Space of nilpotent orbits and extension of period maps (I): The weight 3 Calabi-Yau types
Haohua Deng
Department of mathematics, Duke University, Durham, North Carolina, 27708-0320

In Kato-Nakayama-Usui's theory, a certain space of nilpotent orbits can be constructed and serve as a completion of a given period map. This can be regarded as a generalization of Mumford's toroidal compactification for locally symmetric varieties.

Kato-Nakayama-Usui's construction requires the existence of a weak fan, which is not known in general for non-classical cases. In this paper, we show after some slight modifications, such weak fans exist for a large class of period maps of weight $3$ Calabi-Yau type. In particular, for these cases a Kato-Nakayama-Usui type completion can be constructed.

§ INTRODUCTION

§.§ Overview

Theories for compactifying locally symmetric varieties have been developed for decades and are now very rich. For example, in Satake-Baily-Borel's theory <cit.> and <cit.>, for a locally symmetric variety $\Gamma\backslash D$ they constructed a projective but highly singular compact model by adding cusps along the boundary. This is known as the Satake-Baily-Borel compactification $\overline{\Gamma\backslash D}^{\mathrm{BB}}$. To improve the quality of the compactification, Mumford et. al. introduced the theory of toroidal compactification in <cit.> which is not canonical in general, but will provide a smooth projective birational model for $\Gamma\backslash D$ with a decent choice of $\Gamma$-admissable polyhedral decomposition.

As an application of theories on compactifying locally symmetric varietes, Completing period mappings for which the Hodge varieties $\Gamma\backslash D$ are locally symmetric has also been well-investigated. These are called period maps of classical types in Hodge theory, which have been widely used to study the moduli space of polarized abelian varieties (weight $1$ case) and polarized K3 surfaces (weight $2$ case with $h^{2,0}=1$).

One of the major obstructions of using Hodge theory to study families of general algebraic varieties, especially the degenerating behaviors, is the absence of theories completing period maps of non-classical types. Some recent works, for example, <cit.>, <cit.>, showed some critical abstract properties of general period maps, but we still need theories suitable for applications in other fields like algebraic geometry or mirror symmetry.

In recent years, Kato-Usui, or Kato-Nakayama-Usui have developed new theories of extending period maps of general types in <cit.>, <cit.>. Their theory can be regarded as a generalization of Mumford's toroidal compacitification, but the resulting space is essentially different from the classical cases. Generally speaking, for a period mapping $S\xrightarrow[]{\varphi}\Gamma\backslash D$, the resulting enlarged space, which is written as $\Gamma\backslash D_{\Sigma}$, lies in a certain category called logarithmic manifolds, and its construction depends on a choice of a fan, or weak fan $\Sigma$ with certain distinguished properties. This enlarged space is called the space of nilpotent orbits or logarithmic polarized variation of Hodge structures.

One of the critical ingredients, also technical difficulties to apply Kato-Nakayama-Usui's theory, is to construct, or even prove the existence of $\Sigma$. The construction of $\Sigma$ for single-variable period maps is evident. In classical cases, Kato-Usui's theory agree with Mumford's toroidal compactification and $\Sigma$ can be chosen as <cit.> using polyhedral reduction theory. Besides these known cases, the existence of such an object is an open problem in general. Some recent works, for example <cit.> and <cit.> have shown the existence of such a fan or weak fan beyond trivial and classical cases.

§.§ Purpose and organization of the paper

The purpose of this paper is to show that Kato-Nakayama-Usui's theory can be applied to a large class of non-classical (multivariable) period maps. The main result can be summarized as follows:

Main Result (Theorem <ref>): For a period map $\varphi: S\rightarrow \Gamma\backslash D$ of weight $3$ Calabi-Yau type, if along the boundary divisors of $S$ we only acquire type I or type IV degenerations in the sense of <cit.>, then there exists a finite modification of the period map, called $\tilde{\varphi}: \tilde{S}\rightarrow \Gamma\backslash D$, such that this modified period map admits a Kato-Usui type extension.

A weight $3$ Calabi-Yau type limiting mixed Hodge structure $(W(N), F^{\bullet}, N)$ is called type I if:

\begin{equation}
N\neq 0, \ N^2=0, \ h^{3,0}=1,
\end{equation}

or called type IV if:

\begin{equation}
N^3\neq 0, N^4=0.
\end{equation}

They can also be characterized by their Hodge-Deligne diagrams, see Figure <ref> and Figure <ref>.

The meaning of finite modification will be explained in Section 6. Generally speaking, it suffices to prove the existence of $\Sigma$ which can provide such an extension. Indeed, $\Sigma$ can be obtained by taking all local monodromy nilpotent cones arising from the period map $\varphi$, then proceed a finitely generated subdivision. One of the central observations is the finiteness condition on subdivisions we need can be induced from the Siegel properties in the theory of arithmetic groups.

Our methods needs a slight modification on Kato-Nakayama-Usui's theory. Indeed, the object $\Sigma$ we will obtain is neither a fan in the sense of <cit.> nor a weak fan in the sense of <cit.>, but a weak fan of restricted type. However, this modified object still provides the completion we need and the proof is parallel to <cit.> and <cit.>.

In Section 2, we review some necessary Hodge theory background. In Section 3 a summary of Kato-Usui's theory with the minimum requirements will be presented. In Section 4, we will show how Kato-Nakayama-Usui's theory works for the classical weight $1$ case. In Section 5, we will show the main result: For type I and type IV degenerations in the weight $3$ Calabi-Yau case, the construction can be derived from the weight $1$ case, which proves our main result. In Section 6, we conclude the main result by explaining the relation between logarithmic modification and period mappings. We will also review the main results from <cit.> as an example. In Appendix A, we will give a more concrete overview on Kato-Usui's theory. We will also establish the validity of our modified definition on weak fan of restricted types. In Appendix B we will have some technical details properly addressed.

Acknowledgement: The author is grateful to Matt Kerr and Colleen Robles for sharing a lot of motivating ideas. The author also thanks Ben Bakker, Chongyao Chen, Patricio Gallardo, Philip Griffiths, Kazuya Kato, Bruno Klingler, Yilong Zhang and many others for related discussions.

§ PRELIMINARIES ON HODGE THEORY

Let $S$ be a smooth quasi-projective variety, $\bar{S}$ be projective with $\bar{S}\backslash S$ a simple normal crossing divisor. We assume there is a $\mathbb{Z}$-local system $\mathcal{V}$ on $S$ which induces a polarized variation of Hodge structures. This induces a period map:

\begin{equation}\label{periodmapgeneralform}
\varphi: S\rightarrow \Gamma \backslash D
\end{equation}

where $D$ is the classifying space of polarized Hodge structures defined on an integral lattice $H_{\mathbb{Z}}$ with integral polarization form $Q$. Moreover, denote $G=\mathrm{Aut}(H, Q)$, then $\Gamma\leq G_{\mathbb{Z}}$ is just the monodromy group of the local system $\mathcal{V}\rightarrow S$.

Such an abstract model can be realized in algebraic geometry: Suppose

\begin{equation}
\pi: \mathcal{X}\rightarrow S
\end{equation}

is a smooth projective family where $\mathcal{X}$ is smooth and $\pi$ is a holomorphic proper submersion. Moreover suppose for any $s\in S$, the fiber $X_s:=\pi^{-1}(s)$ is a smooth projective variety of fixed dimension $\mathrm{dim}_{\mathbb{C}}X_s=l$. Denote $\mathbb{Z}_{\mathcal{X}}$ as the $\mathbb{Z}$-constant sheaf on $\mathcal{X}$, then

\begin{equation}
\mathcal{V}:=R^l\pi_*(\mathbb{Z}_{\mathcal{X}})
\end{equation}

is a $\mathbb{Z}$-local system over $S$ whose fiber at $s\in S$ is $H:=H^l(X_s, \mathbb{Z})$. Take the Hodge decomposition

\begin{equation}
H_{\mathbb{C}}=H^l(X_s, \mathbb{Z})\otimes \mathbb{C}=\oplus_{p+q=l}H^{p,q}(X_s)
\end{equation}

on each fiber into consideration, this gives a variation of Hodge structure of weight $l$ on $S$ as well as a period map of the form (<ref>).

Moreover for $0\leq p\leq l$, let $\mathcal{F}^p$ be the subsheaves of $\mathcal{V}\otimes O_S$ which is known as Hodge bundles. Under the Gauss-Manin connection $\nabla$ of $\mathcal{V}$, we have the Griffiths transversality condition:

\begin{equation}\label{griffithstrans}
\nabla (\mathcal{F}^p) \subset \Omega^1_S\otimes \mathcal{F}^{p-1}, \ 0\leq p\leq l.
\end{equation}

In this paper we will try to extend $\varphi$ over $\bar{S}$ with image in a properly enlarged space of $\Gamma\backslash D$. We first consider the local model of period map, i.e., suppose the period map is defined over a product of puntured disks:

\begin{equation}\label{periodmaplocalform}
\varphi: (\Delta^*)^n\rightarrow \Gamma \backslash D
\end{equation}

where the local monodromy group $\Gamma$ is abelian and generated by $T_1,...,T_n$ where $T_j$ is obtained by the analytic continuation over a loop around the $j$-th coordinate divisor. If locally $S$ is isomorphic to $(\Delta^*)^k\times \Delta^{n-k}$, we can normalize it to the case $(\Delta^*)^n$ by setting $T_j=\mathrm{Id}$ for $j\geq k+1$. The following classical result is known as the Monodromy theorem.

All $T_i$'s are quasi-unipotent, which means there exist positive integers $w_i, v_i$ such that $(T_i^{w_i}-I)^{v_i}= 0$ for every $i$. Moreover, $v_i\leq l+1$ for each $i$.

Consider the Jordan decomposition $T_j=T_j^{\mathrm{ss}}T_j^{\mathrm{u}}$, the monodromy theorem implies $T_j^{\mathrm{ss}}$ has finite order. Therefore after a base change we can assume all $T_j$'s are unipotent.
Denote $\mathfrak{g}=\mathrm{Lie}(G)$ and $N_j=\log(T_j)$, then $N_j\in \mathfrak{g}_{\mathbb{Q}}$ are nilpotent elements defined over $\mathbb{Q}$, and $\{N_1,...,N_n\}$ generates a nilpotent abelian subalgebra of $\mathfrak{g}_{\mathbb{Q}}$.

Consider the universal covering map:

\begin{equation}
\mathrm{exp}(2\pi i\bullet): \mathfrak{H}^n\rightarrow (\Delta^*)^n,
\end{equation}

where $\mathfrak{H}:=\{z\in \mathbb{C}| \mathfrak{Im}(z)>0\}$ is the upper-half plane. We have the following commutative diagram:

\begin{equation}
\begin{tikzcd}
\mathfrak{H}^n \arrow[d, "\mathrm{exp}(2\pi i\bullet)"] \arrow[r, "\tilde{\varphi}"] & D \arrow[d] \\
B=(\Delta^*)^n \arrow[r, "\varphi"] & \Gamma \backslash D
\end{tikzcd}
\end{equation}

and the action of $T_1,\ldots,T_n$ on $D$ can be interpreted as:

\begin{equation}
\tilde{\varphi}(z_1,...,z_j+1,...,z_n)=T_j\tilde{\varphi}(z_1,...,z_n).
\end{equation}

Let $\Psi: \mathfrak{H}^n\rightarrow D$ and $\psi: (\Delta^*)^n\rightarrow D$ be defined as:

\begin{align}
\Psi(z_1,...,z_n)&:= \mathrm{exp}(-\sum_{1\leq j\leq n}z_jN_j)\tilde{\varphi}(z_1,...,z_n),\\
\psi(e^{2\pi iz_1},...,e^{2\pi iz_n})&:= \Psi(z_1,...,z_n).
\end{align}

Clearly $\psi$ is a well-defined map, and we have the following classical result which is known as Schmid's nilpotent orbit theorem.

The map $\psi$ can be extended holomorphically as:

\begin{equation}
\psi: \Delta^n\rightarrow \check{D}
\end{equation}

where $\check{D}$ is the compact dual of $D$. Denote $\psi(0)=:F_0^{\bullet}\in \check{D}$, the following properties hold:

(i) $N_jF_0^{p}\subset F_0^{p-1}$ for any $1\leq j\leq n$ and $0\leq p\leq l$;

(ii) $\mathrm{exp}(\sum_{j=1}^{n}z_jN_j)F_0^{\bullet}\in D$ for $z_j\in \mathbb{C}$ and $\mathfrak{Im}(z_j)>>0$ for any $j$.

In general, we have the following definition:

Suppose $N_1,...,N_n\in \mathfrak{g}_{\mathbb{Q}}$ are commuting nilpotent elements, and $F^{\bullet}\in \check{D}$. We call $\sigma:=\sum_{j=1}^{n}\mathbb{Q}_{\geq 0}N_j$ a nilpotent cone[Strictly speaking, $\sigma$ should be defined as the convex hull of $\{N_1,...,N_n\}$. For simplicity we assume $\sigma$ is not degenerate, equivalently $N_1,...,N_n$ are linearly independent.] and $Z:=\exp(\sigma_{\mathbb{C}})F^{\bullet}\subset \check{D}$ (resp. $Z^{\#}:=\exp(i\sigma_{\mathbb{R}})F^{\bullet}\subset \check{D}$) a nilpotent orbit (resp. nilpotent $i$-orbit) if the following conditions hold:

(i) $NF^{p}\subset F^{p-1}$ for any $N\in \sigma$ and $0\leq p\leq l$;

(ii) $\mathrm{exp}(\sum_{j=1}^{n}z_jN_j)F^{\bullet}\in D$ for $z_j\in \mathbb{C}$ and $\mathfrak{Im}(z_j)>>0$ for any $j$.

In Definition <ref> (i) is known as the Griffiths transversality condition and (ii) is called the positivity condition.

For any nilpotent element $N\in \mathfrak{g}_{\mathbb{Q}}$, there is a naturally associated Jacobson-Morozov weight filtration $W(N)_{\bullet}$ which is increasing and defined on $H_{\mathbb{Q}}$, see for example <cit.>. In particular, if the nilpotency index of $N$ is $\mu$, then $W(N)_{-\mu}=\mathrm{Img}(N^{\mu})$ and $W(N)_{\mu}=H_{\mathbb{Q}}$. In <cit.> Cattani and Kaplan proved the following result:

If $(\sigma, F^{\bullet})$ is a nilpotent orbit, then there is a weight filtration $W_{\bullet}$ on $H_{\mathbb{Q}}$ such that $W_{\bullet}=W(N)_{\bullet}$ for any $N$ in the relative interior of $\sigma$.

As a consequence, For a nilpotent cone $\sigma$, it make sense to define the weight filtration associated to $\sigma$ as $W(\sigma)_{\bullet}$. Notice if $\tau\leq \sigma$ is a proper face, $W(\sigma)_{\bullet}$ and $W(\tau)_{\bullet}$ are different in general. In the rest of this paper, we should use the following convention:

By a convex polyhedral cone $\sigma$, we should mean $\sigma$ is relatively open. In other words, $\sigma=\overline{\sigma}\backslash \{\hbox{Union of all proper faces of }\sigma\}$.

Let $W_{\bullet}$ be an increasing (weight) filtration defined on $H_{\mathbb{Q}}$ and $F^{\bullet}\in \check{D}$. We say $(W_{\bullet}, F^{\bullet})$ is a mixed Hodge structure (of weight $l$) polarized by $N$ on $(H,Q)$ if:

(i) $W_{\bullet} = W(N)_{\bullet}[-l]$.

(ii) $N(F^p)\subset F^{p-1}$.

(iii) Writing $P_{l+k}:=\mathrm{ker}\{N^{k+1}:\mathrm{Gr}^W_{l+k}\rightarrow\mathrm{Gr}^W_{l-k-2}\}$ and the non-degenerate bilinear form $Q_k(v,w):=Q(v,N^kw)$ on $P_{l+k}$, $F^{\bullet}$ induces a weight $l+k$ Hodge structure on $P_{l+k}$ polarized by $Q_k$.

The following theorem regarding the relation between nilpotent orbits and limiting mixed Hodge structures is classical, see for example <cit.>

$(\sigma, F^{\bullet})$ gives a nilpotent orbit if and only if $(W(\sigma)[-l]_{\bullet}, F^{\bullet})$ is a mixed Hodge structure polarized by any $N\in \sigma$.

The triple $(W(\sigma)[-l]_{\bullet}, F^{\bullet}, \sigma)$[In the rest of the paper, we will denote $W(\sigma)_{\bullet}$ as $W(\sigma)$ for convenience.] associated to a nilpotent orbit $(\sigma, F^{\bullet})$ is called the associated limiting mixed Hodge structure (LMHS).

§ WEAK FAN OF RESTRICTED TYPE AND KATO-USUI SPACES

In this section we briefly overview Kato-Usui's theory <cit.>, and introduce a modification of the theory which is suitable for applications.

§.§ Kato-Usui's main theorems

Let $\Sigma$ be a collection of rational nilpotent cones in $\mathfrak{g}_{\mathbb{Q}}$, and $|\Sigma|$ be its support in $\mathfrak{g}_{\mathbb{Q}}$. For $\sigma\in \Sigma$, set $\Gamma(\sigma):=\Gamma\cap \mathrm{exp}(\sigma_{\mathbb{R}})$. This is a multiplicative monoid, let $\Gamma(\sigma)^{\mathrm{gp}}$ be its associated group.

We say $\Sigma$ is strongly compatible with $\Gamma\leq G_{\mathbb{Z}}$ if the following holds:

(i) $\Sigma$ is closed under the action by $\mathrm{Ad}(\Gamma)$;

(ii) $\sigma_{\mathbb{R}}= \mathbb{R}_{\geq 0}<\mathrm{log}(\Gamma(\sigma))>$ for any $\sigma\in \Sigma$.

We say $\Sigma$ is a fan (resp. weak fan) if the following (i) and (ii) (resp. (i) and (ii')) hold:

(i) $\Sigma$ is closed under taking faces;

(ii) For any $\sigma, \tau\in \Sigma$, either $\sigma\cap \tau=\emptyset$ or $\sigma=\tau$;

(ii') For any $\sigma, \tau\in \Sigma$, suppose $\sigma\cap \tau$ is not empty, and there exists $F^{\bullet}\in \check{D}$ such that both $(\sigma, F^{\bullet})$ and $(\tau, F^{\bullet})$ are nilpotent orbits, then $\sigma=\tau$.

Now we suppose $\Sigma$ is a collection of rational nilpotent cones strongly compatible with $\Gamma$. Define the spaces of nilpotent orbits and nilpotent $i$-orbits:

\begin{align}\label{defspaceofnilpotentorbits}
D_{\Sigma}:&=\{(\sigma, Z) \ | \ \sigma\in \Sigma, Z \ \text{is a} \ \sigma-\text{nilpotent orbit} \ \}\\
\nonumber &=\sqcup_{\sigma\in \Sigma}B(\tau)
\end{align}

$B(\sigma, \Phi):=\{\text{$\sigma$-nilpotent orbits}\}$,

\begin{align}\label{defspaceofnilpotentiorbits}
D_{\Sigma}^{\#}:&=\{(\sigma, Z^{\#}) \ | \ \sigma\in \Sigma, Z^{\#} \ \text{is a} \ \sigma-\text{nilpotent $i$-orbit} \ \}\\
\nonumber &=\sqcup_{\sigma\in \Sigma}B^{\#}(\tau),
\end{align}

$B^{\#}(\sigma):=\{\text{$\sigma$-nilpotent $i$-orbits}\}$.

For $\sigma\in \Sigma$, we define $D_{\sigma}$ and $D_{\sigma}^{\#}$ by regarding {all faces of $\sigma$} as a collection of rational nilpotent cones strongly compatible with $\Gamma(\sigma)^{\mathrm{gp}}$. Notice that $D$ embeds into both $D_{\sigma}$ and $D_{\sigma}^{\#}$ by $F^{\bullet}\rightarrow (\{0\}, F^{\bullet})$, and $\Gamma$ acts on $D_{\Sigma}$ by:

\begin{equation}\label{gammaactiononnilporbit}
\gamma: (\sigma, F^{\bullet})\rightarrow (\mathrm{Ad}(\gamma)\sigma, \gamma F^{\bullet}).
\end{equation}

The main result of <cit.> can be summarized as follows.

Suppose $\Sigma$ is a fan which is strongly compatible with $\Gamma\leq G_{\mathbb{Z}}$ which is assumed to be neat, then:

(i) $\Gamma \backslash D_{\Sigma}$ admits a structure of logarithmic manifold which is also Hausdorff.

(ii) For any $\sigma\in \Sigma$, the natural map $\Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma}\rightarrow \Gamma\backslash D_{\Sigma}$ is locally a homeomorphism.

Let $\Gamma$ and $\Sigma$ be the same as above. For any period map $\varphi: S\rightarrow \Gamma \backslash D$ over a quasi-projective variety $S$ whose monodromy group is contained in $\Gamma$, $\varphi$ extends uniquely to:

$\overline{\varphi}: \bar{S}\rightarrow \Gamma\backslash D_{\Sigma}$

with $\varphi$ a morphism between logarithmic manifolds.

(i) We refer readers to <cit.> for the more general statement. The theorems stated here are special cases;

(ii) In <cit.>, $\Sigma$ is assumed to be a $\Gamma$-strongly compatible weak fan (instead of a fan), and the same results still hold.

We will give a more detailed introduction to Kato-Usui's theory as well as a sketch of proof for Theorem <ref> in Appendix A.

§.§ An alternative definition of fan

Besides the space of $\sigma$-nilpotent orbits (resp. $i$-orbits) $B(\sigma)$ (resp. $B^{\#}(\sigma)$), we also define:

$\tilde{B}(\sigma):=\{F^{\bullet}\in \check{D} | (\sigma, F^{\bullet}) \ \hbox{is a nilpotent orbit}\}$.

By Theorem <ref> and Theorem <ref>, we have:

$\tilde{B}(\sigma)=\{F^{\bullet}\in \check{D} | (W(\sigma)[-l], F^{\bullet}, \sigma) \ \hbox{is a limiting mixed Hodge structure}\}$.

Regarding the structure of $\tilde{B}(\sigma)\subset \check{D}$, <cit.> says for $F^{\bullet}\in \tilde{B}(\sigma)$:

\begin{equation}\label{tangentspaceofbn}
T_{F^{\bullet}}\tilde{B}(\sigma)=\mathfrak{z}(\sigma)\cap T_{F^{\bullet}}\check{D}
\end{equation}

\begin{equation}
\mathfrak{z}(\sigma):=\cap_{N\in \sigma_{\mathbb{R}}}\mathrm{Ker}(\mathrm{ad}(N))\subset \mathfrak{g}_{\mathbb{C}}.
\end{equation}

We also denote

\begin{equation}
Z(\sigma):=\{g\in G_{\mathbb{C}}|\mathrm{Ad}(g)N=N, \ \forall N\in \sigma_{\mathbb{Q}}\}\subset G_{\mathbb{C}}.
\end{equation}

as the centralizer of $\sigma$. Indeed, $Z(\sigma)\subset P_{W(\sigma)}(\mathbb{C})$ where $P_{W(\sigma)}$ is the $\mathbb{Q}$-parabolic subgroup of $G_{\mathbb{Q}}$ stabilizing $W(\sigma)$. Clearly $Z(\sigma)$ acts locally transitively on $\tilde{B}(\sigma)$.

The global structure of $\tilde{B}(\sigma)$ is more complicated. Fix $F^{\bullet}\in \tilde{B}(\sigma)$, and let $\tilde{B}(\sigma)^{\circ}$ be the $Z(\sigma)$-orbit containing $F^{\bullet}$. <cit.> says:

\begin{equation}\label{structureofbn}
\tilde{B}(\sigma)^{\circ}\cong Z(\sigma)\cap(U_{W(\sigma)}(\mathbb{C})\rtimes P_{W(\sigma)}(\mathbb{R}))/Z(\sigma)\cap \mathrm{Stab}_{F^{\bullet}}G_{\mathbb{C}}.
\end{equation}

However as <cit.> has pointed out, $\tilde{B}(\sigma)$ may not be connected, and may have more than one (but finitely many) $Z(\sigma)$-orbits. These orbits can be classified by the type of limiting mixed Hodge structures (LMHS) induced by the corresponding nilpotent orbit.

We give an example to illustrate the point. Suppose $D$ is the period domain parametrizing polarized Hodge structures of type $(1,a+b,a+b,1)$ on $(H_{\mathbb{Q}}, Q)$ (hence the dimension of $H$ is $2(1+a+b)$), then for a nilpotent element $N\in \mathfrak{g}_{\mathbb{Q}}$ such that

\begin{equation}
N\neq 0,  \ N^2=0, \ \mathrm{rank}(N)=a,
\end{equation}

$\tilde{B}(N)$ may contain $2$ disjoint $Z(\sigma)$-orbits parametrizing the following two types of LMHS described by their Hodge-Deligne diagrams, see Figure <ref>.

[-,line width=1.0pt] (0,0) – (0,3.5);
[-,line width=1.0pt] (0,0) – (3.5,0);
(0,3) circle (2pt);
at (-0.3,3) $1$;

(1,1) circle (2pt);
at (0.7,1) $a$;

(1,2) circle (2pt);
at (0.7,2.3) $b$;

(3,0) circle (2pt);
at (3,-0.3) $1$;

(2,1) circle (2pt);
at (2.3,0.7) $b$;

(2,2) circle (2pt);
at (2.3,2) $a$;

[-stealth](1.8,1.8) – (1.2,1.2) [line width = 1.0pt];

\mspace{50mu}
[-,line width=1.0pt] (0,0) – (0,3.5);
[-,line width=1.0pt] (0,0) – (3.5,0);
(1,3) circle (2pt);
at (1,3.3) $1$;

(0,2) circle (2pt);
at (-0.3,2) $1$;

(1,1) circle (2pt);
at (1,0.7) $a-2$;

(1,2) circle (2pt);
at (1,2.3) $b+1$;

(3,1) circle (2pt);
at (3.3,1) $1$;

(2,0) circle (2pt);
at (2,-0.3) $1$;

(2,1) circle (2pt);
at (2,0.7) $b+1$;

(2,2) circle (2pt);
at (2,2.3) $a-2$;

[-stealth](1.8,1.8) – (1.2,1.2) [line width = 1.0pt];
[-stealth](0.8,2.8) – (0.2,2.2) [line width = 1.0pt];
[-stealth](2.8,0.8) – (2.2,0.2) [line width = 1.0pt];

Type I and II LMHS for $(1,a+b,a+b,1)$

Here we use the type of LMHS defined by <cit.>. Particularly for the $(1,a+b,a+b,1)$ period domain, type $X$ means the Hodge numbers $\{h^{p,q}\}_{p,q\in \mathbb{Z}}$ satisfy $h^{4-X, 0}=1$. A quick way to distinguish these $2$ types of LMHS is checking whether $N$ annihilates $F^{3}$.

Let $\Phi$ be an index set of limiting mixed Hodge structure types (corresponds to some fixed Hodge numbers $h^{p,q}$). We say a nilpotent orbit $(\sigma, F^{\bullet})$ is of type $\Phi$ if the LMHS induced by the nilpotent orbit has type included by $\Phi$.

For example, in the example above, if a nilpotent orbit $(\sigma, F^{\bullet})$ gives type I LMHS, we say the nilpotent orbit is of type I. In this case if we let the index set $\Phi=\{\text{Pure LMHS}, \text{Type I LMHS}\}$, then $(\sigma, F^{\bullet})$ is of type $\Phi$.

Next we introduce alternative definitions to (<ref>) and (<ref>). Define:

\begin{align}
D_{\Sigma, \Phi}:&=\{(\sigma, Z) \ | \ \sigma\in \Sigma, Z \ \text{is a} \ \sigma-\text{nilpotent orbit of type $\Phi$} \ \}\\
\nonumber &=\bigsqcup_{\sigma\in \Sigma}B(\sigma, \Phi),
\end{align}

$B(\sigma, \Phi):=\{\text{$\sigma$-nilpotent orbits of type $\Phi$}\}$,

\begin{align}
D_{\Sigma, \Phi}^{\#}:&=\{(\sigma, Z^{\#}) \ | \ \sigma\in \Sigma, Z^{\#} \ \text{is a} \ \sigma-\text{nilpotent $i$-orbit of type $\Phi$} \ \}\\
\nonumber &=\bigsqcup_{\sigma\in \Sigma}B^{\#}(\sigma, \Phi),
\end{align}

$B^{\#}(\sigma, \Phi):=\{\text{$\sigma$-nilpotent $i$-orbits of type $\Phi$}\}$.

The action of $\Gamma$ on nilpotent orbits preserves the associated LMHS type, therefore the natural inclusions

\begin{align}
D_{\Sigma, \Phi} &\hookrightarrow D_{\Sigma},\\
\nonumber D^{\#}_{\Sigma, \Phi} &\hookrightarrow D^{\#}_{\Sigma}
\end{align}

descend to inclusions

\begin{align}
\Gamma\backslash D_{\Sigma, \Phi} &\hookrightarrow \Gamma\backslash D_{\Sigma},\\
\nonumber \Gamma\backslash D^{\#}_{\Sigma, \Phi} &\hookrightarrow \Gamma\backslash D^{\#}_{\Sigma}.
\end{align}

In Definition <ref>, if we keep condition (i) but replace condition (ii) or (ii') by the following:

(ii”)  For any $\sigma, \tau\in \Sigma$, suppose $(\sigma\cap \tau)^{\circ}$ is not empty, and there exists $F^{\bullet}\in \check{D}$ such that both $(\sigma, F^{\bullet})$ and $(\tau, F^{\bullet})$ are nilpotent orbits with type $\Phi$ LMHS, then $\sigma=\tau$.

then we say $\Sigma$ is a type $\Phi$ weak fan. When there is no need to emphasize the set $\Phi$, we also say $\Sigma$ is a weak fan of restricted type.

The condition (ii”) in Definition <ref> is generally weaker than (ii) and (ii') in Definition <ref> unless $\Phi$ contains all possible LMHS types polarized by any $\sigma\in \Sigma$.

Now suppose for a given period map

$\varphi: S\rightarrow \Gamma \backslash D$,

$\Phi$ denotes all LMHS types arising via Schmid's nilpotent orbit theorem. We present the following variation of Kato-(Nakayama)-Usui's theorems.

Suppose $\Sigma$ is a type-$\Phi$ weak fan strongly compatible with $\Gamma\leq G_{\mathbb{Z}}$ which is assumed to be neat, then:

(i) $\Gamma \backslash D_{\Sigma, \Phi}$ admits a structure of logarithmic manifold as well as Hausdorff topological space.

(ii) For any $\sigma\in \Sigma$, the canonical projection $\Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma, \Phi}\rightarrow \Gamma\backslash D_{\Sigma, \Phi}$ is locally an isomorphism of logarithmic manifolds.

Let $\Gamma$ and $\Sigma$ be the same as before. For a period map $\varphi: S\rightarrow \Gamma \backslash D$ over a quasi-projective variety $S$ whose monodromy group is contained in $\Gamma$. If all local monodromy nilpotent orbits along $\partial S$ have types contained in $\Phi$, then $\varphi$ extends uniquely to:

$\overline{\varphi}: \bar{S}\rightarrow \Gamma\backslash D_{\Sigma, \Phi}$

with $\overline{\varphi}$ a morphism between logarithmic manifolds.

The proof will be left to the Appendix A. The motivation for this definition is the existence of type $\Phi$ weak fan relies on $\Phi$. This feature is not captured by the original definitions in <cit.> or <cit.>.

In the following sections, we will prove a weak fan (in the sense of <cit.>) exists for a period map with $D$ parametrizing weight $1$ polarized Hodge structures. This can be regarded as an alternative approach to <cit.>'s toroidal compactification of locally symmetric varieties.

§ WEAK FAN: THE CLASSICAL WEIGHT $1$ CASE

§.§ Basic settings
We consider the period domain $D_{(g,g)}$ parametrizing weight $1$ polarized Hodge structures on $(H_{\mathbb{Z}}, Q)$ with rank$(H_{\mathbb{Z}})=2g$. We have:

\begin{equation}
D_{(g,g)}\cong \mathrm{Sp}(2g, \mathbb{R})/\mathrm{U}(g)
\end{equation}

Regarding the Hodge-theoretic degenerations, the limiting mixed Hodge structures $H_{\mathbb{C}}=\oplus_{p,q}H^{p,q}$ are characterized by $0\leq a=h^{1,1}\leq g$.

[-,line width=1pt] (0,0) – (0,3.5);
[-,line width=1pt] (0,0) – (3.5,0);
(0,3) circle (3pt);
at (-0.8,3) $g-a$;

(0,0) circle (3pt);
at (-0.3,-0.3) $a$;

(3,0) circle (3pt);
at (3,-0.6) $g-a$;

(3,3) circle (3pt);
at (3.3,3.3) $a$;

[-stealth](2.8,2.8) – (0.2,0.2) [line width = 1.5pt];

\mspace{50mu}
[-,line width=1.0pt] (-3.5,0) – (3.5,0);
[-,line width=1.0pt] (0,-3.5) – (0,3.5);

(0,0) circle (3pt);
(2.5,0) circle (3pt);
(-2.5,0) circle (3pt);
(2.5,-2.5) circle (3pt);
(0,-2.5) circle (3pt);
(-2.5,-2.5) circle (3pt);
(2.5,2.5) circle (3pt);
(0,2.5) circle (3pt);
(-2.5,2.5) circle (3pt);

[-stealth](-0.2,2.3) – (-2.3,0.2) [line width = 1.5pt];
[-stealth](2.3,2.3) – (0.2,0.2) [line width = 1.5pt];

[-stealth](2.3,-0.2) – (0.2,-2.3) [line width = 1.5pt];
[-stealth](-0.2,-0.2) – (-2.3,-2.3) [line width = 1.5pt];

Type $\mathrm{I}_a$ LMHS for $(g,g)$

Suppose $\varphi: S\rightarrow \Gamma\backslash D$ is a period map of type $(g,g)$ where $S$ is quasi-projective with normal crossing boundary divisors, and $\Gamma\leq \mathrm{Sp}(2g, \mathbb{Z})$ is assumed to be neat.

Let $\Sigma$ be the collection of all local monodromy nilpotent cones arising from $\varphi$. A description of $\Sigma$ is as follows: Since $\bar{S}$ is compact and $\bar{S}\backslash S$ is a normal crossing divisor, there exists a finite open cover $\{U_m\}_{1\leq m\leq M}$ such that $\cup_{1\leq m\leq M}U_s=\bar{S}$, and for each $1\leq m\leq M$, there exists $0\leq k_m\leq n$ such that

\begin{equation}
U_m\cap S\cong (\Delta^{*})^{k_m}\times \Delta^{n-k_m}.
\end{equation}

Fix a base point $s_0\in S$, each of $U_m$ gives a $k_m$-dimensional monodromy nilpotent cone $\sigma_m$ and all of its faces, and for any point $s\in U_M\cap (\bar{S}\backslash S)$, by Schmid's nilpotent orbit theorem there is a nilpotent orbit associated to $s$ well-defined up to monodromy.

By definition, $\Sigma$ is obtained by taking all $\mathrm{Ad}(\Gamma)$-orbits of $\{\sigma_m \ | \ 1\leq m\leq M\}$ and all of their faces. Since changing the base point $s_0$ is equivalent to conjugating by an element in $G_{\mathbb{Q}}$, which does not change the topology of $\Sigma$, we may assume $\Sigma$ is canonically associated to the period map.

In this chapter we will prove the following theorem.

There exists a polyhedral subdivision of $\Sigma$, denoted $\hat{\Sigma}$, such that $\hat{\Sigma}$ is a weak fan strongly compatible with $\Gamma$ and finitely generated under the action by $\mathrm{Ad}(\Gamma)$. In other words, for any $\sigma, \tau\in \hat{\Sigma}$, $\tilde{B}(\sigma)\cap \tilde{B}(\tau)\neq \emptyset$ and $\sigma\cap \tau\neq \emptyset$ imply $\sigma=\tau$.

In Appendix B, we will see $\hat{\Sigma}$ constructed in Theorem <ref> is actually a fan. Moreover, by Proposition <ref>, fan and weak fan are equivalent in the classical cases.

§.§ The Hodge-Tate degeneration

In this subsection we shall consider the Hodge-Tate degenerations, i.e. In Figure <ref> we set $a=g$.

Let $\Sigma_{\mathrm{HT}}\subset \Sigma$ be the subcollection generated by all local monodromy nilpotent cones arising from $\varphi$ whose corresponding LMHS are of Hodge-Tate type. Note that under the assumption cones in $\Sigma$ are relatively open, $\Sigma_{\mathrm{HT}}\subset \Sigma$ is not closed under taking faces in general.

Let $\sigma, \tau\in \Sigma_{\mathrm{HT}}$ be two nilpotent cones. Suppose there exists a sequence $\{\gamma_i\}_{i\geq 0}\subset \Gamma$ such that there exists $0\neq N_i\in \sigma\cap \mathrm{Ad}(\gamma_i)\tau$ and $\{F_i^{\bullet}\}\subset \check{D}$, we have

\begin{equation}\label{busterclassicalforHT}
(\sigma, F_i^{\bullet}), (\mathrm{Ad}(\gamma_i)\tau, F_i^{\bullet}), (N_i, F_i^{\bullet})
\end{equation}

are all nilpotent orbits producing Hodge-Tate type LMHS. Theorem <ref> implies $\sigma$ and all $\mathrm{Ad}(\gamma_i)\tau$ generate the same Jacobson-Morozov weight filtration $W(\sigma)$, therefore $\gamma_i\in \Gamma\cap P_{W(\sigma)}$ for any $i$. Moreover, we can assume $\gamma_0=\mathrm{Id}$, otherwise we shift the sequence $\{\gamma_i\}$ by $1$.

Next, we will prove the following proposition.

We can choose all $F_i^{\bullet}=F_0^{\bullet}$. In other words,

$(\sigma, F_0^{\bullet}), (\mathrm{Ad}(\gamma_i)\tau, F_0^{\bullet}), (N_i, F_0^{\bullet})$

are all nilpotent orbits.

\begin{align}
H_{\mathbb{C}}&=\oplus H_i^{p,q}=H_i^{1,1}\oplus H_i^{0,0}, \\
\nonumber \mathfrak{g}_{\mathbb{C}}&=\oplus \mathfrak{g}_i^{p,q}=\mathfrak{g}_i^{1,1}\oplus \mathfrak{g}_i^{0,0}\oplus \mathfrak{g}_i^{-1,-1}
\end{align}

as the Deligne splittings of the LMHS $(W(\sigma)[-1], F_i^{\bullet}, \sigma)$ and its adjoint. In this case (<ref>) is read as:

\begin{equation}\label{localtangentspaceareequal}
T_{F_i^{\bullet}}\tilde{B}(\sigma)= T_{F_i^{\bullet}}\tilde{B}(\mathrm{Ad}(\gamma_i)\tau)= T_{F_i^{\bullet}}\tilde{B}(N_i)\cong \mathfrak{g}_i^{-1,-1}.
\end{equation}

We also notice that $F_0^{\bullet}\in \tilde{B}(\sigma)$. Therefore (<ref>) and (<ref>) imply there exists $g_i\in Z(\sigma)\cap Z(\mathrm{Ad}(\gamma_i)\tau)\cap Z(N_i)$ such that $F_0^{\bullet}=g_iF_i^{\bullet}$, hence we can replace $F_i^{\bullet}$ by $F_0^{\bullet}$ without destroying the properties of the triple (<ref>).

From now on we denote:

\begin{align}
H_{\mathbb{C}}&=\oplus H^{p,q}=H^{1,1}\oplus H^{0,0}\\
\nonumber \mathfrak{g}_{\mathbb{C}}&=\oplus \mathfrak{g}^{p,q}=\mathfrak{g}^{1,1}\oplus \mathfrak{g}^{0,0}\oplus \mathfrak{g}^{-1,-1}
\end{align}

as the Hodge-Deligne splittings for $(W(\sigma)[-1], F_0^{\bullet}, \sigma)$ and its adjoint LMHS. Since for any $i$ we have $W(\mathrm{Ad}(\gamma_i)\tau)=W(\sigma)$, we have $\mathrm{Ad}(\gamma_i)\tau \subset \mathfrak{g}^{-1,-1}$.

The following proposition can be regarded as the local version of Theorem <ref>:

Regarding the collection of triples in Proposition <ref>, there exists a finite polyhedral decomposition $\Sigma(\sigma)$ of $\sigma$ such that $\sigma \cap \mathrm{Ad}(\gamma_i)\tau$ is the union of some $\{\sigma_k\}\subset \Sigma(\sigma)$ for any $i$.

We claim that to prove Proposition <ref>, it suffices to prove the following:

For the set $\{\gamma_i\}\in \Gamma$ given in Proposition <ref>, its image under the projection

\begin{equation}
g\in G_{\mathbb{C}} \rightarrow \overline{g}\in  Z(\sigma)\backslash G_{\mathbb{C}}/ Z(\tau)
\end{equation}

is finite, where $Z(\sigma)\leq G_{\mathbb{C}}$ is the centralizer of $\sigma$.

The reason is the intersection $\sigma \cap \mathrm{Ad}(\gamma_i)\tau\subset \sigma$ is invariant under left multiplication on $\gamma_i$ by $Z(\sigma)$ and right multiplication on $\gamma_i$ by $Z(\tau)$, therefore the choice of subdivision $\Sigma(\sigma)$ is determined only by the class of $\gamma_i$ in $Z(\sigma)\backslash G_{\mathbb{C}}/Z(\tau)$, and we only have to consider a finite cell complex as a consequence of Proposition <ref>. In Appendix B the way to obtain such a finite polyhedral decomposition will be explained.

The remainder of this section will be dedicated to proving Proposition <ref>.

\begin{align}
&\mathcal{Z}(\sigma):=\mathrm{exp}(\sigma_{\mathbb{C}})F_0^{\bullet}=:(\sigma, F_0^{\bullet})\\
\nonumber  &\mathcal{Z}^{\#}(\sigma):=\mathrm{exp}(i\sigma_{\mathbb{R}})F_0^{\bullet}=:(\sigma, F_0^{\bullet})^{\#}
\end{align}

as the nilpotent orbit and nilpotent $i$-orbit associated to $(W(\sigma)[-1], F_0^{\bullet}, \sigma)$. For different LMHS's we use the similar notation. The triplet of nilpotent orbit:

\begin{align}
&\mathcal{Z}(\sigma):=(\sigma, F_0^{\bullet}),\\
\nonumber &\mathcal{Z}_i(\tau):=(\mathrm{Ad}(\gamma_i)\tau, F_0^{\bullet}),\\
\nonumber &\mathcal{Z}_i(N_i):=(0\neq N_i\in \sigma\cap \mathrm{Ad}(\gamma_i)\tau, F_0^{\bullet})
\end{align}

induces the triplet of nilpotent $i$-orbit:

\begin{align}\label{gangstatriple2}
&\mathcal{Z}^{\#}(\sigma):=(\sigma, F_0^{\bullet})^{\#},\\
\nonumber &\mathcal{Z}_i^{\#}(\tau):=(\mathrm{Ad}(\gamma_i)\tau, F_0^{\bullet})^{\#},\\
\nonumber &\mathcal{Z}_i^{\#}(N_i):=(0\neq N_i\in \sigma\cap \mathrm{Ad}(\gamma_i)\tau, F_0^{\bullet})^{\#}
\end{align}

which satisfies the third is contained in the intersection of the first two.

There exist open convex polyhedral cones $\mathcal{C}(\sigma), \mathcal{C}(\tau)\subset \mathfrak{g}_{\mathbb{Q}}\cap\mathfrak{g}^{-1,-1}$ such that

\begin{align}
\sigma&\subset \mathcal{C}(\sigma),\\
\nonumber \tau&\subset \mathcal{C}(\tau),\\
\nonumber F_0^{\bullet}&\in \tilde{B}(\mathcal{C}(\sigma))\cap \tilde{B}(\mathcal{C}(\tau)).
\end{align}

For a fixed $F_0^{\bullet}$, the Griffiths transversality condition in Definition <ref> is trivial in the classical case.
Let $\mathcal{C}\subset \mathfrak{g}^{-1,-1}$ be the subset containing all $N\in \mathfrak{g}^{-1,-1}$ such that $(N, F_0^{\bullet})$ is a nilpotent orbit of Hodge-Tate type. Moreover,  $\mathcal{C}$ is open convex according to <cit.>[$\mathcal{C}$ is a homogeneous self-adjoint cone in the sense of <cit.>.]. It is possible to choose a set of vectors $\{v_i\}\subset \overline{\mathcal{C}}$ containing all dimension $1$ faces of $\sigma$, and whose convex hull is open in $\overline{\mathcal{C}}$, hence we can just take $\mathcal{C}(\sigma)$ to be the interior of this convex hull. $\mathcal{C}(\tau)$ can be constructed in the same way.

Another way to see Proposition <ref> is using the fact that in $\mathcal{C}$, polyhedral cones and Siegel sets are cofinal. See <cit.>.

Moreover, we may assume $\mathcal{C}(\sigma)$ and $\mathcal{C}(\tau)$ are both simplicial. Otherwise suppose $\mathcal{C}(\sigma)$ is not simplicial, we proceed a finite simplicial decomposition on $\mathcal{C}(\sigma)$[Such a simplicial decomposition can be chosen as a star subdivision introduced in <cit.>]. If the sequence $\{N_i\}$ in Proposition <ref> is infinite, there exists an open simplicial cone $\mathcal{C}(\sigma)^{'}$ in the simplicial decomposition of $\mathcal{C}(\sigma)$ and an infinite subsequence of $\{N_i\}$ lie in $\mathcal{C}(\sigma)^{'}$, then we can replace $\sigma$ by $\sigma\cap \mathcal{C}(\sigma)^{'}$ without destroying the properties of triples in (<ref>). The argument for $\mathcal{C}(\tau)$ is similar.

Now we suppose the open simplicial cones $\mathcal{C}(\sigma), \mathcal{C}(\tau)\subset \mathfrak{g}_{\mathbb{Q}}\cap\mathfrak{g}^{-1,-1}$ are generated by:

\begin{align}
\mathcal{C}(\sigma)&=<N_1,...,N_k>_{\mathbb{Q}>0}\\
\nonumber \mathcal{C}(\tau)&=<M_1,...,M_k>_{\mathbb{Q}>0}.
\end{align}

with $k=\mathrm{dim}_{\mathbb{Q}}(\mathfrak{g}^{-1,-1})$. Denote

\begin{align}
U(\mathcal{C}(\sigma))&:=\{\mathrm{exp}(\sum_{j=1}^k z_jN_j)F_0^{\bullet} \ | \ 0\leq \mathfrak{Re}(z_j)\leq 1\}\\
\nonumber U_i(\mathcal{C}(\tau))&:=\{\mathrm{exp}(\sum_{j=1}^k z_j(\gamma_iM_j\gamma_i^{-1}))F_0^{\bullet} \ | \ 0\leq \mathfrak{Re}(z_j)\leq 1\}
\end{align}

as the neighborhood of $\mathcal{Z}^{\#}(\mathcal{C}(\sigma))$ (resp. $\mathcal{Z}^{\#}_i(\mathcal{C}(\tau))$) in $\mathcal{Z}(\mathcal{C}(\sigma))$ (resp. $\mathcal{Z}_i(\mathcal{C}(\tau))$) bounded in the real directions. Clearly $\mathcal{Z}^{\#}_i(N_i)\subset U(\mathcal{C}(\sigma))$.

There exists

\begin{equation}
g_i\in \mathrm{Sp}(2g, \mathbb{Z})\cap \mathrm{exp}(\mathfrak{g}^{-1,-1})
\end{equation}

such that

\begin{equation}
\mathcal{Z}_i^{\#}(N_i)\subset U(\mathcal{C}(\sigma))\cap g_iU_0(\mathcal{C}(\tau)).
\end{equation}

According to the assumptions, we have:

\begin{equation}
F_0^{\bullet}, \ \gamma_iF_0^{\bullet}\in \tilde{B}(\mathrm{Ad}(\gamma_i)\tau).
\end{equation}

Since $\mathrm{Ad}(\gamma_i)\mathcal{C}(\tau)$ is open in $\mathfrak{g}^{-1,-1}=T_{\gamma_iF_0^{\bullet}}\tilde{B}(\mathrm{Ad}(\gamma_i)\tau)$ (see (<ref>)), we can assume

\begin{equation}
F_0^{\bullet}=\mathrm{exp}(\sum_{j=1}^{k} z^i_j(\gamma_iM_j\gamma_i^{-1}))\gamma_iF_0^{\bullet}
\end{equation}

for some $\{z^i_j\}\subset \mathbb{C}$, hence we can just take

\begin{equation}
\mu_i:=\mathrm{exp}(-\sum_{j=1}^{k} \lfloor\mathfrak{Re}(z^i_j)\rfloor (\gamma_iM_j\gamma_i^{-1}))\in \mathrm{Sp}(2g, \mathbb{Z})\cap \mathrm{exp}(\mathfrak{g}^{-1,-1}),
\end{equation}

if we set

\begin{equation}
\end{equation}

since $\mu_i\in \mathrm{exp}(\mathfrak{g}^{-1,-1})$ centralizes $\mathfrak{g}^{-1,-1}$, we have

\begin{equation}
F_0^{\bullet}=\mathrm{exp}(\sum_{j=1}^{k} w^i_j(\gamma_iM_j\gamma_i^{-1}))g_iF_0^{\bullet}
\end{equation}

with $\{w^i_j\}\subset \mathbb{C}$ and $0\leq \mathfrak{Re}(w^i_j)\leq 1$, this implies we have

\begin{equation}
\mathcal{Z}^{\#}_i(N_i)\subset g_iU_0(\mathcal{C}(\tau))
\end{equation}

as $F_0^{\bullet}\in g_iU_0(\mathcal{C}(\tau))$ and $N_i\in \mathrm{Ad}(\gamma_i)\tau\subset \mathrm{Ad}(\gamma_i)\mathcal{C}(\tau)$.

To sum up, we must have for any $i$,

\begin{equation}\label{intersection2}
U(\mathcal{C}(\sigma))\cap g_iU_0(\mathcal{C}(\tau))\neq \emptyset
\end{equation}

and it contains a nilpotent $i$-orbit (i.e., $(N_i, F_0^{\bullet})^{\#}$).

On the other hand, for any $K>0$, define:

\begin{align}
U(\mathcal{C}(\sigma))_{>K}&:=\{\mathrm{exp}(\sum_{j=1}^{k} z^i_jN_j)F_0^{\bullet} \ | \ 0\leq \mathfrak{Re}(z^i_j)\leq 1, \mathfrak{Im}(z^i_j)>K\}\\
\nonumber U_0(\mathcal{C}(\tau))_{>K}&:=\{\mathrm{exp}(\sum_{j=1}^{k} z^i_jM_j)F_0^{\bullet} \ | \ 0\leq \mathfrak{Re}(z^i_j)\leq 1, \mathfrak{Im}(z^i_j)>K\}
\end{align}

We fix a $K_0>0$ such that $U(\mathcal{C}(\sigma))_{>K_0}$, $U_0(\mathcal{C}(\tau))_{>K_0}\subset D$. We know from <cit.> for any $\epsilon>0$, there exist two finite collections of Siegel sets $\{\mathfrak{S}_p\}$, $\{\mathfrak{T}_q\}$ of $D$
such that

\begin{align}
U(\mathcal{C}(\sigma))_{> K_0+\epsilon}&\subset \cup \mathfrak{S}_p,\\
\nonumber U_0(\mathcal{C}(\tau))_{> K_0+\epsilon}&\subset \cup \mathfrak{T}_q.
\end{align}

Also by the basic Siegel properties[See for example, <cit.>], for any arithmetic subgroup $\Gamma\leq \mathrm{Sp}(2g,\mathbb{Q})$, the set

\begin{equation}\label{siegelpropertyarithmeticgroup}
S(\sigma, \tau):=\{\gamma\in \Gamma|(\cup \mathfrak{S}_p)\cap (\cup \gamma\mathfrak{T}_q)\neq \emptyset\}
\end{equation}

is finite. Since the intersection in (<ref>) contains a non-trivial nilpotent $i$-orbit $\mathcal{Z}^{\#}_i(N_i)$, combine with the evident fact that

\begin{align}
\mathcal{Z}^{\#}_i(N_i)&\cap U(\mathcal{C}(\sigma))_{> K_0+\epsilon}\neq \emptyset\\
\nonumber \mathcal{Z}^{\#}_i(N_i)&\cap g_iU_0(\mathcal{C}(\tau))_{> K_0+\epsilon}\neq \emptyset,
\end{align}

we conclude the sequence $\{g_i=\mu_i\gamma_i\}$ must lie in the set $S(\sigma, \tau)$ thus contain only finitely many different elements, therefore $\{\gamma_i\}$ only gives finitely many $Z(\sigma)$-right cosets (as $\gamma_i=\mu_i^{-1}g_i$ and all $\mu_i\in Z(\sigma)$). To sum up, we get the following proposition, which is stronger than Proposition <ref>:

For the set $\{\gamma_i\}\in \Gamma$ given in Proposition <ref>, its image under the projection

\begin{equation}
g\in G_{\mathbb{C}} \rightarrow \overline{g}\in  Z(\sigma)\backslash G_{\mathbb{C}}
\end{equation}

is finite, where $Z(\sigma)\leq G_{\mathbb{C}}$ is the centralizer of $\sigma$.

The necessity of stating Proposition <ref> with $(Z(\sigma), Z(\tau))$-double cosets instead of only $Z(\sigma)$-right cosets will be clear in future works where we will treat more general cases.

§.§ Arbitrary degenerations

We will generalize results in the last section to all cones in $\Sigma$. The strategy is to translate general cases into the Hodge-Tate case.

Like before, let $\sigma, \tau\in \Sigma$ be two monodromy nilpotent cones. Suppose there exists a (countable) sequence $\{\gamma_i\}_{i\geq 0}\subset \Gamma$ such that there exists $0\neq N_i\in \sigma\cap \mathrm{Ad}(\gamma_i)\tau$ and $\{F_i^{\bullet}\}\subset \check{D}$, we have

$(\sigma, F_i^{\bullet}), (\mathrm{Ad}(\gamma_i)\tau, F_i^{\bullet}), (N_i, F_i^{\bullet})$

are all nilpotent orbits producing type $\mathrm{I}_a$ LMHS with $0\leq a\leq g$, when $a=g$ this is the Hodge-Tate degeneration, see Figure <ref>, and again we assume $\gamma_0=\mathrm{Id}$.

We can choose all $F_i^{\bullet}=F_0^{\bullet}$. In other words,

$(\sigma, F_0^{\bullet}), (\mathrm{Ad}(\gamma_i)\tau, F_0^{\bullet}), (N_i, F_0^{\bullet})$

are all nilpotent orbits.

The proof of Proposition <ref> applies verbatim.

Similar to Proposition <ref>, we need to prove the following proposition:

Regarding the collection of triples in Proposition <ref>, there exists a finite polyhedral decomposition $\Sigma(\sigma)$ of $\sigma$ such that $\sigma \cap \mathrm{Ad}(\gamma_i)\tau$ is the union of some $\{\sigma_k\}\subset \Sigma(\sigma)$ for any $i$.

Now the splitting of $(W(\sigma)[-1], F_0^{\bullet}, \sigma)$ becomes:

\begin{align}
H_{\mathbb{C}}&=H^{1,1}\oplus H^{1,0}\oplus H^{0,1}\oplus H^{0,0},\\
\nonumber \mathfrak{g}_{\mathbb{C}}&=\oplus_{-1\leq p,q\leq 1} \mathfrak{g}^{p,q}.
\end{align}

with $\sigma$, $\mathrm{Ad}(\gamma_i)\tau\subset \mathfrak{g}^{-1,-1}$. Consider the splitting:

\begin{equation}
H_{\mathbb{C}}=H_1\oplus H_2
\end{equation}

\begin{align}
H_1&:=H^{1,0}\oplus H^{0,1};\\
\nonumber H_2&:=H^{0,0}\oplus H^{1,1}.
\end{align}

Clearly $H_2$ is defined over $\mathbb{R}$ and $\mathfrak{g}^{-1,-1}$ acts on $H_1$ trivially. For any $N\in \mathfrak{g}^{-1,-1}$ denote $\bar{N}\in \mathrm{End}(H_2)$ as an nilpotent endomorphism of $H_2$.

We use matrix representations to illustrate. Let

\begin{equation}\label{intbasisforclassical}
\mathcal{B}:=\{e_1,...,e_a,f_1,...,f_{h-a},f^{h-a},...,f^1,e^a,...,e^1\}
\end{equation}

be an integral basis of $(H_{\mathbb{Z}}, Q)$ such that:

\begin{align}
&W(\sigma)_{-1}\cong \mathrm{span}\{e_1,...,e_a\};\\
\nonumber &W(\sigma)_0\cong \mathrm{span}\{e_1,...,e_a, f_1,f^1,...,f_{g-a},f^{g-a}\};\\
\nonumber  &W(\sigma)_1\cong H_{\mathbb{Q}},
\end{align}

and the alternating polarization form $Q$ satisfies:

\begin{align}
&Q(e^i,e_j)=\delta^i_j, 1\leq i,j\leq a\\
\nonumber  &Q(f^i,f_j)=\delta^i_j, 1\leq i,j\leq g-a.
\end{align}

and $Q(\alpha, \beta)=0$ for other cases when $\alpha, \beta\in \mathcal{B}$. The existence of such a basis is explained in <cit.>.

Under the ordered basis (<ref>), We have:

\begin{equation}\label{polarizationformclassical}
& & &-E_a\\ \hline

& &-E_{g-a}&\\ \hline

&E_{g-a} & &\\ \hline
E_a& & &\\
\end{array}\right]
\end{equation}

where $E_n$ is the $n\times n$ matrix defined by
& & 1\\
& ... & \\
1 & &
\end{pmatrix}$
, and

\begin{equation}\label{matrixformofnandgamma}
\left[\begin{array}{c|c|c|c}
& & & *\\ \hline

& & &\\ \hline

& & &\\ \hline
& & &\\
\end{array}\right],\;\;\;
\gamma_i=
\left[\begin{array}{c|c|c|c}
A& * & *& B\\ \hline

& *& *& *\\ \hline

& *&* &*\\ \hline
& & &A^*\\
\end{array}\right].
\end{equation}

for any $N\in \mathfrak{g}^{-1,-1}$ and $\gamma_i$ in the triples provided by Proposition <ref>. The form of $\gamma_i$ is obtained from the evident fact $\gamma_i\in \Gamma\cap P_{W(\sigma)}$. With respect to the basis $\mathcal{B}$, consider the corresponding $\mathbb{Q}$-Levi decomposition:

\begin{equation}
P_{W(\sigma)}=L_{W(\sigma)}\ltimes  U_{W(\sigma)},
\end{equation}

hence for any $\gamma_i$ we have the unique Levi decomposition:

\begin{equation}
\gamma_i=\gamma_i^{l}\cdot \gamma_i^u
\end{equation}

in which $\gamma_i^l, \gamma_i^u$ have the matrix forms as:

\begin{equation}
\gamma_i^l=
\left[\begin{array}{c|c|c|c}
A&  & & \\ \hline

& *&* & \\ \hline

& *&* &\\ \hline
& & &A^*\\
\end{array}\right],\;\;\;
\gamma_i^u=
\left[\begin{array}{c|c|c|c}
1& * & *& B\\ \hline

& 1& & *\\ \hline

& &1 &*\\ \hline
& & &1\\
\end{array}\right].
\end{equation}

Note that it is clear we have:

\begin{align}
\nonumber &\mathrm{Ad}(\gamma_i)N=\mathrm{Ad}(\gamma_i^l)N.
\end{align}

Let $\tilde{Q}:=Q|_{H_2}$ be the induced polarization form on the $\mathbb{R}$-vector space $H_2$. We will associate $(H_2,\tilde{Q})$ an integral structure as follows. There is a ($\tilde{Q}$-isotropic) basis for $H^{2,2}$ of the following form:

\begin{equation}\label{standardbasisH2}
\alpha_i=e^i+g_i+\sum_{j=1}^{a}z_{ij}e_j, 1\leq i\leq a.
\end{equation}

where $g_i\in \mathrm{span}\{f_k,f^k\}_{\mathbb{R}}$ and $z_{ij}\in \mathbb{C}$.

The set $\mathcal{E}:=\{\beta_i:=\frac{\alpha_i+\bar{\alpha_i}}{2}, e_j\}_{1\leq i,j\leq a}$ is a basis of $(H_2)_{\mathbb{R}}$, and $\tilde{Q}$ is integral under the basis $\mathcal{E}$.

$\mathcal{E}$'s realness as well as $\tilde{Q}(e^i, \beta_j)=\delta^i_j$ is clear. We only have to check the isotropic condition $\tilde{Q}(\beta_i,\beta_j)=0$.
We have:

\begin{align}
\nonumber &=\tilde{Q}(e^i+g_i+\sum_{k=1}^{a}z_{ik}e_k, e^j+g_j+\sum_{k=1}^{a}z_{jk}e_k)\\
\nonumber &=z_{ij}-z_{ji}+\tilde{Q}(g_i,g_j).
\end{align}

Therefore $z_{ij}-z_{ji}=-\tilde{Q}(g_i,g_j)$ is real. Moreover we have:

\begin{align}
\nonumber &=\frac{z_{ij}+\bar{z_{ij}}}{2}-\frac{z_{ji}+\bar{z_{ji}}}{2}+\tilde{Q}(g_i,g_j)\\
\nonumber &=\tilde{Q}(e^i+g_i+\sum_{k=1}^{a}\frac{z_{ik}+\bar{z_{ik}}}{2}e_k, e^j+g_j+\sum_{k=1}^{a}\frac{z_{jk}+\bar{z_{jk}}}{2}e_k)\\
\nonumber  &=\tilde{Q}(\beta_i, \beta_j).
\end{align}

As a consequence of Proposition <ref>, $\mathbb{Z}\mathcal{E}$ realizes $((H_2)_{\mathbb{Z}}, \tilde{Q})$ as an integral symplectic lattice, under which $\tilde{Q}$ has the matrix form:

\begin{equation}\label{inducedformforclassical}
\tilde{Q}:=Q|_{H_2}=\left[\begin{array}{c|c}
& E_a\\ \hline

\end{array}\right].
\end{equation}

\begin{equation}
H_2^{'}:=\mathrm{span}_{\mathbb{Z}}\{e_1,...,e_a,e^a,...,e^1\}\subset H_{\mathbb{Z}},
\end{equation}

there is a natural isomorphism of integral symplectic lattices $(H_2^{'}, Q|_{H_2^{'}})\xrightarrow{\sim} ((H_2)_{\mathbb{Z}}, \tilde{Q})$ given by

\begin{equation}\label{identificationintegrallattice}
e_i\rightarrow e_i, \ e^i\rightarrow \beta_i=e^i+g_i+\sum_{j=1}^{a}\mathfrak{Re}(z_{ij})e_j, \ 1\leq i\leq a.
\end{equation}

Under this identification, for any $N\in \mathfrak{g}^{-1,-1}$ and $\gamma_i^l\in \Gamma_L:=\Gamma\cap L_{W(\sigma)}$, both viewed as operators on $(H_2^{'})_{\mathbb{Q}}$, we have the induced $\tilde{N},\tilde{\gamma_i^l}$ as operators on $H_2$ by $N, \gamma_i^l$. If $N$ and $\gamma$ has the matrix forms given by (<ref>), under the integral structure $((H_2)_{\mathbb{Z}}, \tilde{Q})$ and basis $\mathcal{E}$, $\tilde{N}$ and $\tilde{\gamma_i^l}$ has the matrix form:

\begin{equation}
\tilde{N}=
\left[\begin{array}{c|c}
& B\\ \hline

\end{array}\right],\;\;\;
\tilde{\gamma_i^l}=
\left[\begin{array}{c|c}
A & \\ \hline

& A^*
\end{array}\right].
\end{equation}

Clearly under this identification, $\tilde{N}\in \mathfrak{sp}(H_2,\mathbb{Q})$ and $\tilde{\gamma_i^l}\in \mathrm{Sp}(H_2,\mathbb{Z})$.

We now turn to the Hodge filtration $F_0^{\bullet}$. On $H$ we have:

\begin{equation}
F_0^{\bullet}(p)=\oplus_{s\geq p}H^{s,q}.
\end{equation}

Denote the induced Hodge filtration on $H_2$ as:

\begin{align}
&\tilde{F}_0^2:=\mathrm{img}\{H^{1,1}\hookrightarrow H_2\},\\
\nonumber &\tilde{F}_0^1:=H_{2, \mathbb{C}}.
\end{align}

Which can be identified as an element in $\check{D}_{(a,a)}$, the compact dual of the period domain $D_{(a,a)}$ parametrizing weight $1$ polarized $\mathbb{Z}$-Hodge structures on $((H_2)_{\mathbb{Z}}, \tilde{Q})$.

The weight filtration $W(\sigma)$ on $H$ gives a natural weight filtration $W(\tilde{\sigma})$ on $H_{2, \mathbb{Q}}$ by:

\begin{align}
W(\tilde{\sigma})_{-1}=W(\tilde{\sigma})_0&=\mathrm{span}_{\mathbb{Q}}\{e_i\}_{1\leq i\leq a}\\
\nonumber W(\tilde{\sigma})_1&=(H_2)_{\mathbb{Q}}.
\end{align}

which agrees with the Jacobson-Morozov filtration associated to $(H_2, \tilde{\sigma}\subset \mathrm{End}_{\mathbb{Q}}(H_2))$. Therefore we have the following:

$(W(\sigma)[-1]|_{H_2}, \tilde{F}_0^{\bullet},  \sigma|_{H_2})$ gives a limiting mixed Hodge structure on $((H_2)_{\mathbb{Z}}, \tilde{Q})$ of Hodge-Tate type.

From now we denote

\begin{equation}
\end{equation}

\begin{equation}
\tilde{\mathfrak{g}}_{\mathbb{C}}=\oplus_{p,q}\tilde{\mathfrak{g}}^{p,q}.
\end{equation}

as the splittings for LMHS $(W(\sigma)[-1]|_{H_2}, \tilde{F}_0^{\bullet},  \sigma|_{H_2})$ and its adjoint.

Denote $\tilde{\Sigma}$ as the image of $\Sigma$ under $\mathfrak{g}^{-1,-1}\cong \tilde{\mathfrak{g}}^{-1,-1}$. This isomorphism identifies the topology of $\Sigma$ and $\tilde{\Sigma}$. Therefore, the condition of Proposition <ref> implies $\tilde{\sigma}, \tilde{\tau}\in \tilde{\Sigma}$ are two monodromy nilpotent cones, and $\{\tilde{\gamma}_i^l\}\subset  P_{W(\tilde{\sigma})}\leq \mathrm{Sp}(H_2, \mathbb{Z})$ such that there exists

\begin{equation}
0\neq \tilde{N_i}\in \tilde{\mathfrak{g}}^{-1,-1}\cong \mathfrak{g}^{-1,-1},
\end{equation}

and the triple:

$(\tilde{\sigma}, \tilde{F}_0^{\bullet}), (\mathrm{Ad}(\tilde{\gamma}_i^l)\tilde{\tau}, \tilde{F}_0^{\bullet}), (\tilde{N_i}, \tilde{F}_0^{\bullet})$

are all nilpotent orbits rendering Hodge-Tate type LMHS. This completes the transition, and hence Proposition <ref> implies Proposition <ref>.

§.§ Concluding remarks

To complete the proof of Theorem <ref>, we still need to fill the gap between the local statement (i.e., Proposition <ref>) and the global one. We make the following claim:

Proposition <ref> implies Theorem <ref>.

The proof of this claim will be left to Appendix B. This completes the proof of Theorem <ref>.

§ WEAK FAN OF RESTRICTED TYPES: THE WEIGHT $3$ CALABI-YAU CASE

In this section we move to non-classical cases. Suppose $\varphi: S\rightarrow \Gamma\backslash D$ is a period map of type $(1,h,h,1)$ where $\Gamma\leq \mathrm{Sp}(2h+2, \mathbb{Z})$ is assumed to be neat[Any arithmetic subgroup contains a neat subgroup of finite index.].

As before, let $\Sigma$ be the collection of all local monodromy nilpotent cones arising from $\varphi$. We also assume any nilpotent orbit coming from $\varphi$ via Schmid's nilpotent orbit theorem is of type $\Phi$, where

\begin{equation}
\Phi:=\{\hbox{Pure LMHS}, \ \hbox{Type I LMHS}, \ \hbox{Type IV LMHS} \}.
\end{equation}

Here the LMHS types are defined by <cit.>. This is equivalent to say any divisorial degeneration arising from $\varphi$ is not of type II or type III. The family analyzed in <cit.> is such an example, see Appendix B for a summary. In this section we will prove the following theorem.

For the period map $\varphi$ with the above properties, there exists a polyhedral subdivision of $\Sigma$, denoted as $\hat{\Sigma}$, such that $\hat{\Sigma}$ is a type $\Phi$ weak fan strongly compatible with $\Gamma$ and finitely generated under the action by $\mathrm{Ad}(\Gamma)$. In other words, for any $\sigma, \tau\in \hat{\Sigma}$, $\tilde{B}(\sigma, \Phi)\cap \tilde{B}(\tau, \Phi)\neq \emptyset$ and $\sigma\cap \tau\neq \emptyset$ imply $\sigma=\tau$.

This is an analog of Theorem <ref>, and we should proceed in similar ways. Assume $\sigma, \tau\in \Sigma$ are two monodromy nilpotent cones. Suppose there exists a countable sequence $\{\gamma_i\}_{i\geq 0}\in \Gamma$ such that there exists $0\neq N_i\in \sigma\cap \mathrm{Ad}(\gamma_i)\tau$, and there exists $F_i^{\bullet}\in \check{D}$, such that

\begin{equation}\label{busterCY3}
(\sigma, F_i^{\bullet}), (\mathrm{Ad}(\gamma_i)\tau, F_i^{\bullet}), (N_i, F_i^{\bullet})
\end{equation}

are all nilpotent orbits of type $\Phi$. Again we assume $\gamma_0=\mathrm{Id}$. We should check the two non-trivial LMHS types in $\Phi$ separately.

§.§ The type I case

The type $\mathrm{I}_a \ (0\leq a\leq h)$ LMHS for the $(1,h,h,1)$ is characterized by Hodge-Deligne diagrams in Figure <ref>.

[-,line width=1.0pt] (0,0) – (0,3.5);
[-,line width=1.0pt] (0,0) – (3.5,0);
(0,3) circle (2pt);
at (-0.3,3) $1$;

(1,1) circle (2pt);
at (0.7,1) $a$;

(1,2) circle (2pt);
at (0.7,2.3) $h-a$;

(3,0) circle (2pt);
at (3,-0.3) $1$;

(2,1) circle (2pt);
at (2.3,0.7) $h-a$;

(2,2) circle (2pt);
at (2.3,2) $a$;

[-stealth](1.8,1.8) – (1.2,1.2) [line width = 1.0pt];

\mspace{50mu}
[-,line width=1.0pt] (-3.5,0) – (3.5,0);
[-,line width=1.0pt] (0,-3.5) – (0,3.5);

(0,0) circle (3pt);
(1,0) circle (3pt);
(1,1) circle (3pt);
(1,-1) circle (3pt);
(0,1) circle (3pt);
(0,-1) circle (3pt);
(-1,0) circle (3pt);
(-1,1) circle (3pt);
(-1,-1) circle (3pt);

(2,-1) circle (3pt);
(1,-2) circle (3pt);
(2,-2) circle (3pt);
(3,-3) circle (3pt);

(-2,1) circle (3pt);
(-2,2) circle (3pt);
(-1,2) circle (3pt);
(-3,3) circle (3pt);

[-stealth](-1.2,1.8) – (-1.8, 1.2) [line width = 1.0pt];
[-stealth](1.8,-1.2) – (1.2, -1.8) [line width = 1.0pt];

[-stealth](-0.2,0.8) – (-0.8,0.2) [line width = 1.0pt];
[-stealth](0.8,-0.2) – (0.2,-0.8) [line width = 1.0pt];

[-stealth](0.8,0.8) – (0.2,0.2) [line width = 1.0pt];
[-stealth](-0.2,-0.2) – (-0.8,-0.8) [line width = 1.0pt];

Type $\mathrm{I}_a$ LMHS for $(1,h,h,1)$

Suppose $(W(N)[-3], F^{\bullet}, N)$ where $N\in \mathfrak{g}_{\mathbb{Q}}$, $F^{\bullet}\in \check{D}$ gives a type I LMHS. Applying (<ref>) and (<ref>) on type I LMHS, we have

\begin{equation}
(P_N)_{\mathbb{C}}\backslash P_{F^{\bullet}}\cong Z(N)\backslash P_{F^{\bullet}}
\end{equation}

which implies for any monodromy nilpotent cones $\sigma$ and $\tau$, if $W(\sigma)=W(\tau)$ and $\tilde{B}(\sigma, \Phi)\cap \tilde{B}(\tau, \Phi)\neq \emptyset$, then $\tilde{B}(\sigma, \Phi)=\tilde{B}(\tau, \Phi)$.

Let $\Sigma_{\mathrm{I}}\subset \Sigma$ be the subcollection containing all nilpotent cones which could polarize type I LMHS. Now regarding the triples in <ref>, we consider the case when $\sigma, \tau\in \Sigma_{\mathrm{I}}$ and all of

$(\sigma, F_i^{\bullet}), (\mathrm{Ad}(\gamma_i)\tau, F_i^{\bullet}), (N_i, F_i^{\bullet})$

are nilpotent orbits with type I LMHS. Discussion above shows we have an analog of Proposition <ref> for type I degenerations:

We can choose all $F_i^{\bullet}=F_0^{\bullet}$. In other words,

$(\sigma, F_0^{\bullet}), (\mathrm{Ad}(\gamma_i)\tau, F_0^{\bullet}), (N_i, F_0^{\bullet})$

are all nilpotent orbits with type I LMHS.

What we need to prove in this subsection is the following analog of Proposition <ref>:

Regarding the collection of triples in Proposition <ref>, there exists a finite polyhedral decomposition $\Sigma(\sigma)$ of $\sigma$ such that $\sigma \cap \mathrm{Ad}(\gamma_i)\tau$ is the union of some $\{\sigma_k\}\subset \Sigma(\sigma)$ for any $i$.

Consider the LMHS $(W(\sigma)[-3], F_0^{\bullet}, \sigma)$. The corresponding Deligne splittings are denoted as:

\begin{equation}
\end{equation}

\begin{equation}
\mathfrak{g}_{\mathbb{C}}=\oplus_{p,q}\mathfrak{g}^{p,q}.
\end{equation}

with $\sigma$, $\mathrm{Ad}(\gamma_i)\tau\subset \mathfrak{g}^{-1,-1}$. Consider the splitting:

\begin{equation}
H_{\mathbb{C}}=H_1\oplus H_2
\end{equation}

\begin{align}
H_1&:=H^{3,0}\oplus H^{2,1}\oplus H^{1,2}\oplus H^{0,3}\\
\nonumber H_2&:=H^{2,2}\oplus H^{1,1}.
\end{align}

Clearly $H_2$ is defined over $\mathbb{R}$ and $N$ only acts on $H_2$ non-trivially, we denote $\bar{N}\in \mathrm{End}(H_2)$ as an nilpotent endomorphism of $H_2$.

Applying methods in Section <ref> in a similar way, let

\begin{equation}\label{intbasis}
\mathcal{B}:=\{e_1,...,e_a,f_0,...,f_{h-a},f^{h-a},...,f^0,e^a,...,e^1\}
\end{equation}

be an integral basis of $(H_{\mathbb{Q}}, Q)$ such that:

\begin{align}
&W(\sigma)_{-1}\cong \mathrm{span}\{e_1,...,e_a\};\\
\nonumber &W(\sigma)_0\cong \mathrm{span}\{e_1,...,e_a,f_0,f^0,...,f_{h-a},f^{h-a}\};\\
\nonumber &W(\sigma)_1\cong H_{\mathbb{Q}}.
\end{align}

and the (alternating) polarization form $Q$ satisfies:

\begin{align}
&Q(e^i,e_j)=\delta^i_j, 1\leq i,j\leq a\\
\nonumber &Q(f^i,f_j)=\delta^i_j, 0\leq i,j\leq h-a.
\end{align}

and $Q(\alpha, \beta)=0$ for other cases when $\alpha, \beta\in \mathcal{B}$.

Writing $\tilde{Q}:=Q|_{H_2}$. We also consider the induced Hodge filtration of $F_0^{\bullet}$ on $H_2$ as:

\begin{align}
\nonumber &\tilde{F}_0^2:=\mathrm{img}\{H^{2,2}\hookrightarrow H_2\},\\
\nonumber &\tilde{F}_0^1=\tilde{F}_0^0:=H_{2, \mathbb{C}}.
\end{align}

Apply the same procedures in Section <ref>, we obtain similar statements as Proposition <ref> and Proposition <ref> but for the weight $3$ type I degeneration:

There exists a basis of $H_{2,\mathbb{R}}$ under which $\tilde{Q}$ has the standard form (<ref>), and under this basis $(H_2, \tilde{Q})$ can be regarded as an integral symplectic lattice.

By setting the compact dual $\check{D}_{(a,a)}$ of the period domain $D_{(a,a)}$ parametrizing weight $1$ polarized $\mathbb{Z}$-Hodge structures on $((H_2)_{\mathbb{Z}}, \tilde{Q})$, we can identify $\tilde{F}_0^{\bullet}$ as an element in $\check{D}_{(a,a)}$, and $(W(\sigma)[-3]|_{H_2}, \tilde{F}_0^{\bullet},  \sigma|_{H_2})$ gives a limiting mixed Hodge structure on $H_2$ of Hodge-Tate type.

Again we denote

\begin{equation}
\end{equation}

\begin{equation}
\tilde{\mathfrak{g}}_{\mathbb{C}}=\oplus_{p,q}\tilde{\mathfrak{g}}^{p,q}.
\end{equation}

as the splittings for LMHS $(W(\sigma)[-3]|_{H_2}, \tilde{F}_0^{\bullet},  \sigma|_{H_2})$ and its adjoint.

Denote $\tilde{\Sigma}_{\mathrm{I}}$ as the image of $\Sigma_{\mathrm{I}}$ under $\mathfrak{g}^{-1,-1}\cong \tilde{\mathfrak{g}}^{-1,-1}$. Therefore, the condition of Proposition <ref> implies $\tilde{\sigma}, \tilde{\tau}\in \tilde{\Sigma}_{\mathrm{I}}$ are two monodromy nilpotent cones, and $\{\tilde{\gamma}_i\}\subset  P_{W(\tilde{\sigma})}\cap \mathrm{Sp}(H_2, \mathbb{Z})$ such that there exists

\begin{equation}
0\neq \tilde{N_i}\in \tilde{\mathfrak{g}}^{-1,-1}\cong \mathfrak{g}^{-1,-1},
\end{equation}

and the triple:

$(\tilde{\sigma}, \tilde{F}_0^{\bullet}), (\mathrm{Ad}(\tilde{\gamma}_i)\tilde{\tau}, \tilde{F}_0^{\bullet}), (\tilde{N_i}, \tilde{F}_0^{\bullet})$

are all nilpotent orbits with Hodge-Tate type LMHS. This completes the transition from the type I degeneration of $(1,h,h,1)$ to the Hodge-Tate degeneration of weight $1$ case, hence Proposition <ref> implies Proposition <ref>.

§.§ The type IV case

Now we consider the type IV degeneration of $(1,h,h,1)$. Its Hodge-Deligne diagrams are shown in Figure <ref>. In particular, when $a=h$, it gives the Hodge-Tate degeneration (of weight $3$ Calabi-Yau case).

[-,line width=1.0pt] (0,0) – (0,3.5);
[-,line width=1.0pt] (0,0) – (3.5,0);
(0,0) circle (2pt);
at (-0.3,-0.3) $1$;

(1,1) circle (2pt);
at (0.7,1) $a$;

(1,2) circle (2pt);
at (0.7,2.3) $h-a$;

(3,3) circle (2pt);
at (3.3,3.3) $1$;

(2,1) circle (2pt);
at (2.3,0.7) $h-a$;

(2,2) circle (2pt);
at (2.3,2) $a$;

[-stealth](2.8,2.8) – (2.2,2.2) [line width = 1.0pt];
[-stealth](1.8,1.8) – (1.2,1.2) [line width = 1.0pt];
[-stealth](0.8,0.8) – (0.2,0.2) [line width = 1.0pt];

\mspace{50mu}
[-,line width=1.0pt] (-3.5,0) – (3.5,0);
[-,line width=1.0pt] (0,-3.5) – (0,3.5);

(0,0) circle (3pt);
(1,0) circle (3pt);
(1,1) circle (3pt);
(1,-1) circle (3pt);
(0,1) circle (3pt);
(0,-1) circle (3pt);
(-1,0) circle (3pt);
(-1,1) circle (3pt);
(-1,-1) circle (3pt);

(2,1) circle (3pt);
(1,2) circle (3pt);
(2,2) circle (3pt);
(3,3) circle (3pt);

(-2,-1) circle (3pt);
(-2,-2) circle (3pt);
(-1,-2) circle (3pt);
(-3,-3) circle (3pt);

[-stealth](-2.2,-2.2) – (-2.8, -2.8) [line width = 1.0pt];
[-stealth](2.8,2.8) – (2.2, 2.2) [line width = 1.0pt];
[-stealth](1.8,1.8) – (1.2, 1.2) [line width = 1.0pt];
[-stealth](-1.2,-1.2) – (-1.8,-1.8) [line width = 1.0pt];
[-stealth](0.8,0.8) – (0.2,0.2) [line width = 1.0pt];
[-stealth](-0.2,-0.2) – (-0.8,-0.8) [line width = 1.0pt];

[-stealth](-0.2,0.8) – (-0.8,0.2) [line width = 1.0pt];
[-stealth](0.8,1.8) – (0.2,1.2) [line width = 1.0pt];
[-stealth](-1.2,-0.2) – (-1.8,-0.8) [line width = 1.0pt];

[-stealth](1.8,0.8) – (1.2,0.2) [line width = 1.0pt];
[-stealth](0.8,-0.2) – (0.2,-0.8) [line width = 1.0pt];
[-stealth](-0.2,-1.2) – (-0.8,-1.8) [line width = 1.0pt];

Type $\mathrm{IV}_a$ LMHS for $(1,h,h,1)$

Again the corresponding Deligne splittings are denoted as:

\begin{equation}
\end{equation}

\begin{equation}
\mathfrak{g}_{\mathbb{C}}=\oplus_{p,q}\mathfrak{g}^{p,q}.
\end{equation}

Let $\Sigma_{\mathrm{IV}}\subset \Sigma$ be the subcollection containing all nilpotent cones which could polarize type IV LMHS. Now regarding the triples in (<ref>), we consider the case when $\sigma, \tau\in \Sigma_{\mathrm{IV}}$ and all of

$(\sigma, F_i^{\bullet}), (\mathrm{Ad}(\gamma_i)\tau, F_i^{\bullet}), (N_i, F_i^{\bullet})$

are nilpotent orbits with type IV LMHS. In this subsection we need to prove:

There exists a polyhedral subdivision of $\Sigma_{\mathrm{IV}}$, denoted as $\hat{\Sigma}_{\mathrm{IV}}$, such that $\hat{\Sigma}_{\mathrm{IV}}$ is a weak fan strongly compatible with $\Gamma$ and finitely generated under the action by $\mathrm{Ad}(\Gamma)$. In other words, for any $\sigma, \tau\in \hat{\Sigma}_{\mathrm{IV}}$, $\tilde{B}(\sigma, \Phi)\cap \tilde{B}(\tau, \Phi)\neq \emptyset$ and $\sigma\cap \tau\neq \emptyset$ imply $\sigma=\tau$.

If $\sigma\subset \mathfrak{g}_{\mathbb{Q}}$ is a monodromy nilpotent cone polarizing type IV LMHS, its Jacobson-Morozov weight filtration $W(\sigma)$ on $H_{\mathbb{Q}}$ can be described as follows. Let $(H_{\mathbb{Z}}, Q)$ be the underlying lattice. Take an integral basis $\mathcal{B}$ under which $Q$ has the same form as (<ref>), where $\mathfrak{B}$ and $W(\sigma)$ are:

\begin{align}
&W(\sigma)_3\cong H_{\mathbb{Q}}\\
\nonumber W(\sigma)_2=&W(\sigma)_1\cong \mathrm{span}\{\omega_-,e_1,...,e_a,f_1,...,f_{h-a},f^{h-a},...,f^1,e^a,...,e^1\}\\
\nonumber &W(\sigma)_0\cong \mathrm{span}\{\omega_-,e_1,...,e_a,f_1,...,f_{h-a},f^{h-a},...,f^1\}\\
\nonumber &W(\sigma)_{-1}\cong \mathrm{span}\{\omega_-,e_1,...,e_a\}\\
\nonumber &W(\sigma)_{-2}=W(\sigma)_{-3}\cong \mathrm{span}\{\omega_-\}.
\end{align}

Suppose for any two type IV monodromy nilpotent cones $\sigma, \tau$, there exists a (countable) sequence $\{\gamma_i\}_{i\geq 0}\subset \Gamma$ such that there exists $0\neq N_i\in \sigma\cap \mathrm{Ad}(\gamma_i)\tau$ and $\{F_i^{\bullet}\}\subset \check{D}$, we have

$(\sigma, F_i^{\bullet}), (\mathrm{Ad}(\gamma_i)\tau, F_i^{\bullet}), (N_i, F_i^{\bullet})$

are all nilpotent orbits.

\begin{equation}
\tilde{H}:=W(\sigma)_1/W(\sigma)_{-2},
\end{equation}

that is,

\begin{equation}
\tilde{H}\cong\mathrm{span}\{e_1,...,e_a,f_1,...,f_{h-a},f^{h-a},...,f^1,e^a,...,e^1\}
\end{equation}

Denote $\tilde{F}_i^{\bullet}$ as the filtration on $\tilde{H}$ induced by $F_i^{\bullet}$. It is straightforward to check if we regard $\check{D}_{(h,h)}$ as the compact dual of $D_{(h,h)}$ which parametrizes $\mathbb{Z}$-polarized Hodge structures on $(\tilde{H}, \tilde{Q}=Q|_{\tilde{H}})$, then $\tilde{F}_i^{\bullet}\in \check{D}_{(h,h)}$[More precisely, we are identifying $D_{(0,h,h,0)}$ and $D_{(h,h)}$ by forgetting two trivial factors and twisting by $\mathbb{Z}[1]$.].

\begin{equation}
\tilde{\sigma}:= \mathrm{img}\{\sigma\subset \mathfrak{g}_{\mathbb{Q}}\cap \mathfrak{g}^{-1,-1}\rightarrow \mathrm{End}(\tilde{H}_{\mathbb{Q}})\}.
\end{equation}

It is well defined since $\sigma$ annihilates $W(\sigma)_{-2}$. We also denote $W(\tilde{\sigma})$ on $\tilde{H}_{\mathbb{Q}}$ by:

\begin{align}
&W(\tilde{\sigma})_1\cong \mathrm{span}\{e_1,...,e_a,f_1,...,f_{h-a},f^{h-a},...,f^1,e^a,...,e^1\}\\
&\nonumber W(\tilde{\sigma})_0\cong \mathrm{span}\{e_1,...,e_a,f_1,...,f_{h-a},f^{h-a},...,f^1\}\\
&\nonumber W(\tilde{\sigma})_{-1}\cong \mathrm{span}\{e_1,...,e_a\}.
\end{align}

It is now clear that:

On $(\tilde{H}, \tilde{Q}=Q|_{\tilde{H}})$, $(W(\tilde{\sigma})[-1], \tilde{F}_i^{\bullet}, \tilde{\sigma})$ gives a limiting mixed Hodge structure (of classical weight $1$ type).

We denote this LMHS and its adjoint as:

\begin{equation}\label{hodgedecompositiontypeIVquotient}
\tilde{H}_{\mathbb{C}}=\oplus_{p,q}\tilde{H}^{p,q}
\end{equation}

\begin{equation}
\tilde{\mathfrak{g}}_{\mathbb{C}}=\oplus_{p,q}\tilde{\mathfrak{g}}^{p,q}.
\end{equation}

Another observation is

\begin{equation}
\{\gamma_i\}\subset \mathrm{Sp}(2h+2,Q;\mathbb{Z})\cap P_{W(\sigma)}
\end{equation}

has induced actions

\begin{equation}
\{\tilde{\gamma}_i\}\subset \mathrm{Sp}(2h,\tilde{Q};\mathbb{Z})\cap P_{W(\tilde{\sigma})}
\end{equation}

by restricting on $W(\sigma)_2$ and taking quotient by $W(\sigma)_{-2}$.

Let $\tilde{\Sigma}_{\mathrm{IV}}$ be the projection of $\Sigma_{\mathrm{IV}}$ in $\tilde{\mathfrak{g}}^{-1,-1}$. Since for any $\sigma, \tau\in \Sigma_{\mathrm{IV}}$ , any $\{\gamma_i\}\subset \Gamma$, and any triples

$(\sigma, F_i^{\bullet}), (\mathrm{Ad}(\gamma_i)\tau, F_i^{\bullet}), (N_i, F_i^{\bullet})$,

with $N_i\in \sigma\cap \mathrm{Ad}(\gamma_i)\tau$, we have the induced triples

$(\tilde{\sigma}, \tilde{F}_i^{\bullet}), (\mathrm{Ad}(\gamma_i)\tilde{\tau}, \tilde{F}_i^{\bullet}), (\tilde{N}_i, \tilde{F}_i^{\bullet})$.

with the same properties. Therefore, by applying Theorem <ref> to $\tilde{\Sigma}_{\mathrm{IV}}$ and the following lemma, we prove Proposition <ref>.

Suppose $\sigma, \tau\in \Sigma_{\mathrm{IV}}$ such that $\tilde{B}(\sigma, \Phi)\cap \tilde{B}(\tau, \Phi)\neq \emptyset$, $\sigma\cap \tau\neq \emptyset$ and $\tilde{\sigma}=\tilde{\tau}$ in $\tilde{\Sigma}_{\mathrm{IV}}$, then $\sigma=\tau$.

For $F^{\bullet}\in \tilde{B}(\sigma, \Phi)\cap \tilde{B}(\tau, \Phi)$ let $H_{\mathbb{C}}=\oplus H^{p,q}$ be the Hodge decomposition of LMHS $(W(\sigma)[-3], F^{\bullet}, \sigma)$. Suppose the lemma is false, there exist $N_1\in \sigma\backslash \tau$ and $N_2\in \tau$ such that $\tilde{N}_1=\tilde{N}_2$ in $\tilde{\sigma}=\tilde{\tau}$, this implies $N_1-N_2\neq 0$ annihilates $\tilde{H}$. On the other hand, take $0\neq N_3\in \sigma\cap\tau$, we know $[N_1-N_2, N_3]=0$ and $(N_3, F^{\bullet})$ is a nilpotent orbit with type IV LMHS, this leads to a contradiction since for $0\neq \mathbf{v}\in H^{3,3}$, $(N_1-N_2)N_3\mathbf{v}=0$ but $N_3(N_1-N_2)\mathbf{v}\neq 0$.

§.§ Concluding remarks

Theorem <ref> is proved by combining Proposition <ref> and Proposition <ref>. A comment on the necessity of using weak fan of restricted type is elements in $\Sigma_{\mathrm{I}}$ could also polarize type II LMHS, as Figure <ref> shows. We are not able to show the finiteness results like Theorem <ref> if $\Phi$ includes type II LMHS at this time.

§ SUBDIVISION, LOGARITHMIC MODIFICATION AND EXAMPLE

§.§ Overview

Given a period map $\varphi: S\rightarrow \Gamma\backslash D$ which are of types described above, Theorem <ref> and <ref> are combinatorial statements that we can modify the collection of local monodromy nilpotent cones $\Sigma$ in a proper way to make it fit Kato-Nakayama-Usui's theory. To fully establish the main result regarding period maps themselves (see Section <ref>), we shall explain what do Theorem <ref> and <ref> mean in the Hodge-theoretical side.

Indeed, in Theorem <ref> and <ref>, since the subdivision on $\Sigma$ is finitely generated under $\mathrm{Ad}(\Gamma)$-action, it can be generated by a finite subdivision on the set of local monodromy nilpotent cones coming from the period map $\varphi: S\rightarrow \Gamma\backslash D$. This operation is called "logarithmic modification" in Kato-Usui's theory.

The general theory of logarithmic modification is introduced in <cit.>, in which the base space $S$ is assumed to be an object in an abstract category $\mathcal{B}(\mathrm{log})$. In our settings where $S$ is quasi-projective with normal crossing divisors, logarithmic modification is exactly local toric blow-ups[See <cit.> for a comprehensive introduction on toric resolutions and polyhedral subdivisions.]. Indeed, a finite (rational) subdivision on local monodromy cones corresponds to a finite blow-up sequence:

\begin{equation}\label{finiteblowupsequence}
\pi: \overline{S_M}\rightarrow \overline{S_{M-1}}\rightarrow ... \rightarrow \overline{S_1}\rightarrow \overline{S_0}=\overline{S},
\end{equation}

where each arrow is a (weighted) blow-up along some subvariety. As a consequence of finiteness, $S_M:=\pi^{-1}(S)$ is also quasi-projective, and the original period map can be lifted to:

\begin{equation}\label{liftedperiodmap}
\begin{tikzcd}
S_M \arrow[d] \arrow[dr, "\varphi_M"] & \\
S \arrow[r, "\varphi"] & \Gamma \backslash D.
\end{tikzcd}
\end{equation}

Combine this with Theorems <ref> and <ref>, the main result of this paper summarized in Section <ref> can be formally stated as follows:

Suppose $\varphi: S\rightarrow \Gamma\backslash D$ is a period map such that:

(i) $S$ is quasi-projective with normal crossing boundary divisors;

(ii) The period map is either of classical weight $1$ type, or weight $3$ Calabi-Yau type which acquires only type I or type IV degenerations,

then there exists a quasi-projective variety $S_M$ and a finite blow-up sequence (<ref>), such that there exists a $\Gamma$-strongly compatible weak fan (of restricted type) $\Sigma$, and the lifted period map (<ref>) admits an Kato-Usui type extension:

\begin{equation}
\overline{\varphi_M}: \overline{S_M}\rightarrow \Gamma\backslash D_{\Sigma, \Phi}
\end{equation}

in which $\Phi$ is a set of LMHS types, $\Gamma\backslash D_{\Sigma, \Phi}$ is a logarithmic manifold and $\overline{\varphi_M}$ is a morphism of logarithmic manifolds.

§.§ An example on the $2$-dimensional case

In this subsection we work out a $2$-dimensional example to show how logarithmic modification can make changes on the period map.

Consider the local period map:

\begin{equation}
\varphi: (\Delta^{*})^2\rightarrow \Gamma\backslash D,
\end{equation}

where under a fixed coordinate system $(x,y)\in \Delta^2$, the (unipotent) monodromy operators around coordinate divisors $D_x:=\{x=0\}, D_y:=\{y=0\}$ are $T_x, T_y$, and $\Gamma=<T_x, T_y>$ is abelian.

Consider the blow-up of $(\Delta^{*})^2$ at the point $(0,0)$, the blow-up map is read as:

\begin{equation}
\pi: \widehat{\Delta^2}\rightarrow {\Delta^2}.
\end{equation}

Denote the exception divisor $E$. The same method in <cit.> shows for the lifted period map:

\begin{equation}
\hat{\varphi}: \widehat{\Delta^2}\rightarrow \Gamma\backslash D,
\end{equation}

the monodromy operator around $E$ is $T_xT_y$. Passing to the monodromy logarithms, we have $N_E=N_x+N_y$, which implies lifting the period map is derived from subdividing the monodromy nilpotent cone $<N_x,N_y>$ to $<N_x, N_x+N_y>\cup <N_x+N_y, N_y>$.

Under the coordinates $(u:=\frac{\log(x)}{2\pi i},v:=\frac{\log(y)}{2\pi i})$, the period map $\varphi$ can be written as:

\begin{equation}\label{localperiodmap2dim}
\varphi(u,v)=\exp(uN_x+vN_y)\psi(x,y),
\end{equation}

where $\psi(x,y): \Delta^2\rightarrow \check{D}$ is holomorphic. Switch to $\hat{\varphi}$, around the point $o_y:=E\cap D_y$, the local coordinate expression of $\pi$ is:

\begin{equation}
(x,t)\rightarrow (x,xt,[1:t])\in \widehat{\Delta^2}\xrightarrow[]{\pi}(x,xt)\in \Delta^2.
\end{equation}

This implies around $o_y$, the local lift of the period map (<ref>) is lifted to:

\begin{equation}
\hat{\varphi}(x,t)=\varphi(x,xt)=\exp(\frac{\log(x)}{2\pi i}(N_x+N_y)+\frac{\log(t)}{2\pi i}N_y)\psi(x,xt).
\end{equation}

Notice that on $E:\{x=0\}$, we have:

\begin{equation}
\exp(-\frac{\log(x)}{2\pi i}(N_x+N_y)-\frac{\log(t)}{2\pi i}N_y)\hat{\varphi}(0,t)=\psi(0,0)
\end{equation}

is constant. By Schmid's nilpotent orbit theorem, locally for any point $x\in E\backslash (D_x\cup D_y)$, the nilpotent orbit associated to $x$ is $(N_x+N_y, \psi(0,0))$. This implies if we construct the Kato-Usui extension of $\hat{\varphi}$ as Theorem <ref>, the extended map $\overline{\hat{\varphi}}$ is locally constant on exceptional divisors.

Another observation is since in (<ref>), $\psi(0,0)$ is well-defined only up to the action of $\mathbb{C}<N_x, N_y>$, the value $(N_x+N_y, \psi(0,0))$ is well-defined only after fixing a local coordinate system. In general, the Kato-Usui type extended map in Theorem <ref> is well defined only after a coordinate system on $S$ is fixed.

§.§ Summary of the example in <cit.>

In this subsection we recall the construction in <cit.>. The period map used here is the one introduced by <cit.> and <cit.>, which is a period map of type $(1,2,2,1)$ as follows:

\begin{equation}
\hat{\varphi}: \mathbb{P}^2\backslash \mathrm{Dis}\rightarrow \Gamma\backslash D.
\end{equation}

The discriminant locus is given by:

\begin{equation}
\mathrm{Dis}:= D_0\cup D_1\cup D_2\cup C
\end{equation}

where $D_i$ are coordinate divisors in $\mathbb{P}^2$, and $C$ is an irreducible quintic with $6$ self-intersection nodes and $1$ tangent point of order $5$ with each coordinate divisor. The picture of this base can be found at <cit.>. The precise monodromy group $\Gamma$ is not known, but from <cit.> we know the algebraic monodromy group $\overline{\Gamma}^{\mathbb{Q}}$ is $\mathrm{Sp}(6,\mathbb{Q})$.

By blowing-up each tangent point for $5$ times, we get a lifted period map:

\begin{equation}
\varphi: \widetilde{\mathbb{P}^2\backslash \mathrm{Dis}}\rightarrow \Gamma\backslash D.
\end{equation}

where the base $\widetilde{\mathbb{P}^2\backslash \mathrm{Dis}}=:S$ is quasi-projective with normal-crossing boundary divisors.

We mark all local monodromy nilpotent cones as follows:

(i) Let $\sigma_x, \sigma_y, \sigma_z$ be the ($2$-dimensional) local monodromy nilpotent cones around the intersection of each pair of coordinate divisors;

(ii) Let $\sigma_0, \sigma_1, \sigma_2$ be the local monodromy nilpotent cones obtained by blowing-up each of the fifth tangent point. As <cit.> shows, the blow-up process annihilates the order-$5$ semisimple part of the monodromy operators around $C$ and leave the unipotent part invariant;

(iii) Let $\tau_j, \ j=1,...,6$ be the local monodromy nilpotent cones obtained from all self-intersection nodes of $C$.

By computational results in <cit.> and <cit.>, all $2$-dimensional local monodromy nilpotent orbits and their Hodge degeneration types are listed as follows (Again we use the type defined by <cit.>).

(i) $\sigma_x, \sigma_y, \sigma_z$ are all of the type $<\mathrm{IV}_2|\mathrm{IV}_2|\mathrm{IV}_2>$;

(ii) $\sigma_0, \sigma_1, \sigma_2$ are all of the type

(iii) $\tau_j, \ 1\leq j\leq 6$ are all of the type

Where for a $2$-dimensional cone $\sigma=\mathbb{Q}_{\geq 0}<N_1,N_2>$ and its associated nilpotent orbit $(\sigma, F^{\bullet})$, type $<A|B|C>$ means the type of LMHS given by $(N_1, F^{\bullet})$, $(\sigma, F^{\bullet})$ and $(N_2, F^{\bullet})$ respectively.

Let $\mathcal{S}$ be the set of all monodromy nilpotent cones listed above and all of their proper faces. Clearly $\mathcal{S}$ is a finite set. By our definition, the collection $\Sigma$ with $|\Sigma|\subset \mathfrak{g}_{\mathbb{Q}}$ is the union of $\mathrm{Ad}(\Gamma)$-orbits of $\mathcal{S}$. One of the main results in <cit.> is showing that after a finite base change which leads to changing $\Gamma$ to a neat subgroup of finite index $\Gamma_0$ as well as a replacing $\Sigma$ by one of its finitely $\mathrm{Ad}(\Gamma)$-generated subdivisions $\Sigma_0$,  $\Sigma_0$ is a $\Gamma_0$-strongly compatible fan which provides a Kato-Usui style extension. Clearly, the results in this paper provides an alternative proof on the existence of a Kato-Usui type extension.

§ APPENDIX A: KATO-USUI'S THEORY: OVERVIEW AND MODIFICATION

The purpose of this appendix is briefly reviewing Kato-Usui's theory and sketch the proof of Theorems <ref> and <ref>.

§.§ Kato-Usui's theory revisit

§.§.§ Basic constructions

We begin with the local constructions following <cit.>. Given a rational nilpotent cone $\sigma\subset \mathfrak{g}_{\mathbb{Q}}$, let

\begin{align}
\nonumber &\mathrm{torus}_{\sigma}:=\mathrm{Spec}(\mathbb{C}[(\Gamma(\sigma)^{\mathrm{gp}})^{\vee}])_{\mathrm{an}}
\end{align}

be the toric variety and torus embedding associated to $\sigma_{\mathbb{R}}$. For any $q\in \mathrm{toric}_{\sigma}$ we associate the cone $\sigma(q)$ which is the face of $\sigma$ corresponds to the torus orbit $q$ lies in via the orbit-cone correspondence[See for example, <cit.>]. Regarding (evaluating at) $q$ as a regular function over $\mathbb{C}(\Gamma(\sigma)^{\vee})$, it also defines a class $[q]\in \frac{\sigma_{\mathbb{C}}}{\sigma(q)_{\mathbb{C}}+\mathrm{log}\Gamma(\sigma)^{\mathrm{gp}}}$, of which we denote any lift as $\mathrm{log}_{\sigma}q$.

Following <cit.>, define

\begin{align}
\check{E}_{\sigma}&:=\mathrm{toric}_{\sigma}\times D\\
\nonumber \tilde{E}_{\sigma}&:=\{(q,F^{\bullet})\in \check{E}_{\sigma}|NF^{\bullet}\subset F^{\bullet-1} \ \forall  N\in \sigma(q)\}\\
\nonumber E_{\sigma}&:=\{(q,F^{\bullet})\in \tilde{E}_{\sigma}|(\sigma(q), \mathrm{exp}(\mathrm{log}_{\Gamma_{\sigma}}q)F^{\bullet}) \hbox{ is a } \sigma(q)-\hbox{nilpotent orbit } \}
%E_{\sigma, \Phi}&:=\{(q,F^{\bullet})\in \tilde{E}_{\sigma}|(\sigma(q), \mathrm{exp}(\mathrm{log}_{\Gamma_{\sigma}}q)F^{\bullet}) \hbox{ is a } \sigma(q)-\hbox{nilpotent orbit of type $\Phi$ } \},
\end{align}

Consider the map:

\begin{equation}\label{sigmatorsormap}
\Theta_{\sigma}: E_{\sigma}\rightarrow \Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma},
\end{equation}

which is the quotient of the action:

\begin{equation}
a\in \sigma_{\mathbb{C}}: E_{\sigma}\rightarrow E_{\sigma}, \ (a, (q,F^{\bullet}))\rightarrow (e(a)q, e^{-a}F^{\bullet}).
\end{equation}

This map realizes $E_{\sigma}$ as a $\sigma_{\mathbb{C}}$-torsor over $\Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma}$.

Let $|\mathrm{toric}|_{\sigma}$ be the analytic closure of $\mathbb{R}$-points in $\mathrm{toric}_{\sigma}$, then we can similarly define:

\begin{align}
\check{E}^{\#}_{\sigma}&:= |\mathrm{toric}|_{\sigma},\\
\nonumber E^{\#}_{\sigma}&:=E_{\sigma}\cap \check{E}^{\#}_{\sigma},\\
\nonumber \Theta^{\#}_{\sigma}&: E^{\#}_{\sigma}\rightarrow D^{\#}_{\sigma}.
\end{align}

if we replace $\sigma_{\mathbb{C}}$ by $i\sigma_{\mathbb{R}}$, and nilpotent orbits by nilpotent $i$-orbits. There is a commutative diagram

\begin{equation}\label{diagramoftorsors}
\begin{tikzcd}
E^{\#}_{\sigma} \arrow[d] \arrow[d] \arrow[r] & E_{\sigma} \arrow[d]\\
D^{\#}_{\sigma} \arrow[r] & \Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma}
\end{tikzcd}
\end{equation}

which realizes $E^{\#}_{\sigma}$ as an $i\sigma_{\mathbb{R}}$-torsor over $D^{\#}_{\sigma}$.

For a subset $X\subset S$ where $S$ is a complex analytic space, define the strong topology on $X$ as follows: It is the finest topology such that for any morphism of complex analytic spaces $f: T\rightarrow S$ with $f(T)\subset X$, the (set-theoretic) map $f: T\rightarrow X$ is continuous.

Clearly the strong topology is equivalent to or finer than the subspace topology for $X\subset S$, and they are not equivalent in general. For such an example, see <cit.>.

Following <cit.>, we put the strong topology on $E_{\sigma}\subset \check{E}_{\sigma}$, and its induced subspace topology on $E^{\#}_{\sigma}$, then the quotient topology on $D^{\#}_{\Sigma}$ (resp. $\Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma}$) regarding the $i\sigma_{\mathbb{R}}$ (resp. $\sigma_{\mathbb{C}}$) torsor map, and then the strongest topology on $D^{\#}_{\Sigma}$ (resp. $\Gamma\backslash D_{\Sigma}$) such that the inclusion maps:

\begin{align}\label{localembeddingtoglobal}
&D^{\#}_{\sigma} \hookrightarrow D^{\#}_{\Sigma}\\
\nonumber &\Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma} \hookrightarrow \Gamma\backslash D_{\Sigma}
\end{align}

are continuous for every $\sigma\in \Sigma$.

We also need the definition of valuative spaces as follows:

(i)$\mathcal{V}:=$ $\{(A, V) |$  A is a $\mathbb{Q}$-linear subspace of $\mathfrak{g}_\mathbb{Q}$ consisting of mutually commutative nilpotent elements, $V$ is a submonoid of $A^*:=\mathrm{Hom}_\mathbb{Q}(A, \mathbb{Q})$ such that $V\cap (-V)=\{0\}$ and $V\cup (-V)=A^*$}.

(ii)Given $(A,V)\in \mathcal{V}$, let $\mathcal{F}(A,V)$ be the set of rational nilpotent cones $\sigma \subset \mathfrak{g}_{\mathbb{R}}$ such that $\sigma_\mathbb{R}=A_\mathbb{R}$ and $(\sigma \cap A)^{\vee}:= \{h \in A^* | h(\sigma \cap A) \subset \mathbb{Q}_{\geq 0}\} \subset V$.

(i) $\check{D}_{\mathrm{val}}:=\{(A,V,Z)|(A,V)\in \mathcal{V}$ and $Z$ is an $\mathrm{exp}(A_{\mathbb{C}})$-orbit in $\check{D}$}

(ii) ${D}_{\mathrm{val}}:=\{(A,V,Z)|(A,V,Z)\in \check{D}_{\mathrm{val}}$  and there exists $\sigma \in \mathcal{F}(A,V)$ such that $Z$ is a $\sigma$-nilpotent orbit}

(i) Suppose $\Sigma$ is a fan in $\mathfrak{g}_{\mathbb{Q}}$. For $(A,V) \in \mathcal{V}$, let

$X_{A,V,\Sigma}:=\{\sigma \in \Sigma | \sigma \cap A_{\mathbb{R}} \in \mathcal{F}(A,V)\}$

By <cit.>, whenever $X_{A,V,\Sigma}$ is not empty, there is a minimal element $\sigma_0 \in X_{A,V,\Sigma}$.

(ii) Suppose $\Sigma$ is a fan in $\mathfrak{g}_{\mathbb{Q}}$, define:

$D_{\Sigma, \mathrm{val}}:=\{(A,V,Z) \in \check{D}_{\mathrm{val}} | X_{A,V,\Sigma}$ is non-empty, and $\mathrm{exp}(\sigma_{0,\mathbb{C}})Z$ is a $\sigma_0$-nilpotent orbit}

We note that there are similar definitions for the valuative space of (nilpotent) $i$-orbits $D^{\#}_{\mathrm{val}}, D^{\#}_{\Sigma, \mathrm{val}}$. The topology on valuative spaces are defined similarly to their normal counterparts, the details can be found at <cit.>.

§.§.§ Main results

With all notations introduced before, suppose now $\Sigma$ is a fan. We are now ready to state Kato-Usui's main results.

The action of $\sigma_{\mathbb{C}}$ ($i\sigma_{\mathbb{R}}$) on $E_{\sigma}$ ($E_{\sigma}^{\#}$) is proper and free, and as corollary, $D_{\sigma}^{\#}$ and $\Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma}$ are Hausdorff.

$\Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma}$ admits a structure of logarithmic manifold under which $E_{\sigma}\rightarrow \Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma}$ is a $\sigma_{\mathbb{C}}$-torsor of logarithmic manifolds. Moreover,

\begin{equation}
(\Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma})^{\mathrm{log}}\cong \Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma}^{\#}.
\end{equation}

These are results regarding the local structures, or say when the fan $\Sigma$ is just a cone $\sigma$ and its faces. Passing to the global structure, we have the following:

Assuming $\Gamma$ is neat, then $\Gamma\backslash D_{\Sigma}$ admits a structure of logarithmic manifolds as well as Hausdorff topological space, for which the map:

\begin{equation}
\Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma}\rightarrow \Gamma\backslash D_{\Sigma}
\end{equation}

is a local homeomorphism for any $\sigma\in \Sigma$.

The precise defition of logarithmic manifolds can be found in <cit.>. Roughly speaking, a logarithmic manifold is locally isomorphic to an analytic subset of some fs logarithmic analytic space (defined in <cit.>) given by the vanishing locus of a set of logarithmic $1$-forms.

Theorems <ref> and <ref> are about the local model $\Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma}$, for which the proof can be found at <cit.>. In this paper we will briefly outline the idea of proving Theorem <ref>, i.e. how we patch the local logarithmic manifold structures given by $\Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma}$ to a global one on $\Gamma\backslash D_{\Sigma}$ via a (weak) fan $\Sigma$.

§.§.§ Proving main results: First step
<cit.> asserts that the inclusion maps $D_{\sigma}^{\#}\hookrightarrow D_{\Sigma}^{\#}$ as well as $D_{\sigma, \mathrm{val}}^{\#}\hookrightarrow D_{\Sigma, \mathrm{val}}^{\#}$ are open maps, and the canonical map:

\begin{equation}
D_{\Sigma, \mathrm{val}}^{\#}\rightarrow D_{\Sigma}^{\#}
\end{equation}

is a proper map between Hausdorff topological spaces. Particularly, the first argument uses the fact that if $\Sigma$ is a fan, for any $\sigma, \tau\in \Sigma$ we have $D_{\sigma}^{\#}\cap D_{\tau}^{\#}=D_{\sigma\cap \tau}^{\#}$. Hence by the definition of the topology on $D_{\sigma}^{\#}$, it suffices to prove the case when $\Sigma=\{\tau \ \hbox{and its faces}\}$ and $\sigma\leq \tau$. The second argument comes from the commutative diagram:

\begin{equation}
\begin{tikzcd}
E^{\#}_{\sigma} \arrow[d] \arrow[r] & E^{\#}_{\sigma, \mathrm{val}} \arrow[d] \\
D^{\#}_{\sigma} \arrow[r] & D^{\#}_{\sigma, \mathrm{val}}
\end{tikzcd}
\end{equation}

where the vertical arrows are $i\sigma_{\mathbb{R}}$-torsor maps and the horizontal arrows are proper maps by <cit.>.

If we assume $\Sigma$ is a weak fan (instead of a fan), by <cit.>, the condition $D_{\sigma}^{\#}\cap D_{\tau}^{\#}=D_{\sigma\cap \tau}^{\#}$ should be modified by:

\begin{equation}
D_{\sigma}^{\#}\cap D_{\tau}^{\#}=\cup_{\alpha}D_{\alpha}^{\#}
\end{equation}

where $\alpha$ ranges among all common faces of $\sigma$ and $\tau$. The rest of proof is identical to <cit.>.

§.§.§ Proving main results: Second step

Once we prove the natural map

\begin{equation}\label{localchartsofku}
\Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma}\rightarrow \Gamma\backslash D_{\Sigma}
\end{equation}

is a local homeomorphism, the logarithmic manifold structure on $\Gamma\backslash D_{\Sigma}$ will be verified by regarding every $\Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma}$ as a local chart.

The central idea to prove the map (<ref>) is a local homeomorphism is applying <cit.>. For convenience we re-state it in terms of spaces we are considering.

The following conditions hold:

(i) The map $D_{\Sigma}^{\#}\rightarrow \Gamma\backslash D_{\Sigma}^{\#}$ is a local homeomorphism between Hausdorff topological spaces.

(ii) The map $\Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma}^{\#}\rightarrow \Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma}$ is proper.

(iii) For any $(\tau, Z)\in D_{\sigma}$ and $\gamma\in \Gamma$, if $\gamma(\tau, Z)$ and $(\tau, Z)$ lie in the same $\Gamma(\sigma)$-orbit, then $\gamma\in \Gamma(\sigma)$.

Moreover, these properties imply (<ref>) is a local homeomorphism.

We first sketch the proof of conditions (ii) and (iii). For (ii), this comes from the theorem on local charts (<ref>) and basic properties of logarithmic analytic spaces. For (iii), it easily follows from the following proposition:

Assume $\Gamma$ is neat and $(\sigma, Z)\in D_{\Sigma}^{\#}$ $(\mathrm{resp. } \  D_{\Sigma})$, then $\mathrm{Stab}_{\Gamma}(\sigma, Z)=\{\mathrm{Id}\}$ $(\mathrm{resp. } \ \Gamma(\sigma)^{\mathrm{gp}})$.

The most difficult part is proving the condition (i) in Proposition <ref>. This follows from the following theorem.

(i) The action of $\Gamma$ on $D_{\Sigma}^{\#}$ and $D_{\Sigma, \mathrm{val}}^{\#}$ are proper, and the quotient spaces $\Gamma\backslash D_{\Sigma}^{\#}$ and $\Gamma\backslash D_{\Sigma, \mathrm{val}}^{\#}$ are Hausdorff;

(ii) Assume $\Gamma$ is neat, the canonical maps $D_{\Sigma}^{\#}\rightarrow \Gamma\backslash D_{\Sigma}^{\#}$ and $D_{\Sigma, \mathrm{val}}^{\#}\rightarrow \Gamma\backslash D_{\Sigma, \mathrm{val}}^{\#}$
are local homeomorphisms.

Suppose (i) of Theorem <ref> holds. Since $\Gamma$ is a discrete group acting properly and freely on the Hausdorff space $D_{\Sigma}^{\#}$, it is well-known the quotient space $\Gamma\backslash D_{\Sigma}^{\#}$ is Hausdorff and the quotient map is a local homeomorphism. The same argument holds for the valuative space, therefore (ii) of Theorem <ref> holds.

§.§.§ Proving main results: Third step

To prove (i) of Theorem <ref>, we need to formulate theory regarding the space of $\mathrm{SL}_2$-orbits $D_{\mathrm{SL}_2}$. Regarding the general theory of $\mathrm{SL}_2$-orbits, see for example <cit.> or <cit.>.

We shall begin with the classical $\mathrm{SL}_2$-orbit theorem, which was proved in <cit.> for $1$-variable case and <cit.> for general case. We sketch these classical results in a form convenient for our use.

For the field $\mathbb{F}= \mathbb{Q}, \mathbb{R}$ or $\mathbb{C}$, denote

\begin{equation}
\mathbf{n}_-:=\begin{pmatrix}
0 & 1\\
0 & 0
\end{pmatrix}, \
\mathbf{y}:=\begin{pmatrix}
-1 & 0\\
0 & 1
\end{pmatrix}, \
\mathbf{n}_+:=\begin{pmatrix}
0 & 0\\
1 & 0
\end{pmatrix}.
\end{equation}

be the standard generator of $\mathfrak{sl}_2(\mathbb{F})$ with the relations:

\begin{equation}
[\mathbf{y},\mathbf{n}_{\pm}]=\pm 2\mathbf{n}_{\pm}, \ [\mathbf{n}_+, \mathbf{n}_-]=\mathbf{y}.
\end{equation}

Using Cayley transform we can associate a natural $\mathbb{F}$-Hodge structure of weight $0$ on $\mathfrak{sl}_2(\mathbb{F})$ as follows:

\begin{align}
\nonumber (\mathfrak{sl}_2(\mathbb{C}))^{0,0}&=\mathbb{C}(\mathbf{n}_+-\mathbf{n}_-).
\end{align}

Consider an $\mathfrak{sl}_2(\mathbb{F})$-representation:

\begin{equation}
\rho: \mathfrak{sl}_2(\mathbb{F})\rightarrow \mathfrak{g}_{\mathbb{F}}
\end{equation}

which has a lift to the algebraic group level:

\begin{equation}
\tilde{\rho}: \mathrm{SL}_2(\mathbb{F})\rightarrow G_{\mathbb{F}}.
\end{equation}

We denote

\begin{equation}
[N, Y, N^+]:=[\rho(n_-), \rho(y), \rho(n_+)]
\end{equation}

as the associated $\mathfrak{sl}_2$-triple. We also say a pair of $\mathfrak{sl}_2$-triples, namely

\begin{equation}
[N_i, Y_i, N^+_i], \ i=1,2
\end{equation}

commute, if the subalgebras generated by two triples commute.

$\mathfrak{sl}_2$-representations arise in Hodge theory in a natural way. Suppose $(W(N), F^{\bullet}, N)$ is an $\mathbb{R}$-split mixed Hodge structure polarized by $N\in \mathfrak{g}_{\mathbb{Q}}$, let $Y\in \mathfrak{g}_{\mathbb{R}}$ be the semisimple element grading the LMHS, then there exists a unique $\mathfrak{sl}_2$-representation $\rho$ such that

\begin{equation}\label{sl2morphism}
\rho(n_-)=N, \ \rho(y)=Y.
\end{equation}

We complete it into an $\mathfrak{sl}_2$-triple $[N, Y, N^+]$. There is a very special class of $\mathfrak{sl}_2$-representations which induce horizontal $\mathrm{SL}_2$-orbits, see for example <cit.>.

Fix a point $F^{\bullet}\in D$, it induces a weight-$0$ Hodge structure $F_{\mathfrak{g}}^{\bullet}$ on $\mathfrak{g}=\mathrm{End}(V)$. We say $\rho$ gives a horizontal $\mathrm{SL}_2$-orbit at $F^{\bullet}$ if the following holds:

(i) $\tilde{\rho}$ is defined on $\mathbb{R}$;

(ii) $\rho$ is a morphism of Hodge structures with respect to $F_{\mathfrak{g}}^{\bullet}$. Equivalently,

\begin{equation}
\rho(\mathfrak{sl}_2(\mathbb{C}))^{-k,k})\subset F_{\mathfrak{g}}^{-k,k.}
\end{equation}

We also say $\rho$ gives a horizontal $\mathrm{SL}_2$-orbit if it gives one at some point $F^{\bullet}\in D$.

Let $(W(N)[-l], F^{\bullet}, N)$ be an $\mathbb{R}$-split mixed Hodge structure polarized by $N$ and $\rho$ be an $\mathfrak{sl}_2$-representation as (<ref>), then:

(i) $\rho$ is a morphism of Hodge structures at $\exp(iN)F^{\bullet}$;

(ii) $\rho$ gives $\tilde{\rho}$ which is defined over $\mathbb{R}$, and

\begin{equation}
\hat{\rho}: \mathbb{P}^1\rightarrow \check{D}
\end{equation}

an equivariant $\mathrm{SL}_2$-embedding by $\hat{\rho}(gi)=\tilde{\rho}(g)\exp(iN)F^{\bullet}$.

Now we are ready to state the $\mathrm{SL}_2$-orbit theorem of <cit.> and <cit.>. Consider a nilpotent orbit

\begin{equation}
(z_1,..,z_n)\xrightarrow{\theta} \exp(\sum_{j=1}^nz_jN_j)F^{\bullet}.
\end{equation}

<cit.>, i.e. the multivariable $\mathrm{SL}_2$-orbit theorem can be summarized as follows.

There exists a unique $\mathbb{R}$-representation:

\begin{equation}\label{sl2representation}
\rho: (\mathfrak{sl}_2(\mathbb{C}))^n\rightarrow \mathfrak{g}_{\mathbb{C}}
\end{equation}

and its lift:

\begin{equation}
\rho: (\mathrm{SL}_2(\mathbb{C}))^n\rightarrow G_{\mathbb{C}}
\end{equation}

with the following properties:

(i) Let

\begin{equation}
[\hat{N}_j, \hat{Y}_j, \hat{N}^+_j], \ 1\leq j\leq n
\end{equation}

be the $\mathfrak{sl}_2$-triple associated to the $j$-factor of $\rho$, then these $n$-triples commute;

(ii) $W(\sum_{k=1}^{j}N_k)=W(\sum_{k=1}^j\hat{N}_k)$;

Moreover, there exists a unique $F_0^{\bullet}\in \check{D}$ such that:

(iii) $\rho$ gives an $n$-dimensional horizontal $\mathrm{SL}_2$-orbit at $F_0^{\bullet}$;

(iv) Denote $\rho_r, \ 1\leq r\leq n $ as the restriction of $\rho$ on the first $r$ factors, then $\rho_r$ is a morphism of Hodge structures at the point $\exp(i\sum_{k=r+1}^{n}\hat{N}_k)F_0^{\bullet}$;

(v)For $y_1,...,y_n\in \mathbb{R}$, there exists a real analytic function $g(y_1,...,y_n): \mathbb{R}^n\rightarrow G_{\mathbb{R}}$ such that

\begin{equation}\label{sl2approximation}
\exp(i\sum_{j=1}^ny_jN_j)F^{\bullet}=g(y_1,...,y_n)\exp(i\sum_{j=1}^ny_j\hat{N}_j)F_0^{\bullet};
\end{equation}

(vi) Use the convention $y_{n+1}=1$, $g(y_1,...,y_n)$ can be written as a Laurent series of $\frac{y_{j}}{y_{j+1}}, \ 1\leq j\leq n$ with non-positive powers and constant term $1$. Therefore, the two sides of (<ref>) has the same limit as $\frac{y_{j}}{y_{j+1}}\rightarrow \infty, \ 1\leq j\leq n$.

(vii) Again use the convention $y_{n+1}=1$, we have:

\begin{equation}
\exp(i\sum_{j=1}^ny_j\hat{N}_j)F^{\bullet}=\exp(-\frac{1}{2}\sum_{j=1}^{n}(\log(\frac{y_j}{y_{j+1}})\sum_{k=1}^{j}\hat{Y}_k))\exp(i\sum_{j=1}^n\hat{N_{j}})F_0^{\bullet}.
\end{equation}

We call $(\rho, \hat{\rho})$ given by Theorem <ref> as the $\mathrm{SL}_2$-orbit associated to the nilpotent orbit $\theta: (z_1,...,z_n)\rightarrow\exp(\sum_{j=1}^nz_jN_j)F^{\bullet}$. The idea of patching all of these "distinguished $\mathrm{SL}_2$-orbits" to a space leads to the following definition.

The space of $\mathrm{SL}_2$-orbits, denoted as $D_{\mathrm{SL}_2}$, be the collection of all $(\rho, \hat{\rho})$ modulo the equivalence relation $\sim$, such that:

(i) $(\rho, \hat{\rho})$ is an $\mathrm{SL}_2$-orbit;

(ii) The weight filtrations $W(\sum_{k=1}^j\hat{N}_k)$ are rational for $1\leq j\leq n$;

(iii) $(\rho, \hat{\rho})\sim (\rho^{'}, \hat{\rho}^{'})$ if there exists $(t_1,...,t_n)\in \mathbb{R}_{>0}^n$ such that $\hat{Y}_j=\hat{Y}_j^{'}$ and $\hat{N}_j=t_j\hat{N}_j^{'}$.

In <cit.>, the rank of an $\mathrm{SL}_2$-orbit is defined. In this paper we always assume the rank of an $\mathrm{SL}_2$-orbit equals to the number of independent variables $n$, or equivalently (<ref>) is injective.

The topology of the space $D_{\mathrm{SL}_2}$ is constructed in <cit.> and we will omit it here. The point we would like to highlight is the existence of a natural map:

\begin{equation}\label{valuativetosl2}
\psi: D_{\Sigma, \mathrm{val}}^{\#}\rightarrow D_{\mathrm{SL}_2}
\end{equation}

which sends a (valuative) nilpotent orbit to its associated $\mathrm{SL}_2$-orbit('s equivalent class) by Theorem <ref>. Well-defineness of this map can be seen from <cit.>. The critical result is the following:

The map $\psi$ is continuous, and is the unique continuous extension of the identity map on $D$.

Now we return to Theorem <ref> (i). By Theorem <ref>, Proposition <ref> and the fact that $\Gamma$ acts properly on $D_{\mathrm{SL}_2}$ (see <cit.>), the result follows and hence Proposition <ref> is proved.

§.§.§ Final step: Conclusion

Proposition <ref> says the local (Hausdorff) topological and logarithmic analytic structure of $\Gamma\backslash D_{\Sigma}$ are completely inherited from those for $\Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma}$, which are proved in Theorems <ref> and <ref>. This certifies Theorem <ref>.

A remark is that the main Theorems <ref> and <ref> remain valid if we only require $\Sigma$ to be a weak fan. This is proved in <cit.> for the polarized mixed Hodge structure case.

§.§ Weak fan of restricted types

In this subsection we shall go through the same process as the last section, but for a weak fan of restricted type (Definition <ref>). In particular, we shall show Theorems <ref> and <ref>.

Continue using notations from Section 3 and the last section, and let $\Phi$ be an index set containing a certain selection of LMHS types. In this section we shall always assume $\Sigma$ is a weak fan of type $\Phi$. Define:

\begin{equation}
E_{\sigma, \Phi}:=\{(q,F^{\bullet})\in \tilde{E}_{\sigma}|(\sigma(q), \mathrm{exp}(\mathrm{log}_{\Gamma_{\sigma}}q)F^{\bullet}) \hbox{ is a } \sigma(q)-\hbox{nilpotent orbit of type $\Phi$}\}.
\end{equation}

Clearly $E_{\sigma, \Phi}\subset E_{\sigma}$ and we endow it with the subspace topology. The map (<ref>) has the restriction

\begin{equation}
\Theta_{\sigma}|_{ E_{\sigma, \Phi}}: E_{\sigma, \Phi}\rightarrow \Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma, \Phi},
\end{equation}

which is clearly a $\sigma_{\mathbb{C}}$-torsor if we endow $\Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma, \Phi}$ the quotient topology. Similarly we can define $E^{\#}_{\sigma, \Phi}$ with the subspace topology and the $i\sigma_{\mathbb{R}}$-torsor

\begin{equation}
\Theta_{\sigma}^{\#}|_{ E^{\#}_{\sigma, \Phi}}: E^{\#}_{\sigma, \Phi}\rightarrow D^{\#}_{\sigma, \Phi}.
\end{equation}

and hence the diagram (<ref>) can be enlarged to

\begin{equation}\label{enlargeddiagramoftorsors}
\begin{tikzpicture}[scale=1.2]

\node at (0,3) {$E_{\sigma}^{\#}$};
\node at (0,0) {$D_{\sigma}^{\#}$};
\node at (6,0) {$\Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma}$};
\node at (6,3) {$E_{\sigma}$};

\node at (2,1) {$D^{\#}_{\sigma, \Phi}$};
\node at (4,2) {$E_{\sigma, \Phi}$};
\node at (4.5,0.95) {\small $\Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma, \Phi}$\normalsize};
\node at (2,2) {$E^{\#}_{\sigma, \Phi}$};

\draw [->](0.5,0) -- (5,0) [line width = 0.3pt];
\draw [->](0.5,3) -- (5.5,3) [line width = 0.3pt];
\draw [->](0,2.5) -- (0,0.5) [line width = 0.3pt] node[midway,left] {$\Theta^{\#}_{\sigma}$};
\draw [->](6,2.5) -- (6,0.5) [line width = 0.3pt] node[midway, right] {$\Theta_{\sigma}$};

\draw [->](2.5,1) -- (3.5,1) [line width = 0.3pt];
\draw [->](2.5,2) -- (3.5,2) [line width = 0.3pt];
\draw [->](2,1.7) -- (2,1.3) [line width = 0.3pt] node[midway, left] {\small{$\Theta^{\#}_{\sigma}|_{ E^{\#}_{\sigma, \Phi}}$}};
\draw [->](4,1.7) -- (4,1.3) [line width = 0.3pt] node[midway, right] {\small{$\Theta_{\sigma}|_{ E_{\sigma, \Phi}}$}};

\draw [->](1.5,2.3) -- (0.5,2.7) [line width = 0.3pt];
\draw [->](4.5,2.3) -- (5.5,2.7) [line width = 0.3pt];
\draw [->](1.5,0.7) -- (0.5,0.3) [line width = 0.3pt];
\draw [->](4.5,0.7) -- (5.5,0.3) [line width = 0.3pt];

\end{tikzpicture}
\end{equation}

Now we prove Theorem <ref> for the local case where $\Sigma=\{\sigma \ \hbox{and its faces}\}$. Following <cit.>, we need to show:

(i) $E_{\sigma, \Phi}$ admits a structure of logarithmic manifold.

(ii) The action of $\sigma_{\mathbb{C}}$ (resp. $i\sigma_{\mathbb{R}}$) on $E_{\sigma, \Phi}$ (resp. $E^{\#}_{\sigma, \Phi}$) is proper and free.

Indeed, we already know (ii) holds as the restricted maps $\Theta_{\sigma}|_{ E^{\#}_{\sigma, \Phi}}$ and $\Theta^{\#}_{\sigma}|_{ E^{\#}_{\sigma, \Phi}}$ clearly give torsors in the category of Hausdorff topological spaces. We shall now show (i). Indeed, since for any $\sigma$ and any index set $\Phi$, $\tilde{B}(\sigma, \Phi)\subset \tilde{B}(\sigma)$ is a union of connected components (possibly empty), <cit.> applies verbatim, as long as we replace nilpotent orbit by "nilpotent orbit of type $\Phi$".

With (i) and (ii) established, by applying <cit.> and <cit.> verbatim we shall get the desired result, i.e. $\Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma, \Phi}$ admits the structure of logarithmic manifold as well as Hausdroff topological space. Moreover, this structure has two interpretations: Either the one from $E_{\sigma, \Phi}$ using <cit.> or the one obtained from $\Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma, \Phi}\subset \Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma}$.

We turn to the proof of Theorem <ref> for a general $\Sigma$. For $D^{\#}_{\sigma, \Phi}\subset D^{\#}_{\sigma}$ and $\Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma, \Phi}\subset \Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma}$ we endow the subspace topology, then endow $D^{\#}_{\Sigma, \Phi}$ and $\Gamma\backslash D_{\Sigma, \Phi}$ the finest topology such that for any $\sigma\in \Sigma$, the maps:

\begin{align}\label{restrictedlocalembeddingtoglobal}
&D^{\#}_{\sigma, \Phi} \hookrightarrow D^{\#}_{\Sigma, \Phi}\\
\nonumber &\Gamma(\sigma)^{\mathrm{gp}}\backslash D_{\sigma, \Phi} \hookrightarrow \Gamma\backslash D_{\Sigma, \Phi}
\end{align}

are continuous. We also need to define the restricted valuative space $D_{\Sigma, \Phi, \mathrm{val}}$ and $D^{\#}_{\Sigma, \Phi, \mathrm{val}}$:

\begin{align}
D_{\Sigma, \Phi, \mathrm{val}}&:=\{(A,V,Z) \in D_{\Sigma, \mathrm{val}} \ | \ Z \  \hbox{is a nilpotent orbit of type} \  \Phi\},\\
\nonumber D^{\#}_{\Sigma, \Phi, \mathrm{val}}&:=\{(A,V,Z^{\#}) \in D^{\#}_{\Sigma, \mathrm{val}} \ | \ Z \  \hbox{is a nilpotent} \ i\hbox{-orbit of type} \  \Phi\}.
\end{align}

As before, the first step is to obtain an analog of <cit.>. More precisely, $D^{\#}_{\sigma, \Phi}\hookrightarrow D^{\#}_{\Sigma, \Phi}$ and $D^{\#}_{\sigma, \Phi, \mathrm{val}}\hookrightarrow D^{\#}_{\Sigma, \Phi, \mathrm{val}}$ are open maps, and

\begin{equation}
D^{\#}_{\sigma, \Phi, \mathrm{val}}\rightarrow D^{\#}_{\sigma, \Phi}
\end{equation}

is a proper map between Hausdorff topological spaces. We note since $\Sigma$ is a weak fan of type $\Phi$, the statement in Remark <ref> should be replaced by:

\begin{equation}
D_{\sigma, \Phi}^{\#}\cap D_{\tau, \Phi}^{\#}=\cup_{\alpha}D_{\alpha, \Phi}^{\#}
\end{equation}

where $\sigma, \tau\in \Sigma$ and $\alpha$ ranges among all common faces of $\sigma$ and $\tau$. By applying <cit.> verbatim we get the desired results.

Now Theorem <ref>, Proposition <ref>, <ref> and Theorem <ref> can be directly generalized to the restricted case. In particular, Theorem <ref> holds for the restricted case as long as $\{\hbox{Pure LMHS}\}\subset \Phi$. If we consider the map (<ref>) in the restricted case:

\begin{equation}\label{restrictedvaluativetosl2}
\psi|_{D_{\Sigma, \Phi, \mathrm{val}}}: D_{\Sigma, \Phi, \mathrm{val}}^{\#}\rightarrow D_{\mathrm{SL}_2},
\end{equation}

since $\Sigma$ is a weak fan of type $\Phi$, it is enough to prove for any $\sigma$ we have

\begin{equation}\label{restrictedvaluativetosl2local}
\psi|_{D_{\sigma, \Phi, \mathrm{val}}}: D_{\sigma, \Phi, \mathrm{val}}^{\#}\rightarrow D_{\mathrm{SL}_2, \Phi},
\end{equation}

is continuous. Here $D_{\mathrm{SL}_2, \Phi}$ is defined to be the subspace of $D_{\mathrm{SL}_2}$ containing all $\mathrm{SL}_2$ orbits (indeed their equivalent classes) whose associated nilpotent orbits are of type $\Phi$.  The rest of the proof can be obtained by applying <cit.> verbatim.

Finally, Theorem <ref> is obtained by patching up every results we have for the restricted case. Theorem <ref> can be seen as follows: Suppose for a period map $\varphi: S\rightarrow \Gamma\backslash D$, $\Phi$ is the index set containing all LMHS types accessed by $\varphi$. It is enough to check on local charts, i.e.,

\begin{equation}
(\Delta^{*})^k\times \Delta^l\rightarrow \Gamma(\sigma)\backslash D_{\sigma, \Phi}
\end{equation}

is a morphism of logarithmic manifolds for any $\sigma\in \Sigma$, but this is immediate as

\begin{equation}
(\Delta^{*})^k\times \Delta^l\rightarrow \Gamma(\sigma)\backslash D_{\sigma}
\end{equation}

is a morphism of logarithmic manifolds with image in $\Gamma(\sigma)\backslash D_{\sigma, \Phi}$, and

\begin{equation}
\Gamma(\sigma)\backslash D_{\sigma, \Phi}\hookrightarrow \Gamma(\sigma)\backslash D_{\sigma}
\end{equation}

is clearly a morphism of logarithmic manifolds. This concludes the proof.

§ APPENDIX B: COMBINATORICS OF WEAK FAN

The purpose of this section is to investigate the collection $\Sigma$ of nilpotent cones generated by local monodromy nilpotent cones coming from a period map $\varphi: S\rightarrow D$. As a conseuqnce, we shall prove Claim <ref> which completes the proof of Theorem <ref>.

A critical observation is the notions of fan and weak fan are equivalent in the classical cases, as a consequence of the following Lemma <ref> and Proposition <ref>.

Suppose $\hat{\sigma}\subset \sigma$ are nilpotent cones and of same dimension. If $\tilde{B}(\sigma)\neq \emptyset$, then $\tilde{B}(\sigma)=\tilde{B}(\hat{\sigma})$.

As long as there exists $F^{\bullet}\in \tilde{B}(\sigma)\subset \tilde{B}(\hat{\sigma})$, by (<ref>) we have $\tilde{B}(\sigma)= \tilde{B}(\hat{\sigma})$ as $\sigma_{\mathbb{C}}=\hat{\sigma}_{\mathbb{C}}$ by the assumption, where the superscript $\circ$ means the connected components containing the $Z(\sigma)$-orbit of $F^{\bullet}$. However, in the classical weight $1$ case, $Z(\sigma)$ acts transitively on $\tilde{B}(\sigma)$, hence the lemma follows.

Suppose $\sigma, \tau$ are two nilpotent cones polarize LMHS of classical weight $1$ type. If $\sigma\cap \tau\neq \emptyset$, then $\tilde{B}(\sigma)=\tilde{B}(\tau)\neq \emptyset$.

Fix an $F^{\bullet}\in \tilde{B}(\sigma)$, and denote $\mathfrak{g}=\oplus \mathfrak{g}^{p,q}$ as the adjoint Hodge decomposition of $(\sigma, F^{\bullet})$. We also denote

\begin{equation}
\mathcal{C}_{F^{\bullet}}:=\{N\in \mathfrak{g}^{-1,-1}\cap \mathfrak{g}_{\mathbb{Q}} \ | \ (N, F^{\bullet}) \ \hbox{is a nilpotent orbit} \ \}.
\end{equation}

Since $\sigma\cap \tau\neq \emptyset$, we must have $W(\sigma)=W(\tau)$, therefore $\tau\subset W_{\mathfrak{g}}(\sigma)_{-2}$, in the classical weight $1$ case this means $\tau\subset \mathfrak{g}^{-1,-1}$ and thus $\mathcal{C}_{F^{\bullet}}\cap \tau_{\mathbb{Q}}$ is open in $\tau_{\mathbb{Q}}$. In other words, there exists a $\tau^{'}\subset \tau$ such that $\sigma\cap \tau\subset \tau^{'}\subset \tau$, $\tau^{'}_{\mathbb{Q}}=\tau_{\mathbb{Q}}$ and $(\tau^{'}, F^{\bullet})$ is a nilpotent orbit. Combine Lemma <ref> and $\tilde{B}(\tau)\neq \emptyset$ we know $(\tau, F^{\bullet})$ is a nilpotent orbit, hence $\tilde{B}(\sigma)\subset \tilde{B}(\tau)$. Similarly we have $\tilde{B}(\sigma)\supset \tilde{B}(\tau)$ and thus $\tilde{B}(\sigma)= \tilde{B}(\tau)$.

We continue using definitions and notations in Section 3. Let $\Sigma$ be a collection of nilpotent cones with $\mathrm{Ad}(\Gamma)$-generating set:

\begin{equation}
\mathcal{S}:=\{\sigma_0,\sigma_1,...,\sigma_K\},
\end{equation}

here we can just take $\sigma_i$ to be all local monodromy nilpotent cones coming from $\varphi$ (with a chosen order).

For any ordered pair

\begin{equation}\label{orderedpairofcones}
(\sigma_i, \sigma_j)\in \mathcal{S}\times \mathcal{S}, \ 0\leq i,j\leq K,
\end{equation}

let $\{\gamma^{ij}_l\}_{l\geq 0}\subset \Gamma$ be the countable sequence such that

\begin{equation}
\sigma_i\cap \mathrm{Ad}(\gamma^{ij}_k)\sigma_j \neq \emptyset, \ \tilde{B}(\sigma_i)\cap \tilde{B}(\mathrm{Ad}(\gamma^{ij}_k)\sigma_j)\neq \emptyset.
\end{equation}

If the sequence is indeed finite (or even empty), we make it an infinite one by adding infinitely many $\mathrm{Id}$. By Proposition <ref>, the image of $\{\gamma^{ij}_l\}_{l\geq 0}$ under the projection

\begin{equation}
G_{\mathbb{C}}\rightarrow Z(\sigma_i)\backslash G_{\mathbb{C}}/Z(\sigma_j)
\end{equation}

is finite. We assume the set $\{\gamma^{ij}_0,...,\gamma^{ij}_{l_{ij}}\}$ covers all of these classes. By this procedure, for each ordered pair $(i,j)$ and the corresponding ordered pair in (<ref>), we obtain a subset

\begin{equation}
\{\gamma^{ij}_0,...,\gamma^{ij}_{l_{ij}}\}\subset \Gamma, \ 0\leq i,j\leq K.
\end{equation}

By choosing $L:=\mathrm{Max}\{l_{ij}\}$ and extending each sequences to $\{\gamma^{ij}_0,...,\gamma^{ij}_L\}$, we obtain the following $(K+1)^2(L+1)$ cones:

\begin{equation}\label{allpossiblebusters}
\{\sigma_{i}\}\cup \{\mathrm{Ad}(\gamma^{ij}_l)\sigma_j\}, \ 0\leq i,j\leq K, \ 0\leq l\leq L.
\end{equation}

Next we prove a lemma regarding the combinatorics of polyhedral cones.

For a finite-dimensional $\mathbb{Q}$-vector space $V$ and a finite collection of sharp polyhedral cones $\{\sigma_i\}_{0\leq i\leq N}$ in $V$ which is closed under taking faces, there exists a finite subdivision on $\cup_{0\leq i\leq N}|\sigma_i|$ such that the resulting new collection of cones $\{\tau_j\}_{0\leq j\leq M}$ satisfy:

(i) $\cup_{0\leq i\leq N}|\sigma_i|=\cup_{0\leq j\leq M}|\tau_j|$;

(ii) The collection  $\{\tau_j\}_{0\leq j\leq M}$ is a fan;

(iii) Each $\tau_j$ is a simplicial cone.

Fix a coordinate system $(x_1,..,x_n)$ on $V$. For each cone $\sigma_i$, there exists finitely many linear forms

\begin{equation}
l^i_j:=\sum_{\lambda=1}^{n_i} a^{ij}_{\lambda}x_{\lambda}, \ 0\leq j\leq n_i,
\end{equation}

each of which is defined over $\mathbb{Q}$, such that

\begin{equation}
|\sigma_i|=\cap_{0\leq j\leq n_i}l^i_j\circ 0
\end{equation}

where each $\circ$ should be replaced by one of $>, \ <, \ =$. We shall call this a (rational) hyperplane representation of $\sigma_i$. For each $\sigma_i$ we fix a hyperplane representation. Running over all of $0\leq i\leq N$, these linear forms give finitely many hyperplanes in $V$:

\begin{equation}
\{l^i_j=0 \ | \ 0\leq i\leq N, \ 0\leq j\leq n_i\}.
\end{equation}

Which seperate the whole space $V$ into finitely many relatively open chambers $\{\tau_j\ | \ j \in \mathcal{J}\}$ (with different dimensions). A direct observation is the support of $\{\tau_j\ | \ j \ \in \mathcal{J}\}$ is $V$, and $\{\tau_j\ | \ j \in \mathcal{J}\}$ gives a refinement of $\{\sigma_i\}$ in the sense that each $\sigma_i$ is the disjoint union of finitely many elements in $\{\tau_j\ | \ j \in \mathcal{J}\}$.

We take the chambers $\{\tau_j \ | \ 1\leq j\leq M\}$ such that the condition (i) in the lemma is satisfied. The condition (ii) is immediately justified by the following observation: Each $\tau_j$ is a relatively open convex polyhedral cone, and $\tau_j\cap \tau_k$ is either $\emptyset$ or $\tau_j=\tau_k$, in which the latter case happens if and only if $\tau_j$ and $\tau_k$ have the same hyperplane representation.

Next we show after a finite subdivision, $\{\tau_j \ | \ 1\leq j\leq M\}$ satisfies (iii). This uses the simplification of a cell complex by proceeding star subdivision, see <cit.>. Particularly, it is a classical result that there exists a finite sequence of star subdivisions on $\{\tau_j \ | \ 1\leq j\leq M\}$ such that the resulting fan $\{\tau_j^{'} \ | \ 1\leq j\leq M^{'}\}$ is a simplicial cone complex, see <cit.> or <cit.>, thus the lemma is proved.

To apply the lemma, we shall denote $\{\tau_j \ | \ 1\leq j\leq M\}$ as the simplicial cone complex constructed from the set of cones in (<ref>), and $\hat{\Sigma}$ be the union of $\mathrm{Ad}(\Gamma)$-orbits of $\{\tau_j \ | \ 1\leq j\leq M\}$. Clearly $|\hat{\Sigma}|=|\Sigma|$ and $\hat{\Sigma}$ is a refinement of $\Sigma$, and the subdivision is finitely generated up to $\mathrm{Ad}(\Gamma)$-action.

Therefore, to prove Claim <ref>, it suffices to prove $\hat{\Sigma}$ is a $\Gamma$-strongly compatible weak fan. By Proposition <ref>, we shall prove $\hat{\Sigma}$ is actually a fan.

§.§.§ Proof of Claim <ref>

Suppose for some $\hat{\sigma}, \hat{\tau}\in \hat{\Sigma}$ and $\gamma\in \Gamma$, we have $\hat{\sigma} \cap \hat{\tau}\neq \emptyset$. After conjugating by $\Gamma$ we can assume there exists $\sigma\in \mathcal{S}$ such that $\hat{\sigma}\subset \sigma$.

According to Proposition <ref> and Proposition <ref>, there exists some $\tau_0\in \mathcal{S}$ and a finite set $\{\gamma_1,...,\gamma_L\}\in \Gamma$ such that $\hat{\tau}\subset \tau=\mathrm{Ad}(\gamma_0)\tau_0$ for some $\gamma_0\in \Gamma$, and $\gamma_0$ induce the same class as $\gamma_l$ in $Z(\sigma)\backslash G_{\mathbb{C}}/Z(\tau_0)$ for some $1\leq l\leq L$. Since replacing $\gamma_0$ by anything in its own $(Z(\sigma), Z(\tau_0))$-coset does not change $\sigma\cap \mathrm{Ad}(\gamma_0)\tau_0$, it is enough to consider when $(\sigma, \mathrm{Ad}(\gamma_0)\tau_0)$ is one of the cases in (<ref>).

By the construction of Lemma <ref>, support of the (finitely many) cones in (<ref>) also supports the simplicial cone complex $\{\tau_j \ | \ 1\leq j\leq M\}$ which is a fan, and $\hat{\Sigma}$ is taken to be the union of $\{\tau_j \ | \ 1\leq j\leq M\}$ and their $\mathrm{Ad}(\Gamma)$-translations. As a consequence $\hat{\sigma},  \hat{\tau}\in \{\tau_j \ | \ 1\leq j\leq M\}$, but $\{\tau_j \ | \ 1\leq j\leq M\}$ by construction is a fan, therefore we must have $\hat{\sigma}=\hat{\tau}$ and the claim follows.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

Cambridge University Press

titleSmooth Compactifications of Locally Symmetric Varieties

journaltitleAnnals of Mathematics
titleCompactification of arithmetic quotients of bounded symmetric domains

journaltitleInventiones mathematicae
titleo-minimal GAGA and a conjecture of Griffiths

seriesMathematics: Theory & Applications
titleCompactifications of Symmetric and Locally Symmetric Spaces

journaltitleJ. Amer. Math. Soc.
titleTame topology of arithmetic quotients and algebraicity of Hodge loci

noteWork in progress
titleExtension of a 2 dimensional period image of Hodge type (1,2,2,1)

journaltitleInventiones mathematicae
titlePolarized mixed Hodge structures and the local monodromy of a variation of Hodge structure

titleDegenerating variations of Hodge structure

journaltitleAnnals of Mathematics
titleDegeneration of Hodge Structures

American Mathematical Society

seriesGraduate Studies in Mathematics
titleToric Varieties

journaltitleAdvances in Mathematics
titleExtension of period maps by polyhedral fans

seriesGraduate Texts in Mathematics
titleCombinatorial Convexity and Algebraic Geometry

journaltitleContemp. Math
titleNéron models and boundary components for degenerations of Hodge structure of mirror quintic type

journaltitleBull. Amer. Math. Soc.
titlePeriods of integrals on algebraic manifolds: Summary of main results and discussion of open problems

journaltitleCommunications in Mathematical Physics
titleDeterminantal Quintics and Mirror Symmetry of Reye Congruences

journaltitleSymmetry, Integrability and Geometry: Methods and Applications
titleMovable vs Monodromy Nilpotent Cones of Calabi–Yau Manifolds

journaltitleJ. Algebraic Geom.
titleClassifying spaces of degenerating mixed Hodge structures, III: Spaces of nilpotent orbits

journaltitleDuke Math. J.
titleBoundary components of Mumford–Tate domains

journaltitleDocumenta Mathematica
titlePolarized relations on horizontal SL(2)'s

journaltitleAdv. Stud. Pure Math.
titleBorel-Serre Spaces and Spaces of SL(2)-Orbits

Princeton University Press

seriesAnnals of Mathematics Studies
titleClassifying Spaces of Degenerating Polarized Hodge Structures

journaltitleCompositio Mathematica
titleClassification of horizontal SL(2)'s

journaltitleAnn. of Math.
titleOn compactifications of the quotient spaces for arithmetically defined discontinuous groups

journaltitleInventiones mathematicae
titleVariation of Hodge Structure: The Singularities of the Period Mapping