G\times S\longrightarrow M$. The tangent map of 
$p$ at 
$(1,x)$ is an isomorphism 
$p_{\ast }:T_{1}G\oplus T_{x}S\cong T_{x}(Gx)\oplus T_{1}=T_{x}M$. Thus, 
$p_{\ast }$ is an isomorphism at any points in certain neighborhood of 
$x$ in 
$G\times S$. If 
$p_{\ast }$ is an isomorphism at 
$(1,y)$ for 
$y\in S$, then 
$p_{\ast }$ is also an isomorphism at every point in 
$G\times \{y\}$. Actually, for any 
$g\in G$, we can consider the commutative diagram
where the map in the first column is multiplying the first factor with 
$g^{-1}$ from the left. Thus, we have 
$p=g\circ p\circ g^{-1}$. Taking derivatives at 
$(g,y)$, the above equation implies that 
$p_{\ast }$ is an isomorphism at 
$(g,y)$.
Thus, we may shrink 
$S$ such that 
$p_{\ast }$ is an isomorphism at every point in 
$G\times S$. As a summary of the above argument, there exists a 
$G_{x}$-invariant submanifold 
$S$ of 
$M$ containing 
$x$ such that 
$T_{x}S\oplus T_{x}(Gx)=T_{x}M$, and 
$p:G\times S\longrightarrow M$ is open. In particular, 
$GS$ is an open subset of 
$M$.
The map 
$G\times S\longrightarrow GS$ is surjective and factors through 
$G\times ^{G_{x}}S$. It suffices to show that we can suitably shrink 
$S$ such that 
$G\times ^{G_{x}}S\longrightarrow GS$ is an isomorphism. We assume that this cannot be achieved and try to conclude contradiction.
We can find 
$(g,s),(g^{\prime },s^{\prime })\in G\times S$ such that 
$gs=g^{\prime }s^{\prime }$ and 
$g^{-1}g^{\prime }\notin G_{x}$. Denote 
$g_{1}=g^{-1}g^{\prime }$ and 
$s_{1}=s^{\prime }$. Then we obtain a pair 
$(g_{1},s_{1})\in G\times S$ such that 
$g_{1}\notin G_{x}$ and 
$g_{1}s_{1}\in S$. We shrink 
$S$ to obtain 
$x\in S_{2}\subset S$ such that 
$S_{2}$ is a 
$G_{x}$-invariant open submanifold of 
$S$ and 
$s_{1}\notin S_{2}$. By our assumption, there exists 
$(g_{2},s_{2})\in G\times S_{2}$ such that 
$g_{2}\notin G_{x}$ and 
$g_{2}s_{2}\in S_{2}$.
Continuing to do this, we obtain a sequence of pairs 
$(g_{i},s_{i})_{i\in \mathbb{N}_{+}}$ such that 
$g_{i}\notin G_{x}$, 
$g_{i}s_{i}\in S_{i}\subset S$. We may require that the limit of 
$\overline{S_{i}}$ is the point 
$x$, then we have 
$s_{i}\rightarrow x$ as 
$i\rightarrow \infty$. The morphism 
$G\times M\longrightarrow M\times M$, 
$(g,x)\longmapsto (gx,x)$ is proper; hence, the preimage of 
$\overline{S}\times \overline{S}\subset M\times M$ is compact. Thus, there exists a subsequence 
$(g_{i_{k}},s_{i_{k}})$ of 
$(g_{i},s_{i})$ such that 
$(g_{i_{k}},s_{i_{k}})$ has a limit as 
$k\rightarrow \infty$. The limit of 
$(s_{i_{k}})$ must be 
$x$. Assume that 
$g_{i_{k}}\rightarrow g_{0}\in G$. Since 
$g_{i_{k}}s_{i_{k}}\in S_{i_{k}}$, we have 
$g_{0}x=\text{lim}(g_{i_{k}}s_{i_{k}})$ equals 
$x$. Thus, 
$g_{0}\in G_{x}$.
The differential of the morphism 
$G\times S\longrightarrow M$ at 
$(g_{0},x)$ is an isomorphism 
$T_{g_{0}}G\oplus T_{x}S\cong T_{x}(Gx)\oplus T_{x}S\cong T_{x}M$. Therefore, 
$G\times S\longrightarrow M$ is a local isomorphism at 
$(g_{0},x)$. This implies that 
$g_{i_{k}}=g_{0}$ for 
$k$ large enough. But by our choices, we have 
$g_{i_{k}}\notin G_{x}$, which is a contradiction.◻
In this paper, we call a submanifold 
$S$ with all the properties in Proposition 2.2 a slice for the action of 
$G$ on 
$M$ at 
$x$.
Proof of Proposition 2.1.
We consider the action of 
$G=\text{PGL}(n+2,\mathbb{C})$ on 
$M=\mathbb{P}{\mathcal{C}}^{n,d}$. By [Reference Mumford, Fogarty and KirwanMFK94, Proposition 0.8], this action is proper in the sense that 
$G\times M\longrightarrow M\times M$ is proper. By Proposition 2.2, we can take 
$S$ to be a slice containing 
$x$. We next show that 
$S$ satisfies the properties we require.
(i) Take 
$U$ to be an open neighborhood of 
$x^{\prime }$ in 
$GS$. Consider the covering map 
$G\times S\longrightarrow G\times ^{G_{x}}S\cong GS$, we have a unique morphism 
$h:U\longrightarrow G\times S$ with 
$h(x^{\prime })=(f^{-1},x)$ such that the following diagram commutes:
For 
$y^{\prime }\in U$, we denote 
$h(y^{\prime })=(g^{-1},y)$. Then we have 
$g^{-1}y=y^{\prime }$; hence, 
$gy^{\prime }=y$. Thus, the lifting 
$h$ gives rise to a morphism 
$\widetilde{f}:\mathscr{X}_{U}\longrightarrow \mathscr{X}_{S}$ as required. The uniqueness of the lifting implies the uniqueness of 
$\widetilde{f}$.
(ii) Recall that 
$G_{x}=\text{Aut}(X)$. Since 
$S$ is 
$G_{x}$-invariant, the automorphism 
$a$ acts on 
$S$. The pullback 
$\widetilde{a}:\mathscr{X}_{S}\longrightarrow \mathscr{X}_{S}$ of 
$a$ satisfies the requirement.
(iii) Consider the covering map 
$G\times S\longrightarrow G\times ^{G_{x}}S\cong GS$. For any 
$h\in G_{x}$, the pair 
$(h,h^{-1}x_{2})$ is a point in 
$G\times S$ over 
$x_{2}\in GS$. Since 
$gx_{1}=x_{2}$, the pair 
$(g,x_{1})$ is also a point over 
$x_{2}$. Since 
$G\times S\longrightarrow GS$ is of degree 
$|G_{x}|$, one must have 
$(g,x_{1})\in \{(h,h^{-1}x_{2})\big|h\in G_{x}\}$; hence, 
$g\in G_{x}$.◻
3 Moduli of smooth hypersurfaces with markings
In this section, all hypersurfaces are assumed to be smooth. We still assume that 
$n\geqslant 2$ and 
$d\geqslant 3$. We are going to construct the moduli space of marked degree 
$d$
$n$-folds as a complex manifold.
Consider a point 
$x\in M=\mathbb{P}{\mathcal{C}}^{n,d}$ with 
$X=\mathscr{X}_{x}$ the corresponding degree 
$d$
$n$-fold. It is known that 
$H^{n}(X,\mathbb{Z})$ is free. We have a unimodular bilinear form 
$b_{x}:H^{n}(X,\mathbb{Z})\times H^{n}(X,\mathbb{Z})\longrightarrow \mathbb{Z}$ given by the cup product. For 
$n$ even, we denote by 
$\unicode[STIX]{x1D702}_{x}\in H^{n}(X,\mathbb{Z})$ the 
$(n/2)$th power of the hyperplane class. By a symmetric (symplectic) lattice, we mean a free abelian group of finite rank together with an integral symmetric (symplectic) bilinear form which is nondegenerate. Denote by 
$(\unicode[STIX]{x1D6EC}^{n,d},b)$ an abstract lattice isomorphic to 
$(H^{n}(X,\mathbb{Z}),b_{x})$. For 
$n$ even, we fix 
$\unicode[STIX]{x1D702}\in \unicode[STIX]{x1D6EC}^{n,d}$ such that 
$(\unicode[STIX]{x1D6EC}^{n,d},b,\unicode[STIX]{x1D702})\cong (H^{n}(X,\mathbb{Z}),b_{x},\unicode[STIX]{x1D702}_{x})$.
A marking of 
$X$ is an isomorphism 
$\unicode[STIX]{x1D719}:(H^{n}(X,\mathbb{Z}),b_{x})\cong (\unicode[STIX]{x1D6EC}^{n,d},b)$ which sends 
$\unicode[STIX]{x1D702}_{x}$ to 
$\unicode[STIX]{x1D702}$ when 
$n$ is even. Two pairs 
$(x_{1},\unicode[STIX]{x1D719}_{1})$ and 
$(x_{2},\unicode[STIX]{x1D719}_{2})$ are said to be equivalent if there exists 
$g\in G=\text{PGL}(n+2,\mathbb{C})$ such that 
$g(x_{1})=x_{2}$ and 
$\unicode[STIX]{x1D719}_{2}=\unicode[STIX]{x1D719}_{1}\circ g^{\ast }$.
We define 
${\mathcal{N}}^{n,d}$, the moduli space of marked smooth degree 
$d$
$n$-folds, first as a set, consisting of equivalence classes of 
$(x,\unicode[STIX]{x1D719})$. We want to endow 
${\mathcal{N}}^{n,d}$ with the structure of a complex manifold. We first identify the topology on 
${\mathcal{N}}^{n,d}$.
Consider 
$(x,\unicode[STIX]{x1D719})\in {\mathcal{N}}^{n,d}$. We take 
$S$ to be a slice for the action of 
$G$ on 
$M$ at 
$x$. Recall that 
$G_{x}=\text{Aut}(X)$ is the automorphism group of 
$X=\mathscr{X}_{x}$ and 
$\unicode[STIX]{x1D70B}:\mathscr{X}_{S}\longrightarrow S$ is the tautological family of degree 
$d$
$n$-folds over 
$S$. Since 
$S$ is contractible, the local system 
$R^{n}\unicode[STIX]{x1D70B}_{\ast }(\mathbb{Z})$ is trivializable. Thus, 
$\unicode[STIX]{x1D719}$ induces a marking for every fiber of the local system. This gives rise to a map 
$q:S\longrightarrow {\mathcal{N}}^{n,d}$.
Proposition 3.1. The map 
$q$ is injective.
Proof. Suppose there are two different points 
$x_{1},x_{2}\in S$ with 
$q(x_{1})=q(x_{2})$. We denote by 
$\unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{2}$ the induced markings on 
$\mathscr{X}_{x_{1}},\mathscr{X}_{x_{2}}$. Then there exists a linear transformation 
$g:\mathscr{X}_{x_{1}}\longrightarrow \mathscr{X}_{x_{2}}$ with 
$\unicode[STIX]{x1D719}_{2}=\unicode[STIX]{x1D719}_{1}\circ g^{\ast }$.
We have 
$g\in G_{x}$ by Proposition 2.1. By Proposition 1.2, 
$g^{\ast }$ acts nontrivially on 
$H^{n}(X,\mathbb{Z})$. This implies that 
$\unicode[STIX]{x1D719}$ and 
$\unicode[STIX]{x1D719}\circ g^{\ast }$ are two different markings of 
$X$; hence, 
$\unicode[STIX]{x1D719}_{2}$ and 
$\unicode[STIX]{x1D719}_{1}\circ g^{\ast }$ are two different markings of 
$\mathscr{X}_{x_{2}}$, a contradiction! We showed the injectivity of 
$q$.◻
Now we take those slices as charts on 
${\mathcal{N}}^{n,d}$. To make 
${\mathcal{N}}^{n,d}$ a complex manifold, we still need to show that it has the Hausdorff property.
Proposition 3.2. With the topology given as above, 
${\mathcal{N}}^{n,d}$ is Hausdorff.
Proof. Suppose two pairs 
$(x_{1},\unicode[STIX]{x1D719}_{1})$, 
$(x_{2},\unicode[STIX]{x1D719}_{2})$, as points in 
${\mathcal{N}}^{n,d}$, are nonseparated. By [Reference Matsusaka and MumfordMM64b, Theorem 2], the moduli space of degree 
$d$
$n$-folds, as a 
$\text{GIT}$-quotient of 
$\mathbb{P}{\mathcal{C}}^{n,d}$ by 
$\text{PGL}(n+2,\mathbb{C})$, is separated. This implies that 
$\mathscr{X}_{x_{1}}$ and 
$\mathscr{X}_{x_{2}}$ are linearly isomorphic. Without loss of generality, we assume that 
$x_{1}=x_{2}$.
Take a slice 
$S$ containing 
$x_{1}$. Since 
$(x_{1},\unicode[STIX]{x1D719}_{1}),(x_{1},\unicode[STIX]{x1D719}_{2})\in {\mathcal{N}}^{n,d}$ are nonseparated, there exist two points 
$x_{3},x_{4}\in S$ such that 
$(x_{3},\unicode[STIX]{x1D719}_{3}),(x_{4},\unicode[STIX]{x1D719}_{4})$ represent the same point in 
${\mathcal{N}}^{n,d}$ (here we write 
$\unicode[STIX]{x1D719}_{3}$ for the marking on 
$\mathscr{X}_{x_{3}}$ induced by 
$\unicode[STIX]{x1D719}_{1}$ and 
$\unicode[STIX]{x1D719}_{4}$ the marking on 
$\mathscr{X}_{x_{4}}$ induced by 
$\unicode[STIX]{x1D719}_{2}$). Then there exists 
$g:\mathscr{X}_{x_{3}}\cong \mathscr{X}_{x_{4}}$ with 
$\unicode[STIX]{x1D719}_{4}=\unicode[STIX]{x1D719}_{3}\circ g^{\ast }$. By Proposition 2.1, we have 
$g\in G_{x}$. Then 
$\unicode[STIX]{x1D719}_{2}=\unicode[STIX]{x1D719}_{1}\circ g^{\ast }$ as markings on 
$\mathscr{X}_{x_{1}}$. Therefore, 
$(x_{1},\unicode[STIX]{x1D719}_{1})$ and 
$(x_{1},\unicode[STIX]{x1D719}_{2})$ represent the same point in 
${\mathcal{N}}^{n,d}$. This implies that 
${\mathcal{N}}^{n,d}$ is Hausdorff.◻
Corollary 3.3. The set 
${\mathcal{N}}^{n,d}$, with local charts given as above, is a complex manifold.
The space 
${\mathcal{N}}^{n,d}$ may be disconnected. For a complete understanding, we recall some works by Beauville on monodromy group of the universal family of degree 
$d$
$n$-folds. Take a point 
$x\in {\mathcal{C}}^{n,d}$ and denote by 
$X=\mathscr{X}_{x}$ the corresponding smooth degree 
$d$
$n$-fold; there is a representation
of the fundamental group 
$\unicode[STIX]{x1D70B}_{1}({\mathcal{C}}^{n,d},x)$ of 
${\mathcal{C}}^{n,d}$. The image of 
$\unicode[STIX]{x1D70C}$, denoted by 
$\unicode[STIX]{x1D6E4}_{n,d}$, is called the monodromy group of the universal family of smooth degree 
$d$
$n$-folds. From [Reference BeauvilleBea86], we have the following.
Theorem 3.4. (Beauville)
(i) For
$n$ even, and $(n,d)\neq (2,3)$, we have $\unicode[STIX]{x1D6E4}_{n,d}\subset \text{Aut}(H^{n}(X,\mathbb{Z}),b_{x},\unicode[STIX]{x1D702}_{x})$ of index $2$.(ii) For
$n=2$ and $d=3$, we have $\unicode[STIX]{x1D6E4}_{n,d}=\text{Aut}(H^{2}(X,\mathbb{Z}),\unicode[STIX]{x1D702}_{x})$ equals the Weyl group of the $\text{E}_{6}$ lattice.(iii) For
$n$ odd and $d$ even, we have $\unicode[STIX]{x1D6E4}_{n,d}=\text{Aut}(H^{n}(X,\mathbb{Z}),b_{x})$.(iv) For
$n$ odd and $d$ odd, there exists a quadratic form such that$$\begin{eqnarray}q_{x}:H^{n}(X,\mathbb{Z})\longrightarrow \mathbb{Z}/2\mathbb{Z}\end{eqnarray}$$$q_{x}(u+v)=q_{x}(u)+q_{x}(v)+b_{x}(u,v)$ (for any $u,v\in H^{n}(X,\mathbb{Z})$) and $\unicode[STIX]{x1D6E4}_{n,d}=\text{Aut}(H^{n}(X,\mathbb{Z}),b_{x},q_{x})$.
Since 
$\mathbb{P}{\mathcal{C}}^{n,d}$ is connected, the connected components of 
${\mathcal{N}}^{n,d}$ are in bijection with the cosets of the monodromy group in the target automorphism group. Thus, we have the following.
Corollary 3.5. The moduli space 
${\mathcal{N}}^{n,d}$ of marked degree 
$d$
$n$-folds has finitely many connected components, precisely,
(i) it is connected if
$(n,d)=(2,3)$, or $n$ odd and $d$ even,(ii) it has two components if
$n$ even and $(n,d)\neq (2,3)$, and(iii) for
$n$ odd and $d$ odd, the number of its connected components is equal to $[\text{Aut}(\unicode[STIX]{x1D6EC},b):\text{Aut}(\unicode[STIX]{x1D6EC},b,q)]$, where $q$ is the $\mathbb{Z}/2\mathbb{Z}$-valued quadratic form on $\unicode[STIX]{x1D6EC}$ corresponding to $q_{x}$.
4 Automorphism group of cubic fourfold
In this section, we apply Proposition 2.1 to investigate the relation between the automorphism group of a smooth cubic fourfold 
$X$ and that of the polarized Hodge structure of 
$X$. We will prove Proposition 1.3.
We first review some basic facts on Hodge theory of cubic fourfolds. Take 
$x\in \mathbb{P}{\mathcal{C}}^{4,3}$ and denote by 
$X$ the corresponding cubic fourfold, then 
$H^{4}(X,\mathbb{Z})$ is a free abelian group of rank 
$23$, and the natural intersection pairing
$$\begin{eqnarray}b_{x}:H^{4}(X,\mathbb{Z})\times H^{4}(X,\mathbb{Z})\longrightarrow \mathbb{Z}\end{eqnarray}$$ is unimodular and of signature 
$(21,2)$. Recall from Section 3 that we have 
$\unicode[STIX]{x1D702}_{x}\in H^{4}(X,\mathbb{Z})$, and 
$(\unicode[STIX]{x1D6EC}^{4,3},b,\unicode[STIX]{x1D702})\cong (H^{4}(X,\mathbb{Z}),b_{x},\unicode[STIX]{x1D702}_{x})$.
Let 
$L$ be the orthogonal complement of 
$\unicode[STIX]{x1D702}$ in 
$\unicode[STIX]{x1D6EC}^{4,3}$, which is a lattice of signature 
$(20,2)$. Let 
$D$ be the projectivization of the set of points 
$x\in L_{\mathbb{C}}$ with 
$b(x,x)=0$ and 
$b(x,\overline{x})<0$. This is called the period domain of cubic fourfolds. The map 
$\mathscr{P}:{\mathcal{N}}^{4,3}\longrightarrow D$ taking 
$(x,\unicode[STIX]{x1D719})$ to 
$\unicode[STIX]{x1D719}(H^{3,1}(\mathscr{X}_{x}))$ is the period map for cubic fourfolds.
Proposition 4.1. (Local Torelli theorem for cubic fourfolds)
The period map 
$\mathscr{P}$ for cubic fourfolds is locally biholomorphic.
Proof. The dimensions of 
${\mathcal{N}}^{4,3}$ and 
$D_{0}$ are both equal to 
$20$. By Flenner’s infinitesimal Torelli theorem (see [Reference FlennerFle86], Theorem 3.1), the differential of 
$\mathscr{P}$ has full rank everywhere in 
${\mathcal{N}}^{4,3}$. We conclude that 
$\mathscr{P}$ is locally biholomorphic.◻
Proof of Proposition 1.3.
Take 
$x\in \mathbb{P}{\mathcal{C}}^{4,3}$ and denote by 
$X$ the corresponding cubic fourfold. Denote by 
$\unicode[STIX]{x1D70E}$ an automorphism of 
$H^{4}(X,\mathbb{Z})$ which preserves 
$b_{x}$, 
$\unicode[STIX]{x1D702}_{x}$ and the Hodge structure.
Take a slice 
$S$ containing 
$x$. Take 
$\unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{2}$ to be two markings of 
$X$ such that 
$\unicode[STIX]{x1D719}_{2}^{-1}\unicode[STIX]{x1D719}_{1}=\unicode[STIX]{x1D70E}$. For any 
$y\in S$, there are induced markings (from 
$\unicode[STIX]{x1D719}_{1}$, 
$\unicode[STIX]{x1D719}_{2}$) on 
$\mathscr{X}_{y}$, still denoted by 
$\unicode[STIX]{x1D719}_{1}$, 
$\unicode[STIX]{x1D719}_{2}$. Define two holomorphic maps 
$f_{1},f_{2}$ from 
$S$ to 
$D$ by 
$f_{i}(y)=\mathscr{P}(y,\unicode[STIX]{x1D719}_{i})$ for 
$i=1,2$.
By Proposition 4.1, we may assume 
$f_{1},f_{2}$ to be open embeddings (shrink 
$S$ if necessary). Since 
$\unicode[STIX]{x1D70E}$ preserves Hodge structures, we have 
$f_{1}(x)=f_{2}(x)$. Then there exist two points 
$x_{1},x_{2}$ in 
$S$ such that 
$f_{1}(x_{1})=f_{2}(x_{2})$ and this value in 
$D$ can be chosen generically. By Theorem 1.4, 
$\mathscr{X}_{x_{1}}$ and 
$\mathscr{X}_{x_{2}}$ are linearly isomorphic. We can choose a linear isomorphism 
$g:\mathscr{X}_{x_{1}}\cong \mathscr{X}_{x_{2}}$. By Proposition 2.1, we have 
$g\in \text{Aut}(X)$. Since 
$f_{1}(x_{1})=f_{2}(x_{2})$ is generic, it (as Hodge structures on 
$(L,b)$) admits no nontrivial automorphisms; hence, 
$\unicode[STIX]{x1D719}_{2}=\unicode[STIX]{x1D719}_{1}\circ g^{\ast }$ as markings of 
$\mathscr{X}_{x_{2}}$. Then we have also 
$\unicode[STIX]{x1D719}_{2}=\unicode[STIX]{x1D719}_{1}\circ g^{\ast }$ as markings of 
$X$. Thus, 
$\unicode[STIX]{x1D70E}=(g^{-1})^{\ast }$.◻
Corollary 4.2. The period map 
$\mathscr{P}:{\mathcal{N}}^{4,3}\longrightarrow D$ is an open embedding.
Proof. Suppose 
$(x_{1},\unicode[STIX]{x1D719}_{1}),(x_{2},\unicode[STIX]{x1D719}_{2})\in {\mathcal{N}}^{4,3}$ have the same image in 
$D$. Then by Theorem 1.4, there exists 
$g\in \text{PGL}(6,\mathbb{C})$ with 
$g:\mathscr{X}_{x_{1}}\cong \mathscr{X}_{x_{2}}$. We have 
$(g^{\ast })^{-1}\unicode[STIX]{x1D719}_{1}^{-1}\unicode[STIX]{x1D719}_{2}$ an automorphism of 
$H^{4}(\mathscr{X}_{x_{2}},\mathbb{Z})$ preserving 
$b_{x_{2}},\unicode[STIX]{x1D702}_{x_{2}}$ and the Hodge structure; hence, it is induced by an automorphism of 
$\mathscr{X}_{x_{2}}$. This implies that 
$\unicode[STIX]{x1D719}_{2}^{-1}\unicode[STIX]{x1D719}_{1}$ is induced by a linear isomorphism between 
$\mathscr{X}_{x_{1}}$ and 
$\mathscr{X}_{x_{2}}$. Thus, 
$(x_{1},\unicode[STIX]{x1D719}_{1})=(x_{2},\unicode[STIX]{x1D719}_{2})$ in 
${\mathcal{N}}^{4,3}$. We showed the injectivity of 
$\mathscr{P}$; hence, 
$\mathscr{P}$ is an open embedding.◻
5 Automorphism group of cubic threefold
In this section, we deal with the case of cubic threefolds and prove Proposition 1.6.
We first introduce the intermediate Jacobians of smooth cubic threefolds. Take 
$x\in \mathbb{P}{\mathcal{C}}^{3,3}$ and denote by 
$X$ the corresponding cubic threefold, then 
$H^{3}(X,\mathbb{Z})$ is a free abelian group of rank 
$10$. There is a symplectic unimodular bilinear form 
$b_{x}$ on 
$H^{3}(X,\mathbb{Z})$. The intermediate Jacobian of 
$X$ is defined to be 
$J(X)=H^{2,1}(X)\backslash H^{3}(X,\mathbb{C})/H^{3}(X,\mathbb{Z})$, which is a priori a complex torus. The symplectic form 
$b_{x}$ makes 
$J(X)$ a principally polarized abelian variety. We have the following theorem; see [Reference Clemens and GriffithsCG72, Theorem 13.11] or [Reference BeauvilleBea82].
Theorem 5.1. (Global Torelli for cubic threefolds)
Cubic threefolds are determined by their intermediate Jacobians. Precisely, if two cubic threefolds 
$X,Y$ have isomorphic intermediate Jacobians (as principal polarized abelian varieties), then they are isomorphic.
We recall Griffiths’ theory of integral of rational differentials on hypersurfaces; see [Reference GriffithsGri69].
Take 
$F\in {\mathcal{C}}^{n,d}$ a degree 
$d$ polynomial of 
$n+2$ variables 
$Z_{0},\ldots ,Z_{n+1}$ and denote by 
$Z(F)$ the zero locus of 
$F$ in 
$\mathbb{P}^{n+1}$. We write
$$\begin{eqnarray}\unicode[STIX]{x1D6FA}=\mathop{\sum }_{i=0}^{i=n+1}(-1)^{i}Z_{i}dZ_{0}\wedge \cdots \wedge \widehat{dZ_{i}}\wedge \cdots \wedge dZ_{n+1}.\end{eqnarray}$$ Take an integer 
$a>0$ such that 
$ad\geqslant n+2$ and take a degree 
$ad-n-2$ polynomial 
$L$. We have a homogeneous rational differential 
$L\unicode[STIX]{x1D6FA}/F^{a}$ on 
$\mathbb{C}^{n+2}$, with its residue along 
$Z(F)$ giving rise to an 
$n$-form on 
$Z(F)$. Define 
$R:\mathbb{C}[Z_{0},\ldots ,Z_{n+1}]_{ad-n-2}\longrightarrow H^{n}(Z(F),\mathbb{C})$ to be the map taking 
$L$ to 
$Res_{Z(F)}(L\unicode[STIX]{x1D6FA}/F^{a})$. We denote by
$$\begin{eqnarray}F^{n}(Z(F))\subset \cdots \subset F^{0}(Z(F))=H^{n}(Z(F),\mathbb{C})\end{eqnarray}$$ the Hodge filtration on 
$H^{n}(Z(F),\mathbb{C})$. By [Reference GriffithsGri69], we have the following.
Theorem 5.2. The map 
$R$ has image in 
$F^{n-a+1}(Z(F))$, and the composition of
$$\begin{eqnarray}\mathbb{C}[Z_{0},\ldots ,Z_{n+1}]_{ad-n-2}\xrightarrow[{}]{R}F^{n-a+1}\rightarrow F^{n-a+1}/F^{n-a}\cong H^{n-a+1,a-1}(Z(F))\end{eqnarray}$$is surjective.
Lemma 5.3. The automorphism 
$-\text{id}$ of 
$J(X)$ is not induced by any automorphism of 
$X$.
Proof. Suppose there is a linear isomorphism 
$g:X\longrightarrow X$ with 
$g^{\ast }=-\text{id}$ on 
$J(X)$. Then 
$g^{\ast 2}=\text{id}$ on 
$H^{3}(X,\mathbb{Z})$. By Proposition 1.2, we have 
$g^{2}=\text{id}$.
We can take a linear transformation 
$\widetilde{g}:\mathbb{C}^{5}\longrightarrow \mathbb{C}^{5}$ representing 
$g$, and choose a coordinate system 
$(Z_{0},\ldots ,Z_{4})$ such that 
$\widetilde{g}(Z_{i})(=Z_{i}\circ \widetilde{g}^{-1})=Z_{i}$ or 
$-Z_{i}$ for 
$i\in \{0,1,\ldots ,4\}$. For each 
$i\in \{0,1,\ldots ,4\}$, there exists a complex number 
$\unicode[STIX]{x1D706}_{i}$ with 
$\widetilde{g}(Z_{i}\unicode[STIX]{x1D6FA}/F^{2})=\unicode[STIX]{x1D706}_{i}(Z_{i}\unicode[STIX]{x1D6FA}/F^{2})$.
Since 
$g^{\ast }=-\text{id}$, the automorphism 
$g$ is nontrivial; hence, there exists 
$i_{1},i_{2}\in \{0,1,\ldots ,4\}$ such that 
$\widetilde{g}(Z_{i_{1}})=Z_{i_{1}}$ and 
$\widetilde{g}(Z_{i_{2}})=-Z_{i_{2}}$. Thus, 
$\unicode[STIX]{x1D706}_{i_{1}}\neq \unicode[STIX]{x1D706}_{i_{2}}$.
On the other hand, by 
$g^{\ast }=-\text{id}$ on 
$J(X)$, we have that 
$g^{\ast }=-\text{id}$ on 
$H^{3}(X,\mathbb{C})$. By taking residues of 
$Z_{i}\unicode[STIX]{x1D6FA}/F^{2}$ along 
$X$, we obtain a basis for 
$H^{2,1}(X)$. Thus, 
$\unicode[STIX]{x1D706}_{i}=-1$ for every 
$i$. This contradicts the previous result 
$\unicode[STIX]{x1D706}_{i_{1}}\neq \unicode[STIX]{x1D706}_{i_{2}}$.◻
Denote by 
$P$ the ambient space of 
$X$. For a linear form 
$l$ (of variables 
$Z_{0},\ldots ,Z_{4}$), the rational differential 
$l\unicode[STIX]{x1D6FA}/F^{2}$ has residue in 
$H^{2,1}(X)$. Recall that 
$\mathbb{P}H^{2,1}(X)$ is the projectivization of 
$H^{2,1}(X)$. We have a map 
$P^{\ast }\longrightarrow \mathbb{P}H^{2,1}(X)$, where 
$P^{\ast }$ is the dual of 
$P$. By Theorem 5.2, every element in 
$H^{2,1}(X)$ comes in this way. Thus, the map 
$P^{\ast }\longrightarrow \mathbb{P}H^{2,1}(X)$ is surjective. Since 
$\dim P^{\ast }=\dim P=\dim \mathbb{P}H^{2,1}(X)=4$, we obtain an isomorphism 
$\unicode[STIX]{x1D705}:P^{\ast }\cong \mathbb{P}H^{2,1}(X)$. Note that 
$\mathbb{P}H^{2,1}(X)$ and 
$\mathbb{P}H^{1,2}(X)$ are naturally dual to each other, we have an isomorphism 
$\unicode[STIX]{x1D705}^{\ast -1}:P\cong \mathbb{P}H^{1,2}(X)$.
Lemma 5.4. For any 
$g\in \text{Aut}(X)$, the following diagram commutes:
Proof. Let 
$\widetilde{g}:\mathbb{C}^{5}\longrightarrow \mathbb{C}^{5}$ be a linear isomorphism representing 
$g$. For an arbitrary linear form 
$l$, we have
$$\begin{eqnarray}\widetilde{g}^{\ast }(l\unicode[STIX]{x1D6FA}_{5}/F^{2})=\widetilde{g}^{\ast }(l)\widetilde{g}^{\ast }(\unicode[STIX]{x1D6FA}_{5})/(\widetilde{g}^{\ast }(F))^{2}=\unicode[STIX]{x1D706}(g)\widetilde{g}^{\ast }(l)(\unicode[STIX]{x1D6FA}_{5}/F^{2}),\end{eqnarray}$$ where 
$\unicode[STIX]{x1D706}(g)$ is a complex number independent of 
$l$. This implies the commutativity of the following diagram:
which implies the commutativity of diagram (3).◻
The theta divisor 
$\unicode[STIX]{x1D6E9}$ of the intermediate Jacobian 
$J(X)$ has a unique singular point (using translation, we may ask the singular point to be 
$0$) of degree 
$3$, and the projectivized tangent cone 
$\mathbb{P}T_{0}\unicode[STIX]{x1D6E9}\subset \mathbb{P}T_{0}J(X)=\mathbb{P}H^{1,2}(X)$ is identified with 
$X$ via 
$\unicode[STIX]{x1D705}^{\ast }:\mathbb{P}H^{1,2}(X)\cong P$; see [Reference BeauvilleBea82] (main theorem) together with the discussion in [Reference Clemens and GriffithsCG72, Chapter 12].
Take 
$\unicode[STIX]{x1D70E}\in \text{Aut}(J(X))$ which induces a linear automorphism 
$\unicode[STIX]{x1D70E}_{\ast }$ of 
$\mathbb{P}T_{0}J(X)$. Since 
$\unicode[STIX]{x1D70E}$ preserves 
$\unicode[STIX]{x1D6E9}$, it must fix the only singular point 
$0$. Thus, the induced automorphism 
$\unicode[STIX]{x1D70E}_{\ast }$ preserves 
$X\subset P$. We obtain a group homomorphism
$$\begin{eqnarray}\unicode[STIX]{x1D6FC}:\text{Aut}(J(X))\longrightarrow \text{Aut}(X)\end{eqnarray}$$ taking 
$\unicode[STIX]{x1D70E}$ to 
$\unicode[STIX]{x1D70E}_{\ast }^{-1}$.
An automorphism 
$g$ of 
$X$ induces 
$g^{\ast }:H^{1,2}(X)\longrightarrow H^{1,2}(X)$ preserving the lattice 
$H^{3}(X,\mathbb{Z})\subset H^{1,2}(X)$. Thus, 
$g^{\ast }$ gives rise to an automorphism of 
$J(X)$. In this way, we obtain a group homomorphism
$$\begin{eqnarray}\unicode[STIX]{x1D6FD}:\text{Aut}(X)\longrightarrow \text{Aut}(J(X)).\end{eqnarray}$$ By Lemma 5.4, we have 
$\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}=\text{id}$. Thus, 
$\text{Aut}(J(X))\cong \text{Aut}(X)\times \text{Ker}(\unicode[STIX]{x1D6FC})$.
Proof of Proposition 1.6.
To prove Proposition 1.6, it suffices to show 
$\text{Ker}(\unicode[STIX]{x1D6FC})=\unicode[STIX]{x1D707}_{2}$.
Suppose we have 
$\unicode[STIX]{x1D70E}\in \text{Aut}(J(X))$ such that 
$\unicode[STIX]{x1D70E}\neq \text{id}$ and 
$\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D70E})=\text{id}$. Then 
$\unicode[STIX]{x1D70E}$ is acting trivially on 
$\mathbb{P}H^{1,2}(X)$; hence, the action of 
$\unicode[STIX]{x1D70E}$ on 
$H^{1,2}(X)$ is, by a scalar, denoted by 
$\unicode[STIX]{x1D701}$. The action of 
$\unicode[STIX]{x1D70E}$ on 
$H^{2,1}$ is then by the scalar 
$\overline{\unicode[STIX]{x1D701}}$. Any automorphisms of a polarized abelian variety must have finite order (see [Reference LangLan59, Proposition 8, Chapter VII]); hence, 
$\unicode[STIX]{x1D70E}$ has finite order. We may then assume that 
$\unicode[STIX]{x1D701}$ is an 
$n$th root of unity. Since 
$H^{3}(X,\mathbb{Q})$ is a vector space over 
$\mathbb{Q}$, all primitive 
$n$th roots of unity should appear as eigenvalues of the automorphism 
$\unicode[STIX]{x1D70E}$ on 
$H^{3}(X,\mathbb{C})$. But we know that only 
$\unicode[STIX]{x1D701}$ and 
$\overline{\unicode[STIX]{x1D701}}$ appear. Thus, 
$n$ equals 
$2$, 
$3$, 
$4$, or 
$6$. To show 
$\text{Ker}(\unicode[STIX]{x1D6FC})=\unicode[STIX]{x1D707}_{2}$, it suffices to show that the cases 
$n=3,4,6$ do not appear.
Denote by 
$D$ the period domain associated with cubic threefolds. In other words, 
$D$ is the moduli space of Hodge structures on 
$\unicode[STIX]{x1D6EC}^{3,3}$ which have type weight 
$3$ and Hodge numbers 
$(0,5,5,0)$ and are principally polarized by 
$b$. Recall from Section 3 that 
${\mathcal{N}}^{3,3}$ is the moduli space of marked smooth cubic threefolds. We have the period map 
$\mathscr{P}:{\mathcal{N}}^{3,3}\longrightarrow D$.
An automorphism (with order 
$3,4$, or 
$6$) of 
$\unicode[STIX]{x1D6EC}^{3,3}$ with only eigenvalues 
$\unicode[STIX]{x1D701}$ and 
$\overline{\unicode[STIX]{x1D701}}$ uniquely determines a Hodge structure on 
$\unicode[STIX]{x1D6EC}^{3,3}$, hence a point in 
$D$. There are only countably many such automorphisms, determining countably many points in 
$D$. We denote by 
$I$ the subset of 
$D$ consisting of such Hodge structures.
Let 
$x\in \mathbb{P}{\mathcal{C}}^{3,3}$ be the corresponding point of a smooth cubic threefold 
$X$. Assume there exists an automorphism 
$\unicode[STIX]{x1D70E}$ of 
$H^{3}(X,\mathbb{Z})$ which preserves 
$b_{x}$ and acts as scalar by 
$\unicode[STIX]{x1D701}$ on 
$H^{1,2}(X)$, where 
$\unicode[STIX]{x1D701}$ is equal to a primitive third, fourth, or sixth root of unity. We are going to derive contradiction.
Take a slice 
$S$ for the action of 
$\text{PGL}(5,\mathbb{C})$ on 
$\mathbb{P}{\mathcal{C}}^{3,3}$ at 
$x$. Let 
$\unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{2}$ be two markings of 
$X$ such that 
$\unicode[STIX]{x1D719}_{2}^{-1}\unicode[STIX]{x1D719}_{1}=\unicode[STIX]{x1D70E}$. For any 
$y\in S$, there are induced markings (from 
$\unicode[STIX]{x1D719}_{1}$, 
$\unicode[STIX]{x1D719}_{2}$) on 
$\mathscr{X}_{y}$, still denoted by 
$\unicode[STIX]{x1D719}_{1}$, 
$\unicode[STIX]{x1D719}_{2}$. Define two holomorphic maps 
$f_{1},f_{2}$ from 
$S$ to 
$D$ by 
$f_{i}(y)=\mathscr{P}(y,\unicode[STIX]{x1D719}_{i})$ for 
$i=1,2$.
Since 
$\unicode[STIX]{x1D70E}$ preserves Hodge structures, we have 
$f_{1}(x)=f_{2}(x)$. By Flenner’s infinitesimal Torelli theorem, we may assume 
$f_{1},f_{2}$ to be injective on 
$S$ (after suitable shrinking of 
$S$). Since 
$\dim (f_{1}(S))=\dim (f_{2}(S))=10$ and 
$\dim (D)=15$, we have
$$\begin{eqnarray}\dim (f_{1}(S)\cap f_{2}(S))\geqslant 5.\end{eqnarray}$$ Then there exist two points 
$x_{1},x_{2}$ in 
$S$ such that 
$f_{1}(x_{1})=f_{2}(x_{2})$, and this value is not in 
$I$. Therefore, Proposition 1.6 holds for 
$\mathscr{X}_{x_{1}}$ and 
$\mathscr{X}_{x_{2}}$. By Theorem 5.1, there exists a linear isomorphism 
$g:\mathscr{X}_{x_{1}}\cong \mathscr{X}_{x_{2}}$. The composition 
$g^{\ast }\unicode[STIX]{x1D719}_{2}^{-1}\unicode[STIX]{x1D719}_{1}$ is an automorphism of 
$H^{3}(\mathscr{X}_{x_{1}},\mathbb{Z})$ preserving 
$b_{x_{2}}$ and the Hodge structure, and hence lies in 
$\text{Aut}(J(\mathscr{X}_{x_{1}}))\cong \text{Aut}(\mathscr{X}_{x_{1}})\times \unicode[STIX]{x1D707}_{2}$. Without loss of generality, we can select 
$g$ such that 
$g^{\ast }\unicode[STIX]{x1D719}_{2}^{-1}\unicode[STIX]{x1D719}_{1}\in \unicode[STIX]{x1D707}_{2}$. By Proposition 2.1, we have 
$g\in G_{x}$. We have 
$g^{\ast }\unicode[STIX]{x1D719}_{2}^{-1}\unicode[STIX]{x1D719}_{1}=g^{\ast }\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D707}_{2}$ as automorphisms of 
$H^{3}(X,\mathbb{Z})$, which implies that 
$(g^{-1})^{\ast }=\pm \unicode[STIX]{x1D70E}$. Then we have 
$g^{-1}=\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6FD}(g^{-1}))=\unicode[STIX]{x1D6FC}((g^{-1})^{\ast })=\unicode[STIX]{x1D6FC}(\pm \unicode[STIX]{x1D70E})=\text{id}$, which is impossible because 
$(g^{-1})^{\ast }=\pm \unicode[STIX]{x1D70E}$ is nontrivial.◻
6 Occult period map: cubics
In the remaining of this paper, we will consider occult period maps for four cases successively and finally confirm some conjectures made by Kudla and Rapoport in [Reference Kudla and RapoportKR12].
6.1 Case of cubic surfaces
In this section, we deal with cubic surfaces. For details of the construction, see [Reference Allcock, Carlson and ToledoACT02].
Take 
$S$ to be a cubic surface and 
$X$ the associated cubic threefold given as the triple cover of the projective space 
$\mathbb{P}^{3}$ branched along 
$S$. Then there is a natural action of the cyclic group of order 
$3$ on 
$X$ (Deck transformations of the ramified covering) and hence also on 
$H^{3}(X,\mathbb{Z})$ and the intermediate Jacobian 
$J(X)$ of 
$X$. Denote by 
$\unicode[STIX]{x1D70E}$ a generator of the group action.
Therefore, we have the group 
$\unicode[STIX]{x1D707}_{6}=\{\pm \text{id},\pm \unicode[STIX]{x1D70E},\pm \unicode[STIX]{x1D70E}^{2}\}$ acting on 
$J(X)$. We denote by 
$A_{0}$ the subgroup of 
$A=\text{Aut}(J(X))$ consisting of elements commuting with 
$\unicode[STIX]{x1D70E}$. Note that 
$\unicode[STIX]{x1D707}_{6}$ lies at the center of 
$A_{0}$.
We can construct a group homomorphism from 
$\text{Aut}(S)$ to 
$A_{0}/\unicode[STIX]{x1D707}_{6}$ as follows. Take 
$a:S\longrightarrow S$ to be an automorphism of 
$S$, we can lift it to an automorphism 
$\widetilde{a}$ of 
$X$, unique up to Deck transformations. The automorphism 
$\widetilde{a}$ of 
$X$ induces an automorphism of 
$J(X)$ which commutes with 
$\unicode[STIX]{x1D70E}$, hence also induces an element in 
$A_{0}/\unicode[STIX]{x1D707}_{6}$. This construction does not depend on the choices of the lifting of 
$a$.
The map attaching 
$J(X)$ (with the action of 
$\unicode[STIX]{x1D707}_{6}$) to the cubic surface 
$S$ is called the occult period map of cubic surfaces, which is an open embedding of the coarse moduli space 
$\text{PGL}(4,\mathbb{C})\backslash \!\backslash \mathbb{P}{\mathcal{C}}^{2,3}$ of smooth cubic surfaces into an arithmetic ball quotient 
$\unicode[STIX]{x1D6E4}\backslash {\mathcal{B}}^{4}$ of dimension 
$4$, where 
$\unicode[STIX]{x1D6E4}=\text{Aut}(\unicode[STIX]{x1D6EC}^{3,3},\unicode[STIX]{x1D70E})/\unicode[STIX]{x1D707}_{6}$; see [Reference Allcock, Carlson and ToledoACT02]. In [Reference Kudla and RapoportKR12, Remark 5.2], a conjecture about the stack aspect of the occult period map for cubic surfaces is made, which is already claimed as an implication of [Reference Allcock, Carlson and ToledoACT02, Theorem 2.20]. We prove (Theorem 6.2) the conjecture in a more straightforward way.
Proposition 6.1. The group homomorphism 
$\text{Aut}(S)\longrightarrow A_{0}/\unicode[STIX]{x1D707}_{6}$ is an isomorphism.
Proof. We first show the surjectivity. Let 
$\unicode[STIX]{x1D701}\in A_{0}$ be an automorphism of 
$J(X)$ commuting with 
$\unicode[STIX]{x1D707}_{6}$. By Proposition 1.6, one element in 
$\{\unicode[STIX]{x1D701},-\unicode[STIX]{x1D701}\}$ is induced by an automorphism of the cubic threefold 
$X$. With the ambiguity of 
$\unicode[STIX]{x1D707}_{6}$ in mind, we may just assume that 
$\unicode[STIX]{x1D701}$ is induced by an automorphism of 
$X$. We denote this automorphism by 
$\widetilde{a}$.
Since 
$\unicode[STIX]{x1D701}=\widetilde{a}^{\ast }$ commutes with 
$\unicode[STIX]{x1D70E}$, by Proposition 1.2, we have that 
$\widetilde{a}$ commutes with the Deck transformations of 
$X\longrightarrow \mathbb{P}^{3}$. Therefore, 
$\widetilde{a}$ is induced by an automorphism 
$a$ of 
$S$. We showed the surjectivity.
Next, we show the injectivity. Let 
$a$ be an automorphism of 
$S$ inducing the trivial element in the group 
$A/\unicode[STIX]{x1D707}_{6}$. Equivalently, there is a lifting 
$\widetilde{a}$ of 
$a$ such that 
$\widetilde{a}^{\ast }\in \unicode[STIX]{x1D707}_{6}$. We can compose 
$\widetilde{a}$ with Deck transformations; hence, we can assume 
$\widetilde{a}^{\ast }\in \{\pm \text{id}\}$. By Lemma 5.3, we must have 
$\widetilde{a}^{\ast }=\text{id}$ and by Proposition 1.2, 
$\widetilde{a}=\text{id}$; hence, 
$a=\text{id}$. We showed the injectivity.◻
Theorem 6.2. The occult period map
$$\begin{eqnarray}\mathscr{P}:\text{PGL}(4,\mathbb{C})\backslash \!\backslash \mathbb{P}{\mathcal{C}}^{2,3}\longrightarrow \unicode[STIX]{x1D6E4}\backslash {\mathcal{B}}^{4}\end{eqnarray}$$ for smooth cubic surfaces identifies the orbifold structures of the 
$\text{GIT}$-quotient 
$\text{PGL}(4,\mathbb{C})\backslash \!\backslash \mathbb{P}{\mathcal{C}}^{2,3}$ and the image in 
$\unicode[STIX]{x1D6E4}\backslash {\mathcal{B}}^{4}$.
Proof. By [Reference Allcock, Carlson and ToledoACT02], 
$\mathscr{P}$ is an isomorphism of analytic spaces onto its image; by Proposition 6.1, it identifies the natural orbifold structures on the source and image.◻
6.2 Case of cubic threefolds
In this section, we deal with cubic threefolds. For details of the construction, see [Reference Allcock, Carlson and ToledoACT11].
Take 
$T$ to be a cubic threefold and 
$X$ the associated cubic fourfold given as triple cover of the projective space 
$\mathbb{P}^{4}$ branched along 
$T$. As in the case of cubic surfaces, one has an action 
$\unicode[STIX]{x1D70E}$ of order 
$3$ on the middle cohomology 
$H^{4}(X,\mathbb{Z})$ of 
$X$, which preserves the intersection pairing and square of the hyperplane class of 
$X$, and acts freely on the primitive part 
$H_{0}^{4}(X,\mathbb{Z})$. Therefore, we have the group 
$\unicode[STIX]{x1D707}_{6}=\{\pm \text{id},\pm \unicode[STIX]{x1D70E},\pm \unicode[STIX]{x1D70E}^{2}\}$ acting on the lattice 
$H_{0}^{4}(X,\mathbb{Z})$ (with intersection pairing of discriminant 3). We then denote by 
$A$ the subgroup of 
$\text{Aut}(H_{0}^{4}(X,\mathbb{Z}))$ consisting of elements preserving Hodge structures and 
$A_{0}$ the subgroup of 
$A$ consisting of elements commuting with 
$\unicode[STIX]{x1D70E}$. The center of 
$A_{0}$ contains 
$\unicode[STIX]{x1D707}_{6}$.
We can construct a group homomorphism from 
$\text{Aut}(T)$ to 
$A_{0}/\unicode[STIX]{x1D707}_{6}$ as follows. Take 
$a:T\longrightarrow T$ to be an automorphism of 
$T$, we can lift it to 
$\widetilde{a}:X\longrightarrow X$, an automorphism of 
$X$, unique up to Deck transformations. The automorphism 
$\widetilde{a}$ of 
$X$ induces an automorphism of 
$H_{0}^{4}(X,\mathbb{Z})$ which commutes with 
$\unicode[STIX]{x1D70E}$, and hence also induces an element in 
$A_{0}/\unicode[STIX]{x1D707}_{6}$ which does not depend on the choices of the lifting of 
$a$.
The map attaching Hodge structures on the lattice 
$H_{0}^{4}(X,\mathbb{Z})$ (preserved by the action of 
$\unicode[STIX]{x1D707}_{6}$) to the cubic threefolds 
$T$ is the occult period map for cubic threefolds, which is an open embedding of the coarse moduli space 
$\text{PGL}(5,\mathbb{C})\backslash \!\backslash \mathbb{P}{\mathcal{C}}^{3,3}$ of smooth cubic threefolds into an arithmetic ball quotient 
$\unicode[STIX]{x1D6E4}\backslash {\mathcal{B}}^{10}$, where 
$\unicode[STIX]{x1D6E4}=\text{Aut}(\unicode[STIX]{x1D6EC}^{4,3},\unicode[STIX]{x1D702},\unicode[STIX]{x1D70E})/\unicode[STIX]{x1D707}_{3}$ (see Section 3 for the notations 
$\unicode[STIX]{x1D6EC}^{4,3},\unicode[STIX]{x1D702}$). We confirm the conjecture in [Reference Kudla and RapoportKR12, Remark 6.2] by the following proposition.
Proposition 6.3. The group homomorphism 
$\text{Aut}(T)\longrightarrow A_{0}/\unicode[STIX]{x1D707}_{6}$ is an isomorphism.
Proof. We first show the surjectivity. Let 
$\unicode[STIX]{x1D701}\in A_{0}$ be an automorphism of 
$H_{0}^{4}(X,\mathbb{Z})$ preserving Hodge structure and commuting with 
$\unicode[STIX]{x1D70E}$. By lattice theory, one of 
$\unicode[STIX]{x1D701},-\unicode[STIX]{x1D701}$ is induced by an automorphism of the whole cohomology 
$H^{4}(X,\mathbb{Z})$ which preserves square of the hyperplane section, and hence, by Proposition 1.3, also induced by an automorphism of the cubic fourfold 
$X$. With the ambiguity of 
$\unicode[STIX]{x1D707}_{6}$ in mind, we may just assume that 
$\unicode[STIX]{x1D701}$ is induced by an automorphism 
$\widetilde{a}$ of 
$X$.
Since 
$\unicode[STIX]{x1D701}=\widetilde{a}^{\ast }$ commutes with 
$\unicode[STIX]{x1D70E}$, by Proposition 1.2, we have that 
$\widetilde{a}$ commutes with the Deck transformations of 
$X\longrightarrow \mathbb{P}^{4}$. Therefore, 
$\widetilde{a}$ is induced by an automorphism 
$a$ of 
$T$. We showed the surjectivity.
Next, we show the injectivity. Let 
$a$ be an automorphism of 
$T$, inducing the trivial element in the group 
$A_{0}/\unicode[STIX]{x1D707}_{6}$. Equivalently, there is a lifting 
$\widetilde{a}$ of 
$a$ such that 
$\widetilde{a}^{\ast }\in \unicode[STIX]{x1D707}_{6}$. We can compose 
$\widetilde{a}$ with Deck transformations; hence, we may assume that 
$\widetilde{a}^{\ast }\big|_{H_{0}^{4}(X,\mathbb{Z})}\in \{\pm \text{id}\}$. Since 
$\widetilde{a}^{\ast }$ preserves square of the hyperplane class, we must have 
$\widetilde{a}^{\ast }=\text{id}$. By Proposition 1.2, 
$\widetilde{a}=\text{id}$; hence, 
$a=\text{id}$. We showed the injectivity.◻
Theorem 6.4. The occult period map
$$\begin{eqnarray}\mathscr{P}_{3,3}:\text{PGL}(5,\mathbb{C})\backslash \!\backslash \mathbb{P}{\mathcal{C}}^{3,3}\longrightarrow \unicode[STIX]{x1D6E4}\backslash {\mathcal{B}}^{10}\end{eqnarray}$$ for smooth cubic threefolds identifies the orbifold structures of the 
$\text{GIT}$-quotient 
$\text{PGL}(5,\mathbb{C})\backslash \!\backslash \mathbb{P}{\mathcal{C}}^{3,3}$ and the image in 
$\unicode[STIX]{x1D6E4}\backslash {\mathcal{B}}^{10}$.
Proof. By [Reference Allcock, Carlson and ToledoACT11, Theorem 1.9], 
$\mathscr{P}_{3,3}$ is an open embedding of analytic spaces. By Proposition 6.3, it identifies the natural orbifold structures on the source and image.◻
7 Occult period map: Kondō’s examples
In this section, we confirm Kudla and Rapoport’s conjectures for nonhyperelliptic curves of genus 
$3$ and 
$4$. First, we collect some results on 
$K3$ surfaces and lattice theory that will be used.
We will use the global Torelli theorem for 
$K3$ surfaces. The original literature is [Reference Burns and RapoportBR75], and one can also see [Reference HuybrechtsHuy16], [Reference Looijenga and PetersLP81].
Theorem 7.1. (Global Torelli theorem for $K3$ surfaces)
Suppose two 
$K3$ surfaces 
$S_{1}$ and 
$S_{2}$ satisfy the following:
(i) there exists an isomorphism
$\unicode[STIX]{x1D711}:H^{2}(S_{1},\mathbb{Z})\cong H^{2}(S_{2},\mathbb{Z})$ preserving the corresponding Hodge structures,(ii)
$\unicode[STIX]{x1D711}({\mathcal{K}}_{S_{1}})\cap {\mathcal{K}}_{S_{2}}\neq \varnothing$, where ${\mathcal{K}}_{S_{1}}$ and ${\mathcal{K}}_{S_{2}}$ are the Kähler cones of $S_{1}$ and $S_{2}$,
then there exists an isomorphism between the two 
$K3$ surfaces, and this isomorphism induces 
$\unicode[STIX]{x1D711}$.
Parallel to Proposition 1.2, one has the following lemma for 
$K3$ surfaces; see [Reference Looijenga and PetersLP81, Proposition 7.5].
Lemma 7.2. For any 
$K3$ surface 
$S$, the action of 
$\text{Aut}(S)$ on 
$H^{2}(S,\mathbb{Z})$ is faithful.
We recall some basic notions in lattice theory. One can refer to [Reference NikulinNik79].
Let 
$M$ be a lattice. Denote 
$M_{\mathbb{Q}}=M\otimes \mathbb{Q}$ and still denote by 
$b_{M}$ the extended bilinear form on 
$M_{\mathbb{Q}}$. One has naturally 
$M{\hookrightarrow}\text{Hom}(M,\mathbb{Z}){\hookrightarrow}M_{\mathbb{Q}}$. The lattice 
$M$ is called unimodular if 
$M\cong \text{Hom}(M,\mathbb{Z})$.
The discriminant group of 
$M$ is defined to be 
$A_{M}=\text{Hom}(M,\mathbb{Z})/M$. There is a quadratic form on 
$A_{M}$ defined as follows:
$$\begin{eqnarray}\displaystyle & q_{M}:A_{M}\longrightarrow \mathbb{Q}/\mathbb{Z} & \displaystyle \nonumber\\ \displaystyle & [x]\longmapsto [b_{M}(x,x)] & \displaystyle \nonumber\end{eqnarray}$$ for 
$x\in \text{Hom}(M,\mathbb{Z})$ and 
$[x]\in A_{M}$ the equivalence class of 
$x$. This quadratic form 
$q_{M}$ is called the discriminant form associated with 
$M$.
If 
$b_{M}(x,x)\in 2\mathbb{Z}$ for any 
$x\in M$, then 
$M$ is called an even lattice. Suppose 
$M$ is even, then we can take values of the discriminant form 
$q_{M}$ in 
$\mathbb{Q}/(2\mathbb{Z})$. Suppose more that 
$M$ is 
$2$-elementary, that is, 
$A_{M}$ is isomorphic to 
$(\mathbb{Z}/2\mathbb{Z})^{l}$ for certain integer 
$l$, then the image of 
$q_{M}$ lies in 
$(\frac{1}{2}\mathbb{Z})/(2\mathbb{Z})$. By [Reference NikulinNik79], we have the following.
Lemma 7.3. Suppose that 
$L$ is a unimodular lattice. Suppose that 
$M,N$ are two sublattices of 
$L$ perpendicular to each other (then both 
$M,N$ are primitive). Then the following hold:
(i) There is a natural isomorphism between
$(A_{M},q_{M})$ and $(A_{N},-q_{N})$.(ii) Suppose there are isomorphisms
$\unicode[STIX]{x1D70E}_{M}:M\longrightarrow M$ and $\unicode[STIX]{x1D70E}_{N}:N\longrightarrow N$ inducing the same action on $A_{M}\cong A_{N}$, then there exists an automorphism of $L$ inducing $\unicode[STIX]{x1D70E}_{M}$ and $\unicode[STIX]{x1D70E}_{N}$.
7.1 Case of curves of genus $3$
In this section, we deal with curves of genus 
$3$. For details of the construction, see [Reference KondōKon00].
Take 
$C$ to be a smooth nonhyperelliptic curve of genus 
$3$, which is embedded as a quartic curve in 
$\mathbb{P}^{2}$ by the canonical linear system. Take 
$S$ to be the associated quartic 
$K3$ surface given as degree 
$4$ cover of the projective space 
$\mathbb{P}^{2}$ branched along 
$C$. There is a natural action of the cyclic group of order 
$4$ (Deck transformations of the ramified covering) on 
$S$, and hence also on 
$H^{2}(S,\mathbb{Z})$. Denote by 
$\unicode[STIX]{x1D70E}$ a generator of the order 
$4$ group.
Define 
$M=\{x\in H^{2}(S,\mathbb{Z})\big|\unicode[STIX]{x1D70E}(x)=x\}$ and 
$N=\{x\in H^{2}(S,\mathbb{Z})\big|\unicode[STIX]{x1D70E}(x)=-x\}$. They are primitive sublattices of 
$H^{2}(S,\mathbb{Z})$, perpendicular to each other, and both have discriminant group isomorphic to 
$(\mathbb{Z}/2\mathbb{Z})^{8}$. The Hodge decomposition on 
$N$ restricted from that on 
$S$ has type 
$(1,14,1)$.
We have the group 
$\unicode[STIX]{x1D707}_{4}=\{\pm \text{id},\pm \unicode[STIX]{x1D70E}\}$ acting on the lattice 
$M$. We then denote by 
$A$ the subgroup of 
$\text{Aut}(N)$ consisting of elements preserving the Hodge structure and by 
$A_{0}$ the subgroup of 
$A$ consisting of elements commuting with 
$\unicode[STIX]{x1D70E}$.
We can construct a group homomorphism from 
$\text{Aut}(C)$ to 
$A_{0}/\unicode[STIX]{x1D707}_{4}$ as follows. Take 
$a:C\longrightarrow C$ to be an automorphism of 
$C$ coming from a linear transformation of the ambient space 
$\mathbb{P}^{2}$. We can lift 
$a$ to an automorphism 
$\widetilde{a}$ of 
$S$, unique up to Deck transformations. The automorphism 
$\widetilde{a}$ of 
$S$ induces an automorphism of 
$N$ which commutes with 
$\unicode[STIX]{x1D70E}$, and hence also induces an element in 
$A_{0}/\unicode[STIX]{x1D707}_{4}$ which does not depend on the choices of the lifting of 
$a$.
The map attaching the Hodge structure on 
$N$ (preserved by the action of 
$\unicode[STIX]{x1D707}_{4}$) to 
$C$ is the occult period map for smooth nonhyperelliptic curves of genus 
$3$, which is an open embedding of the coarse moduli space 
${\mathcal{M}}_{3}^{\circ }$ of smooth nonhyperelliptic curves of genus 
$3$ into an arithmetic ball quotient 
$\unicode[STIX]{x1D6E4}\backslash {\mathcal{B}}^{6}$, where 
$\unicode[STIX]{x1D6E4}=\text{Aut}(N,\unicode[STIX]{x1D70E})/\unicode[STIX]{x1D707}_{4}$ is an arithmetic group acting on 
${\mathcal{B}}^{6}$. We confirm the conjecture in [Reference Kudla and RapoportKR12, Remark 7.2] by the following proposition.
Proposition 7.4. The group homomorphism 
$\text{Aut}(C)\longrightarrow A_{0}/\unicode[STIX]{x1D707}_{4}$ is an isomorphism.
We need the following lemmas.
Lemma 7.5. For an 
$\text{E}_{7}$-lattice 
$P$, we have a quadratic form 
$q:(\frac{1}{2}P)/P\longrightarrow \mathbb{Z}/(2\mathbb{Z})$ taking 
$x\in \frac{1}{2}P$ to 
$[2b_{P}(x,x)]$. Then we have an exact sequence:
$$\begin{eqnarray}1\longrightarrow \{\pm \text{id}\}\longrightarrow \text{Aut}(P)\longrightarrow \text{Aut}(({\textstyle \frac{1}{2}}P)/P,q)\longrightarrow 1.\end{eqnarray}$$Proof. See [Reference BourbakiBou02, Exercise 3 of Section 4, Chapter 6]. ◻
Lemma 7.6. An automorphism of the lattice 
$N$ is induced by an automorphism of 
$H^{2}(S,\mathbb{Z})$ preserving the hyperplane class 
$\unicode[STIX]{x1D702}\in H^{2}(S,\mathbb{Z})$.
This lemma is proved and used in [Reference KondōKon00]. For completeness, we rewrite a proof.
Proof of Lemma 7.6.
Let 
$D$ be the double cover of 
$\mathbb{P}^{2}$ branched along the quartic curve 
$C$, then 
$D$ is a Del Pezzo surface of degree 
$2$ and 
$S$ is a double cover of 
$D$ branched along 
$C$. The middle cohomology 
$H^{2}(D,\mathbb{Z})$ of 
$D$ is a unimodular lattice, and 
$M\cong H^{2}(D,\mathbb{Z})(2)$. Here we use 
$L(n)$ to denote a lattice 
$L$ with a scaled quadratic form by 
$n$. We have the discriminant group 
$A_{M}=(\frac{1}{2}M)/M$ of 
$M$. We have a sublattice 
$(\unicode[STIX]{x1D702}_{0})\oplus P$ in 
$H^{2}(D,\mathbb{Z})$ of index 
$2$, where 
$\unicode[STIX]{x1D702}_{0}$ is the hyperplane class of 
$D$ and 
$P$ is an 
$\text{E}_{7}$-lattice.
Denote by 
$\unicode[STIX]{x1D701}$ an automorphism of 
$N$; it induces an automorphism of 
$(A_{N},q_{N})\cong (A_{M},-q_{M})$. It suffices to construct an automorphism 
$\unicode[STIX]{x1D70C}$ of 
$M$ such that 
$\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D702})=\unicode[STIX]{x1D702}$ and 
$\unicode[STIX]{x1D70C}$, 
$\unicode[STIX]{x1D701}$ induces the same automorphism of 
$(A_{M},q_{M})$.
The finite group 
$(\frac{1}{2}P)/P$ is a subgroup of 
$A_{M}\cong (\frac{1}{2}M)/M$. We are going to show that the induced map of 
$\unicode[STIX]{x1D701}$ on 
$A_{M}$ preserves 
$(\frac{1}{2}P)/P$.
Take an element 
$x\in P$, consider 
$[\frac{1}{2}x]\in A_{M}$, then
$$\begin{eqnarray}q_{M}([{\textstyle \frac{1}{2}}x])=[{\textstyle \frac{1}{4}}b_{M}(x,x)]=[{\textstyle \frac{1}{2}}b_{P}(x,x)]\in \mathbb{Z}/(2\mathbb{Z}),\end{eqnarray}$$ where the last step is because 
$P$ is an 
$\text{E}_{7}$-lattice, which is an even lattice. Since 
$H^{2}(D,\mathbb{Z})$ is an odd lattice, there exists element 
$y\in H^{2}(D,\mathbb{Z})$ with self-intersection an odd number; hence, 
$q_{M}([\frac{1}{2}y])\notin \mathbb{Z}/(2\mathbb{Z})$. Therefore, as a subset of 
$A_{M}$, 
$(\frac{1}{2}P)/P=\{\unicode[STIX]{x1D6FC}\in A_{M}\big|q_{M}(\unicode[STIX]{x1D6FC})\in \mathbb{Z}/(2\mathbb{Z})\}$, which implies that 
$\unicode[STIX]{x1D701}$ preserves 
$(\frac{1}{2}P)/P$.
By Lemma 7.5, there are two automorphisms 
$\unicode[STIX]{x1D70C}_{1}$, 
$-\unicode[STIX]{x1D70C}_{1}$ of 
$P$, both inducing the action 
$\unicode[STIX]{x1D701}$ on 
$(\frac{1}{2}P)/P$. We can extend the action 
$\text{id}\oplus \unicode[STIX]{x1D70C}_{1}$ on 
$(\unicode[STIX]{x1D702}_{0})\oplus P$ uniquely to an automorphism 
$\unicode[STIX]{x1D70C}_{2}$ of 
$H^{2}(D,\mathbb{Z})$ and similarly extend 
$\text{id}\oplus (-\unicode[STIX]{x1D70C}_{1})$ to 
$\unicode[STIX]{x1D70C}_{3}$. The two automorphisms 
$\unicode[STIX]{x1D70C}_{2}$ and 
$\unicode[STIX]{x1D70C}_{3}$ can be regarded as automorphisms of 
$M$, and hence also induce actions on 
$A_{M}$. Consider the automorphisms 
$\unicode[STIX]{x1D709}_{1}=\unicode[STIX]{x1D70C}_{2}^{-1}\circ \unicode[STIX]{x1D701}$ and 
$\unicode[STIX]{x1D709}_{2}=\unicode[STIX]{x1D70C}_{3}^{-1}\circ \unicode[STIX]{x1D701}$ on 
$(A_{M},q_{M})$; they are different and both act as identity on 
$(\frac{1}{2}P)/P$.
Assume that 
$\unicode[STIX]{x1D709}:A_{M}\longrightarrow A_{M}$ is an automorphism preserving 
$q_{M}$ and acting trivially on 
$(\frac{1}{2}P)/P$. Take 
$x\in M$ with 
$[\frac{1}{2}x]\notin (\frac{1}{2}P)/P$ and assume 
$\unicode[STIX]{x1D709}([\frac{1}{2}x])=[\frac{1}{2}y]$ for 
$y\in M$. Then for any 
$z\in P$, we have 
$\unicode[STIX]{x1D709}([(x+z)/2])=[(y+z)/2]$, which implies that 
$q_{M}([(x+z)/2])=q_{M}([(y+z)/2])$. Thus, 
$\frac{1}{2}(b_{M}(x-y,z))\in 2\mathbb{Z}$ for any 
$z\in P$. This implies that either 
$x-y$ or 
$x-y-\unicode[STIX]{x1D702}$ belongs to 
$2M$; hence, 
$\unicode[STIX]{x1D709}([\frac{1}{2}x])=[\frac{1}{2}x]$ or 
$[\frac{1}{2}(x-\unicode[STIX]{x1D702})]$. Therefore, the automorphism 
$\unicode[STIX]{x1D709}$ as required has at most two possibilities. We conclude that either 
$\unicode[STIX]{x1D709}_{1}$ or 
$\unicode[STIX]{x1D709}_{2}$ equals identity; hence, either 
$\unicode[STIX]{x1D70C}_{2}$ or 
$\unicode[STIX]{x1D70C}_{3}$ equals 
$\unicode[STIX]{x1D701}$ as automorphisms of 
$A_{M}$.◻
Lemma 7.7. Suppose there are two automorphisms 
$\unicode[STIX]{x1D701}_{1}$, 
$\unicode[STIX]{x1D701}_{2}$ of the 
$K3$ lattice 
$H^{2}(S,\mathbb{Z})$ such that
$$\begin{eqnarray}\unicode[STIX]{x1D701}_{1}\big|_{N}=\unicode[STIX]{x1D701}_{2}\big|_{N}:N\longrightarrow N\end{eqnarray}$$and both the automorphisms preserve the hyperplane class; then they coincide.
Proof. It suffices to show that any automorphism 
$\unicode[STIX]{x1D701}$ of 
$H^{2}(S,\mathbb{Z})$ which acts identically on 
$(\unicode[STIX]{x1D702})\oplus N$ must be the identity.
Define sublattice 
$P$ of 
$H^{2}(D,\mathbb{Z})$ as in the proof of Lemma 7.6. Since 
$\unicode[STIX]{x1D701}$ acts identically on 
$N$, it also acts identically on 
$A_{N}\cong A_{M}$, and hence also identically on 
$(\frac{1}{2}P)/P$. By Lemma 7.5, we have 
$\unicode[STIX]{x1D701}$ equals 
$\text{id}$ or 
$-\text{id}$ on 
$P$, with the latter possibility excluded by the fact that 
$\unicode[STIX]{x1D701}$ is an automorphism of the whole lattice 
$H^{2}(S,\mathbb{Z})$ preserving 
$\unicode[STIX]{x1D702}$. Thus, 
$\unicode[STIX]{x1D701}=\text{id}$ and we proved the lemma.◻
Proof of Proposition 7.4.
We first show the surjectivity. Let 
$\unicode[STIX]{x1D701}\in A_{0}$ be an automorphism of 
$N$ preserving the Hodge structure and commuting with 
$\unicode[STIX]{x1D70E}$. By Lemma 7.6, 
$\unicode[STIX]{x1D701}$ is induced by an automorphism of the whole lattice 
$H^{2}(S,\mathbb{Z})$ which preserves the hyperplane class. This automorphism apparently preserves the Hodge structure on 
$H^{2}(S,\mathbb{Z})$ and hence comes from an automorphism 
$\widetilde{a}$ of the quartic surface 
$S$.
Since 
$\unicode[STIX]{x1D701}=\widetilde{a}^{\ast }\big|_{N}$ commutes with 
$\unicode[STIX]{x1D70E}$, we have 
$\unicode[STIX]{x1D70E}\widetilde{a}^{\ast }$ and 
$\widetilde{a}^{\ast }\unicode[STIX]{x1D70E}$ coincide on the lattice 
$N$ and both preserve the hyperplane class. By Lemma 7.7, the equality 
$\unicode[STIX]{x1D70E}\widetilde{a}^{\ast }=\widetilde{a}^{\ast }\unicode[STIX]{x1D70E}$ holds on the whole lattice 
$H^{2}(S,\mathbb{Z})$. By Lemma 7.2, we have that 
$\widetilde{a}$ commutes with the Deck transformations of 
$S\longrightarrow \mathbb{P}^{2}$. Therefore, 
$\widetilde{a}$ is induced by an automorphism 
$a$ of 
$C$. We showed the surjectivity.
Next, we show the injectivity. Let 
$a$ be an automorphism of 
$C$ inducing the trivial element in the group 
$A_{0}/\unicode[STIX]{x1D707}_{4}$. Then there is a lifting 
$\widetilde{a}$ of 
$a$ such that 
$\widetilde{a}^{\ast }\big|_{N}\in \unicode[STIX]{x1D707}_{4}$. We can compose 
$\widetilde{a}$ with Deck transformations, and hence we can assume that 
$\widetilde{a}^{\ast }\big|_{N}=\text{id}$. Since 
$\widetilde{a}^{\ast }$ acts as identity on the hyperplane class of 
$S$, by Lemma 7.7, 
$\widetilde{a}^{\ast }=\text{id}$ and by Lemma 7.2, 
$\widetilde{a}=\text{id}$; hence, 
$a=\text{id}$. We showed the injectivity.◻
Theorem 7.8. The occult period map
$$\begin{eqnarray}\mathscr{P}:{\mathcal{M}}_{3}^{\circ }\longrightarrow \unicode[STIX]{x1D6E4}\backslash {\mathcal{B}}^{6}\end{eqnarray}$$ for smooth nonhyperelliptic curves of genus 
$3$ identifies the natural orbifold structure of 
${\mathcal{M}}_{3}^{\circ }$ and the image in 
$\unicode[STIX]{x1D6E4}\backslash {\mathcal{B}}^{6}$.
Proof. By [Reference KondōKon00, Theorem 2.5], 
$\mathscr{P}$ is an open embedding of analytic spaces; combining with Proposition 7.4, we have that 
$\mathscr{P}$ identifies the orbifold structures on the source and image.◻
7.2 Case of curves of genus 4
In this section, we deal with curves of genus 4. For details of the construction, see [Reference KondōKon02].
Take 
$C$ to be a smooth nonhyperelliptic curve of genus 
$4$, which is embedded as a complete intersection of a quadric surface 
$Q$ (smooth or with one node) and a smooth cubic surface in 
$\mathbb{P}^{3}$ via the canonical linear system. Take 
$S$ to be the associated 
$K3$ surface given as triple cover of the quadric surface 
$Q$ branched along 
$C$ (in case 
$Q$ is singular, take its minimal resolution instead). Then there is a natural action of the cyclic group of order 
$3$ on 
$S$ (Deck transformations of the ramified covering) and hence also on 
$H^{2}(S,\mathbb{Z})$. Denote by 
$\unicode[STIX]{x1D70E}$ a generator of this group.
Suppose the quadric surface containing 
$C$ is smooth, then it is isomorphic to 
$\mathbb{P}^{1}\times \mathbb{P}^{1}$; if the quadric surface is singular, then we can blow up the singular point and obtain 
$Q$ a rational surface which is the projectivization of the degree 
$2$ and rank 
$2$ vector bundle on 
$\mathbb{P}^{1}$. In both cases, we have 
$U=H^{2}(Q,\mathbb{Z})$ a hyperbolic lattice with generators 
$x_{1},x_{2}$ such that 
$b_{U}(x_{1},x_{1})=b_{U}(x_{2},x_{2})=0,b(x_{1},x_{2})=1$ and 
$\unicode[STIX]{x1D702}_{0}=x_{1}+x_{2}$ is the hyperplane class of 
$Q$.
Denote 
$M=\{x\in H^{2}(S,\mathbb{Z})\big|\unicode[STIX]{x1D70E}(x)=x\}$ and 
$N=M^{\bot }$. Then 
$M$ contains the hyperplane class. Moreover, 
$M,N$ are primitive sublattices of 
$H^{2}(S,\mathbb{Z})$ perpendicular to each other. Explicitly, 
$M\cong H^{2}(Q,\mathbb{Z})(3)$ is of rank 
$2$, 
$N$ is of rank 
$20$, and they have isomorphic discriminant groups 
$A_{N}\cong A_{M}\cong (\mathbb{Z}/3\mathbb{Z})^{2}$. The induced Hodge decomposition on 
$N$ is of type 
$(1,18,1)$.
We have the group 
$\unicode[STIX]{x1D707}_{6}=\{\pm \text{id},\pm \unicode[STIX]{x1D70E},\pm \unicode[STIX]{x1D70E}^{2}\}$ acting on the lattice 
$N$. We then denote by 
$A$ the subgroup of 
$\text{Aut}(N)$ consisting of elements preserving the Hodge structure and by 
$A_{0}$ the subgroup of 
$A$ consisting of elements commuting with 
$\unicode[STIX]{x1D70E}$.
We can construct a group homomorphism from 
$\text{Aut}(C)$ to 
$A_{0}/\unicode[STIX]{x1D707}_{6}$ as follows. Take 
$a:C\longrightarrow C$ to be an automorphism of 
$C$ coming from a linear transformation of the ambient space 
$\mathbb{P}^{3}$. This linear transformation preserves 
$Q$ and we can lift it to 
$\widetilde{a}:S\longrightarrow S$, an automorphism of 
$S$, unique up to Deck transformations. The automorphism 
$\widetilde{a}$ of 
$S$ induces an automorphism of 
$N$ which commutes with 
$\unicode[STIX]{x1D70E}$, and hence also induces an element in 
$A_{0}/\unicode[STIX]{x1D707}_{6}$ which does not depend on the choices of the lifting of 
$a$.
The map attaching the Hodge structure on 
$N$ (preserved by the action of 
$\unicode[STIX]{x1D707}_{6}$) to 
$C$ is the occult period map for smooth nonhyperelliptic curves of genus 
$4$, which is an open embedding of the coarse moduli space 
${\mathcal{M}}_{4}^{\circ }$ of smooth nonhyperelliptic curves of genus 
$4$ into an arithmetic ball quotient 
$\unicode[STIX]{x1D6E4}\backslash {\mathcal{B}}^{9}$, where 
$\unicode[STIX]{x1D6E4}=\text{Aut}(N,\unicode[STIX]{x1D70E})/\unicode[STIX]{x1D707}_{6}$ is an arithmetic group acting on 
${\mathcal{B}}^{9}$. We confirm the conjecture in [Reference Kudla and RapoportKR12, Remark 8.2] by the following proposition.
Proposition 7.9. The group homomorphism 
$\text{Aut}(C)\longrightarrow A_{0}/\unicode[STIX]{x1D707}_{6}$ is an isomorphism.
We need the following lemmas.
Lemma 7.10. Let 
$U$ be a hyperbolic lattice, that is, with generators 
$x_{1},x_{2}$ such that 
$b_{U}(x_{1},x_{1})=b_{U}(x_{2},x_{2})=0$, 
$b_{U}(x_{1},x_{2})=1$. Then all possible automorphisms 
$\unicode[STIX]{x1D70C}$ of 
$U$ are in the list below:
(i)
$\unicode[STIX]{x1D70C}=\pm \text{id}$,(ii)
$\unicode[STIX]{x1D70C}(x_{1})=x_{2},\unicode[STIX]{x1D70C}(x_{2})=x_{1}$,(iii)
$\unicode[STIX]{x1D70C}(x_{1})=-x_{2},\unicode[STIX]{x1D70C}(x_{2})=-x_{1}$.
Proof. The proof of this lemma is straightforward. ◻
Lemma 7.11. Suppose 
$\unicode[STIX]{x1D701}$ to be an automorphism of the lattice 
$N$, then exact one of 
$\pm \unicode[STIX]{x1D701}$ is induced by an automorphism of 
$H^{2}(S,\mathbb{Z})$ preserving the hyperplane class 
$\unicode[STIX]{x1D702}\in H^{2}(S,\mathbb{Z})$.
Proof. The automorphism 
$\unicode[STIX]{x1D701}$ of 
$N$ induces an action on
$$\begin{eqnarray}A_{N}\cong A_{M}=({\textstyle \frac{1}{3}}U)/U=\{0,\pm [{\textstyle \frac{1}{3}}x_{1}],\pm [{\textstyle \frac{1}{3}}x_{2}],\pm [{\textstyle \frac{1}{3}}(x_{1}+x_{2})],\pm [{\textstyle \frac{1}{3}}(x_{1}-x_{2})]\}.\end{eqnarray}$$ Exactly one of 
$\pm \unicode[STIX]{x1D701}$ preserves 
$[\frac{1}{3}\unicode[STIX]{x1D702}_{0}]=[\frac{1}{3}(x_{1}+x_{2})]$. Without loss of generality, we assume that 
$\unicode[STIX]{x1D701}$ satisfies this property. Then 
$\unicode[STIX]{x1D701}$ must send 
$[\frac{1}{3}x_{1}]$ to 
$[\frac{1}{3}x_{1}]$ or 
$[\frac{1}{3}x_{2}]$, and the value 
$\unicode[STIX]{x1D701}([\frac{1}{3}x_{2}])$ is correspondingly determined. Combining with Lemma 7.10, there exists an automorphism of 
$M=U(3)$ which preserves 
$\unicode[STIX]{x1D702}$ and matches with 
$\unicode[STIX]{x1D701}$ on 
$N$. Thus, by Lemma 7.3, the automorphism 
$\unicode[STIX]{x1D701}$ is induced from an automorphism of the whole lattice 
$H^{2}(S,\mathbb{Z})$ which preserves 
$\unicode[STIX]{x1D702}$. This proves our lemma.◻
Lemma 7.12. Suppose there are two automorphism 
$\unicode[STIX]{x1D701}_{1}$, 
$\unicode[STIX]{x1D701}_{2}$ of the 
$K3$ lattice 
$H^{2}(S,\mathbb{Z})$ such that 
$\unicode[STIX]{x1D701}_{1}\big|_{N}=\unicode[STIX]{x1D701}_{2}\big|_{N}:N\longrightarrow N$. Then they coincide.
Proof. Since 
$\unicode[STIX]{x1D701}_{1},\unicode[STIX]{x1D701}_{2}$ act the same on 
$N$, they also act the same on 
$A_{N}\cong A_{M}$. By Lemma 7.10, we know that 
$\unicode[STIX]{x1D701}_{1},\unicode[STIX]{x1D701}_{2}$ act the same on 
$M$, and hence the same on the whole lattice 
$H^{2}(S,\mathbb{Z})$.◻
Proof of Proposition 7.9.
We first show the surjectivity. Let 
$\unicode[STIX]{x1D701}\in A_{0}$ be an automorphism of 
$N$ commuting with 
$\unicode[STIX]{x1D70E}$ and preserving the Hodge structure. By Lemma 7.11, one element in 
$\{\unicode[STIX]{x1D701},-\unicode[STIX]{x1D701}\}$ is induced by an automorphism of the whole lattice 
$H^{2}(S,\mathbb{Z})$ which preserves Hodge structure and 
$\unicode[STIX]{x1D702}$. By Theorem 7.1, this automorphism is induced by an automorphism of 
$S$. With the ambiguity of 
$\unicode[STIX]{x1D707}_{6}$ in mind, we may just assume that 
$\unicode[STIX]{x1D701}$ is induced by an automorphism 
$\widetilde{a}$ of 
$S$.
Since 
$\unicode[STIX]{x1D701}=\widetilde{a}^{\ast }\big|_{N}$ commutes with 
$\unicode[STIX]{x1D70E}$, by Lemma 7.12, we have 
$\unicode[STIX]{x1D70E}\widetilde{a}^{\ast }=\widetilde{a}^{\ast }\unicode[STIX]{x1D70E}$ on 
$H^{2}(S,\mathbb{Z})$. By Lemma 7.2 we have that 
$\widetilde{a}$ commutes with the Deck transformations of 
$S\longrightarrow Q$. Therefore, 
$\widetilde{a}$ is induced by an automorphism 
$a$ of 
$C$. We showed the surjectivity.
Next we show the injectivity. Let 
$a$ be an automorphism of 
$C$, inducing the trivial element in the group 
$A_{0}/\unicode[STIX]{x1D707}_{6}$. Then there is a lifting 
$\widetilde{a}$ of 
$a$ such that 
$\widetilde{a}^{\ast }\big|_{N}\in \unicode[STIX]{x1D707}_{6}$. We can compose 
$\widetilde{a}$ with Deck transformations; hence, we can assume 
$\widetilde{a}^{\ast }\big|_{N}\in \{\pm \text{id}\}$. Since 
$\widetilde{a}^{\ast }$ acts as identity on the hyperplane class of 
$S$, we must have 
$\widetilde{a^{\ast }}\big|_{N}=\text{id}$ and by Lemma 7.12, 
$\widetilde{a}^{\ast }=\text{id}$. Thus, by Lemma 7.2, 
$\widetilde{a}=\text{id}$, which implies that 
$a=\text{id}$. We showed the injectivity.◻
Theorem 7.13. The occult period map
$$\begin{eqnarray}\mathscr{P}:{\mathcal{M}}_{4}^{\circ }\longrightarrow \unicode[STIX]{x1D6E4}\backslash {\mathcal{B}}^{9}\end{eqnarray}$$ for smooth nonhyperelliptic curves of genus 
$4$ identifies the natural orbifold structures on 
${\mathcal{M}}_{4}^{\circ }$ and the image in 
$\unicode[STIX]{x1D6E4}\backslash {\mathcal{B}}^{9}$.
Proof. By [Reference KondōKon02], 
$\mathscr{P}$ is an open embedding of analytic spaces; combining with Proposition 7.9, we have that 
$\mathscr{P}$ identifies the orbifold structures on the source and image.◻
Acknowledgments
I thank my Ph.D advisor, Professor Eduard Looijenga, for guiding me to related papers and for his help along the way. I thank Professor Michael Rapoport for helpful communication which pointed out the main problems. Thanks to Ariyan Javanpeykar, Radu Laza, Jialun Li, Gregory Pearlstein, and Chenglong Yu for helpful comments. Finally, I am indebted to the anonymous reviewer for careful reading and helpful suggestions.
