

\subsection{Moduli spaces of polarized Hodge structures.}

This section owes much to a lecture by Phillip Griffiths \cite{Gri}.

First of all, we briefly summarize the classical theory of 
the moduli spaces of polarized Hodge structures. 

\subsubsection{The moduli space $M_h=\Gamma\backslash 
D_h.$}\label{subsection1.1}

Let $n$ be an integer, and let $h$ be a sequence of 
positive integers $(h^{n,0},h^{n-1,0},\cdots,h^{0,n})$ 
satisfying $h^{p,q}=h^{q,p},$ called the \textit{Hodge 
numbers}. Let $H_{\mathbb Z}$ be a free abelian group of 
rank $\sum h^{p,n-p},$ with a non-degenerate bilinear 
form $Q:H_{\mathbb Z}\otimes H_{\mathbb Z}\to\mathbb Z,$ 
which is symmetric (resp. anti-symmetric) if $n$ is even 
(resp. odd). Let $G_{\mathbb Z}$ be the group functor 
$\text{Aut}(H_{\mathbb Z},Q)$ on rings, sending a ring $R$ 
to the group of automorphisms on the free $R$-module 
$H_R:=H_{\mathbb Z}\otimes R$ preserving the bilinear form 
$Q.$ It is clearly a group scheme over $\mathbb Z.$ Let 
$\Gamma$ be an arithmetic subgroup of $G_{\mathbb 
Z}(\mathbb Z)$ (\cite{Mil}, $\S3$).

The set of Hodge structures of weight $n$ on $H_{\mathbb 
R}$ with prescribed Hodge numbers $h,$ such that $Q$ induces a 
\textit{polarization} on $H_{\mathbb R}$ (i.e. it induces a 
morphism $H_{\mathbb R}\otimes H_{\mathbb R}\to\mathbb R(-n)$ of 
Hodge structures, and the bilinear form $Q_C(u,v):=Q(u,Cv),$ 
where $C$ is the Weil operator, is symmetric and positive 
definite), is parameterized by the homogeneous space 
$D_h=G_{\mathbb R}/K,$ where $K$ is the stabilizer group of a 
fixed polarized Hodge structure $F_0$ on $H_{\mathbb R}.$  

This homogeneous space $D=D_h=G_{\mathbb R}/K$ has a 
complex structure defined as follows. It is clear that 
$Q:H_{\mathbb R}\otimes H_{\mathbb R}\to\mathbb R(-n)$ is a 
morphism of Hodge structures if and only if 
$Q(F^p,F^{n-p+1})=0$ for all $p.$ Let $f^p=\sum_{r\ge 
p}h^{r,n-r},$ and let $D^{\vee},$ the \textit{compact dual 
of $D,$} to be the subspace of the product of the 
Grassmannians $\prod_p\text{Gr}(f^p,H_{\mathbb R})$ 
consisting of all flags $F^{\bullet}$ 
$$
\cdots\subset F^{p+1}\subset F^p\subset\cdots 
$$
such that $Q(F^p,F^{n-p+1})=0.$ Then $D^{\vee}=G_{\mathbb 
C}/P,$ where $P$ is a parabolic subgroup corresponding to a 
fixed flag. This gives $D^{\vee}$ a complex structure. We 
see that $D\subset D^{\vee}$ is the locus of flags 
satisfying 

(i) $F^p\cap\overline{F}^{n-p+1}=0$ (so that $F^p\oplus 
\overline{F}^{n-p+1}\cong H_{\mathbb C}),$ and 

(ii) $Q(\overline{u},Cu)>0$ for $u\ne0$ in $H_{\mathbb C}.$

They are both open conditions, so $D\subset D^{\vee}$ is an 
open complex submanifold. The group $\Gamma$ acts on $D_h$ 
properly discontinuously, and the quotient $M_h=\Gamma 
\backslash D_h$ is the moduli space of 
\textit{$\Gamma$-equivalence classes of $Q$-polarized Hodge 
structures on $H_{\mathbb C}$ with Hodge type $h.$} See 
(\cite{Kato-Usui-book}, 0.3.6, 0.3.7). 

\subsubsection{Variations of Hodge structure.}

\begin{definition}
Let $S$ be a complex manifold. A \emph{variation of Hodge 
structure $\mathscr H$ of weight $n$ on $S$} is given by 

$\bullet$ a local system $\mathscr H_{\mathbb Z}$ of free 
abelian groups of finite rank on $S;$ 

$\bullet$ a finite decreasing filtration 
$F^{\bullet}\mathscr H_{\mathscr O}$ of the vector bundle 
$\mathscr H_{\mathscr O}:=\mathscr H_{\mathbb Z}\otimes_{ 
\mathbb Z}\mathscr O_S$ by holomorphic sub-bundles, 

such that the following conditions are satisfied: 

1) (Griffiths transversality) the natural flat connection 
$\nabla=d\otimes\emph{id}_{\mathscr H_{\mathbb 
Z}}:\mathscr H_{\mathscr O}\to\Omega^1_S\otimes\mathscr 
H_{\mathscr O}$ takes $F^p\mathscr H_{\mathscr O}$ into 
$\Omega^1_S\otimes F^{p-1}\mathscr H_{\mathscr O},$ for 
every $p;$ 

2) for each point $s\in S,$ the fiber $F^{\bullet}(s)$ over 
$s$ is a Hodge structure of weight $n.$ 

A \emph{polarization} of the variation of Hodge structure 
$\mathscr H$ is a locally constant bilinear form 
$$
\mathscr Q:\mathscr H_{\mathbb Z}\otimes\mathscr H_{\mathbb 
Z}\to\underline{\mathbb Z}
$$ 
such that on each fiber over $s\in S,$ it induces a 
polarization of the fiber Hodge structure. 
\end{definition}

Suppose we have a polarized family of Hodge structures 
$(\mathscr H,\mathscr Q:\mathscr H_{\mathbb Z}\otimes\mathscr 
H_{\mathbb Z}\to\mathbb Z)$ of weight $n$ on $S,$ and a global 
section of the sheaf 
$$
\Gamma\backslash\underline{\text{Isom}}((\mathscr 
H_{\mathbb Z},\mathscr Q),(H_{\mathbb Z},Q)),
$$
where $H_{\mathbb Z}$ is regarded as a constant sheaf on $S,$
and assume that all the monodromies of this family of Hodge 
structures on $S$ are contained in $\Gamma.$ Then there is a 
\textit{period map} 
$$
\varphi:S\to M_h 
$$
inducing this family of Hodge structures. This map is 
locally liftable to $D_h.$ 

If $f:X\to S$ is 
a projective smooth morphism between quasi-projective complex 
algebraic manifolds, with a relative hyperplane section 
$\eta\in H^0(S,R^2f_*\mathbb Z),$ then the family of the 
primitive part $P^n(X_s,\mathbb Z)$ of the cohomology groups 
$H^n(X_s,\mathbb Z)$ modulo torsion form a polarized variation 
of Hodge structure of weight $n$ on $S,$ and it induces a 
period map $S\to M_h.$ To be precise, the family of 
$H^n(X_s,\mathbb C)$'s are the stalks of 
$R^nf_*(f^{-1}\mathscr O_S),$ and the Hodge filtration on 
$R^nf_*(f^{-1}\mathscr O_S)$ is given by the degenerate 
spectral sequence 
$$
E_1^{pq}=R^qf_*\Omega_{X/S}^p\Longrightarrow 
R^{p+q}f_*(f^{-1}\mathscr O_S),
$$
which is induced from the resolution 
$\Omega_{X/S}^{\bullet}$ of $f^{-1}\mathscr O_S$ (the 
relative holomorphic Poincar\'e lemma, see (\cite{Del1}, 
3.4)). Since $\eta$ is a global section, the primitive part 
form a variation of sub-Hodge structures on $S.$ 

\subsection{Logarithmic Hodge structures.}

One can ask the following question. Let $f:X\to S$ be a family 
of projective manifolds, and let $S$ be the complement of a 
normal crossing divisor $D$ in some compact manifold 
$\overline{S},$ and suppose one can extend the family $f$ to a 
family $\overline{f}:\overline{X}\to\overline{S}$ which is log 
smooth (here $\overline{S}$ has the log structure induced by 
the divisor $D$). Is it possible to enlarge the moduli space 
$M_h$ to some $\overline{M}_h$ so that the period map 
extends to $\overline{\varphi}:\overline{S}\to\overline{M}_h?$ 

To study the degenerations of Hodge structures, Kato and 
Usui introduced the notion of logarithmic Hodge structures. 

\subsubsection{The ringed space $X^{\log}.$}

Let $(X,\alpha:M_X\to\mathscr O_X)$ be an fs log analytic 
space over $\mathbb C$ (for instance the $\mathbb 
C$-points of an fs log scheme over $\mathbb C$), and let 
$X^{\log}$ be the set of pairs $(x,u),$ where $x\in 
X$ and $u:M_{X,x}\to S^1$ is a homomorphism of monoids, 
such that $u(f)=f(x)/|f(x)|$ for $f\in\mathscr 
O^*_{X,x}\subset M_{X,x}.$ Here $S^1$ is the unit circle in 
the complex plane. Let $\tau:X^{\log}\to X$ be the 
function $(x,u)\mapsto x.$ For any open $U\subset X$ and 
$f\in M_X(U),$ there is a 
function $arg(f):\tau^{-1}(U)\to S^1$ sending $(x,u)\mapsto 
u(f).$ We give $X^{\log}$ the weakest topology such that the 
functions $\tau$ and $arg(f)$ are continuous. Over the open 
set $X^*\subset X$ where the log structure is trivial, the map 
$\tau$ is a homeomorphism, and the section 
$j^{\log}:X^*\hookrightarrow X^{\log}$ is a homotopy 
equivalence. The map $\tau$ is proper, with fibers 
$\tau^{-1}(x)$ compact tori $(S^1)^m,$ where $m$ is the rank 
of $\overline{M}_{X,x}^{\text{gp}}.$ 

One can define a sheaf of rings $\mathscr O_{X^{\log}}$ on 
$X^{\log}.$ Roughly speaking, this is the subsheaf of rings of 
$j^{\log}_*\mathscr O_{X^*}$ on $X^{\log}$ generated over 
$\tau^{-1}\mathscr O_X$ by $``\log(q)",$ for all $q\in 
M^{\text{gp}}_X.$ See (\cite{Kato-Usui-book}, 2.2.4) for the precise 
definition. 

For example, if $x\in X$ and $y\in\tau^{-1}(x),$ and the free 
abelian group $\overline{M}_{X,x}^{\text{gp}}$ has rank $m$ 
and is generated by $f_1,\cdots,f_m\in M_{X,x}^{\text{gp}},$ 
then the stalk $\mathscr O_{X^{\log},y}$ is isomorphic to the 
polynomial ring $\mathscr 
O_{X,x}[\log(f_1),\cdots,\log(f_m)].$ This shows that in 
general, $(X^{\log},\mathscr O_{X^{\log}})$ is not a locally 
ringed space. 

Let $\Omega_X^1$ be the sheaf of log differential forms on the 
fs log analytic space $X,$ i.e. 
$$
\Omega_X^1=(\Omega_{\underline{X}}^1\oplus(\mathscr 
O_X\otimes_{\mathbb Z}M_X^{\text{gp}}))/\{(-d\alpha(f), 
\alpha(f)\otimes f)|f\in M_X\}. 
$$
For a morphism $f:X\to Y$ of fs log analytic spaces, define 
$$
\Omega_{X/Y}^1=\text{Coker}(f^*\Omega_Y^1\to\Omega_X^1).
$$
They are both coherent $\mathscr O_X$-modules. Let 
$\Omega_{X/Y}^r$ be the $r$-th exterior power of 
$\Omega_{X/Y}^1,$ and let 
$$
\Omega_{X^{\log}/Y^{\log}}^r=\tau^*\Omega_{X/Y}^r=\tau^{-1} 
\Omega_{X/Y}^r\otimes_{\tau^{-1}\mathscr O_X}\mathscr 
O_{X^{\log}}. 
$$
One can define differential maps and have the log de Rham 
complex $(\Omega_{X/Y}^{\bullet},d)$ (resp. 
$(\Omega_{X^{\log}/Y^{\log}}^{\bullet},d)$) on $X$ (resp. 
$X^{\log}$). 

For $y\in X^{\log}$ and $x=\tau(y)\in X,$ let $\text{sp}(y)$ 
be the set of all ring homomorphisms $s:\mathscr O_{X^{\log}, 
y}\to\mathbb C$ that extend the evaluation map $\text{ev}_x: 
\mathscr O_{X,x}\to\mathbb C.$ Since $\mathscr O_{X^{\log},y}$ 
is isomorphic to the polynomial ring over $\mathscr O_{X,x}$ 
generated by log of a basis for $\overline{M}_{X,x},$ if we 
fix an $s_0\in\text{sp}(y),$ then we have a bijection: 
$$
s\mapsto(f\mapsto s(\log(f))-s_0(\log(f))):\text{sp}(y) 
\overset{\sim}{\longrightarrow}Hom_{\text{group}}(\overline 
{M}_{X,x}^{\text{gp}},\mathbb C).
$$

\subsubsection{Log variations of polarized Hodge structure.}

\begin{definition}\label{LVPHS}
Let $X$ be an fs log analytic space. A \emph{log variation of 
polarized Hodge structure of weight $n$} on $X$ is given by 

$\bullet$ a local system of free abelian groups of finite rank 
$\mathscr H_{\mathbb Z}$ on $X^{\log},$ 

$\bullet$ a bilinear form $\mathscr Q:\mathscr H_{\mathbb 
Z}\otimes \mathscr H_{\mathbb Z}\to\underline{\mathbb Z},$ 

$\bullet$ a finite decreasing filtration $F^{\bullet}\mathscr 
H_{\mathscr O}$ of $\mathscr H_{\mathscr O}:=\mathscr 
H_{\mathbb Z}\otimes\mathscr O_{X^{\log}}$ by $\mathscr 
O_{X^{\log}}$-submodules,

such that the following conditions are satisfied: 

1) there exist a locally free $\mathscr O_X$-module $\mathscr 
E$ and a finite decreasing filtration $F^{\bullet}\mathscr E$ 
by $\mathscr O_X$-submodules, such that $\text{Gr}_p(\mathscr 
E)$ is locally free for each $p,$ and 
$$
F^p\mathscr H_{\mathscr O}=\tau^*F^p\mathscr E=\tau^{-1}F^p 
\mathscr E\otimes_{\tau^{-1}\mathscr O_X}\mathscr 
O_{X^{\log}};
$$

2) for $y\in X^{\log}$ and $x=\tau(y)\in X,$ let $s\in\emph 
{sp}(y)$ and let $f_1,\cdots,f_r\in M_{X,x}-\mathscr 
O_{X,x}^*$ generate the monoid $\overline{M}_{X,x}.$ If the 
$|\exp(s(\log(f_i)))|$ are sufficient small for all $i,$ 
then $(\mathscr H_{\mathbb Z,y},\mathscr Q,F^{\bullet}(s))$ is 
a polarized Hodge structure of weight $n;$
 
3) the connection $d\otimes\emph{id}:\mathscr H_{\mathscr 
O}\to\Omega_{X^{\log}}^1\otimes_{\mathscr 
O_{X^{\log}}}\mathscr H_{\mathscr O}$ takes $F^p\mathscr 
H_{\mathscr O}$ into $\Omega_{X^{\log}}^1\otimes 
F^{p-1}\mathscr H_{\mathscr O}.$
\end{definition}

Here $F^{\bullet}(s),$ the \textit{specialization of $F$ at 
$s$}, is the decreasing filtration of $\mathscr H_{\mathbb 
C,y}:=\mathbb C\otimes_{\mathbb Z}\mathscr H_{\mathbb Z,y}$ 
defined by $F^p(s)=\mathbb C\otimes_{s,\mathscr 
O_{X^{\log},y}}F^p\mathscr H_{\mathscr O}.$ For a fixed point 
$y\in X^{\log},$ the family $(\mathscr H_{\mathbb 
Z,y},\mathscr Q,F^{\bullet}(s))_{s\in\text{sp}(y)}$ is called 
a \textit{polarized log Hodge structure} on the log point 
$(x,M_{X,x});$ this is the same as a log variation of 
polarized Hodge structure on the log point $(x,M_{X,x}).$    

Log variations of polarized Hodge structure arise from 
geometry in the following way. Let $f:X\to Y$ be a projective 
log smooth morphism between fs log analytic spaces, and we fix 
a line bundle on $X$ which is relatively ample over $Y.$ By a 
theorem of Kajiwara and Nakayama, for every integer $n,$ the 
sheaf $R^nf^{\log}_*\mathbb Z$ is a local system on 
$Y^{\log}.$ We take $\mathscr H_{Z}$ to be 
$R^nf^{\log}_*\mathbb Z$ modulo torsion, take $\mathscr Q$ to 
be the pairing induced by the fixed ample line bundle, take 
$\mathscr E$ to be $R^nf_*(\Omega_{X/Y}^{\bullet}),$ with 
filtration $F^p\mathscr E=R^nf_*(\Omega_{X/Y}^{\ge 
p})\subset\mathscr E,$ and take $F^p\mathscr H_{\mathscr O}$ 
to be $\tau^*F^p\mathscr E.$ Then by a theorem of Kato, 
Matsubara and Nakayama, this is a log variation of polarized 
Hodge structure on $Y.$ 

\subsection{Kato-Usui spaces.}

We fix $n,h,H_{\mathbb Z},Q,G_{\mathbb Z},D$ and $D^{\vee}$ 
as in (\ref{subsection1.1}). Let $\mathfrak 
g_R=\text{Lie}(G_R).$ A subset $\sigma\subset\mathfrak 
g_{\mathbb R}$ is called a \textit{nilpotent cone} if it is a 
cone 
$$
\sigma=\sum_{i=1}^n\mathbb R_{\ge0}N_i
$$
generated by mutually commutative nilpotent operators 
$N_i\in\mathfrak g_{\mathbb R}\subset\text{End}(H_{\mathbb 
R}).$ Let $\Gamma$ be a \textit{neat subgroup} of 
$G_{\mathbb Z}(\mathbb Z),$ i.e. for every element 
$\gamma\in\Gamma,$ its eigenvalues on $H_{\mathbb C}$ generate 
a torsion-free subgroup of $\mathbb C^*.$

\subsubsection{Nilpotent orbits.}

\begin{definition}\label{nilp-orbit}
Let $\sigma=\sum_i\mathbb R_{\ge0}N_i$ be a nilpotent cone. A 
subset $Z\subset D^{\vee}$ is called a 
\emph{$\sigma$-nilpotent orbit}, if there exists an 
$F_0\in D^{\vee}$ such that 

$\bullet\ Z=\exp(\sum_i\mathbb CN_i)F_0,$

$\bullet\ NF^p_0\subset F^{p-1}_0$ for all $p\in\mathbb Z$ and 
$N\in\sigma,$

$\bullet\ \exp(\sum_iz_iN_i)F_0\in D$ if $\emph{Im}(z_i)\gg0$ 
for all $i.$
\end{definition}

We also call the pair $(\sigma,Z)$ a \textit{nilpotent orbit.} 

Let $\Sigma$ be a \textit{fan} in $\mathfrak g_{\mathbb Q},$ 
i.e. $\Sigma\ne\emptyset$ is a set of rational nilpotent cones 
in $\mathfrak g_{\mathbb R}$ (namely, those generated by 
nilpotent operators in $\mathfrak g_{\mathbb Q}$) such that 

$\bullet$ if $\sigma\in\Sigma,$ then all faces of $\sigma$ are 
in $\Sigma,$

$\bullet$ for $\sigma,\sigma'\in\Sigma,$ the intersection 
$\sigma\cap\sigma'$ is a face of both $\sigma$ and $\sigma',$ 

$\bullet$ for every $\sigma\in\Sigma,$ we have 
$\sigma\cap(-\sigma)=0.$

One can then define the set $D_{h,\Sigma}$ (or just 
$D_{\Sigma},$ if there is no confusion) of \textit{nilpotent 
orbits in the directions in $\Sigma$} to be the set of 
nilpotent orbits $(\sigma,Z)$ where $\sigma\in\Sigma.$ There 
is a natural injection 
$$
F\mapsto(0,\{F\}):D\hookrightarrow D_{\Sigma}.
$$

\subsubsection{The moduli space $M_{\Sigma}$.}

Let $\Sigma$ be a fan in $\mathfrak g_{\mathbb Q}$ and let 
$\Gamma\subset G_{\mathbb Z}(\mathbb Z)$ be a subgroup. Then 
we say that \textit{$\Gamma$ is compatible with $\Sigma$} if 
for every $\gamma\in\Gamma$ and $\sigma\in\Sigma,$ we have 
$Ad(\gamma)(\sigma)\in\Sigma.$ In this case, there is an 
action of $\Gamma$ on $D_{\Sigma}$ given by 
$$
\xymatrix@C=1cm{
(\sigma,Z) \ar@{|->}[r]^-{\gamma} & 
(Ad(\gamma)(\sigma),\gamma Z).}
$$
We say that \textit{$\Gamma$ is strongly compatible with 
$\Sigma$} if every cone $\sigma\in\Sigma$ is generated by 
elements in $\log\Gamma.$ Kato and Usui showed that when 
$\Gamma$ is strongly compatible with $\Sigma$ and the 
arithmetic subgroup $\Gamma$ is neat, the quotient set 
$\Gamma\backslash D_{\Sigma}$ can be given the structure of a 
log locally ringed space over $\mathbb C,$ in fact a 
\textit{log manifold} (see (\cite{Kato-Usui-book}, 3.5.7)). Roughly 
speaking, a log manifold is a log locally ringed space over 
$\mathbb C,$ which is locally isomorphic to the ``zero locus" 
of some log differential forms on a log smooth analytic space.  

Informally speaking, Kato and Usui proved the following. 
First, there is a one-to-one correspondence between 
$D_{\Sigma}$ and the set of polarized log Hodge structures of 
the given type. Second, if $\overline{X}\to\overline{S}$ is a 
log smooth family extending the projective smooth family $X\to 
S,$ where $S\subset\overline{S}$ is the complement of a normal 
crossing divisor, then the period map extends to 
$\overline{S}\to M_{\Sigma}.$ We briefly explain the first 
part in the following. 

We shall show how to get a nilpotent orbit from a polarized 
log Hodge structure on a log point (\cite{Kato-Usui-book}, 0.4.24). Let 
$x$ be an fs log point with log structure $M_x.$ Then 
$\overline{M}_x$ is a sharp fs monoid and 
$\overline{M}_x^{\text{gp}}$ if a free abelian group of finite 
rank, say $r.$ Fix $y\in x^{\log}.$ We have 
$x^{\log}=Hom(\overline{M}_x^{\text{gp}},S^1)\simeq(S^1)^r$ 
and hence $\pi_1(x^{\log})=Hom(\overline{M}_x^{\text{gp}}, 
\mathbb Z)\simeq\mathbb Z^r.$ Let 
$\pi_1^+(x^{\log})\subset\pi_1(x^{\log})$ be the subset 
consisting of those homomorphisms $a:\overline{M}_x^{\text 
{gp}}\to\mathbb Z$ that take $\overline{M}_x$ into $\mathbb 
N;$ this subset is an fs monoid.  

Let $(H_{\mathbb Z},Q,F^{\bullet}H_{\mathscr O})$ be a 
polarized log Hodge structure on $x.$ Let 
$(h_i)_{i=1}^n$ be a family of generators for 
$\pi_1^+(x^{\log})$ and $s_0\in\text{sp}(y).$ Let 
$z_1,\cdots,z_r$ be complex numbers, and let 
$s\in\text{sp}(y)$ be such that 
$$
s\Big(\frac{\log(f)}{2\pi i}\Big)-s_0\Big(\frac{\log(f)}{2\pi 
i}\Big)=\sum_{i=1}^rz_ih_i(f),\quad\text{for 
}f\in\overline{M}_x^{\text{gp}}.
$$
Let $N_i:H_{\mathbb Q,y}\to H_{\mathbb Q,y}$ be the logarithm 
of $h_i.$ Then we have 
$$
F(s)=\exp\Big(\sum_{i=1}^nz_iN_i\Big)F(s_0),
$$
which shows that $(F(s))_{s\in\text{sp}(y)}$ is an orbit of 
filtrations under $\exp(\sigma\otimes\mathbb C)$ for 
$\sigma=\sum_i\mathbb R_{\ge0}N_i.$ Moreover, the condition 2) 
in (\ref{LVPHS}) implies that $F(s)\in D$ if 
$\text{Im}(z_i)\gg 0$ for all $i,$ and the condition 3) in 
(\ref{LVPHS}) implies that $NF(s_0)^p\subset F(s_0)^{p-1}$ for 
all $p\in\mathbb Z$ and $N\in\sigma.$ In other words, the 
family $(F(s))_s$ is a $\sigma$-nilpotent orbit. 
