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\begin{document}
\title{Steenbrink's MHS with non isolated singularities}
\author{EAB, MGV, JMM}
\maketitle
Steenbrink constructed a MHS on the vanishing cohomology of an isolated singular point of a hypersurface. We study the application of Steenbrink's construction to the case of non isolated singularities.
\section{General setting.} Consider $P \in \mathbb{C}[z_0, \dots, z_n]$ such that $P$ is non constant and $P(o)=0$. Denote the set of zeros of $P$ by $V(P)$, i.e., $V(P)= \{ z \in \mathbb{C}^{n+1} \, : \, P(z)=0\}$. Assume that $o$ is a (non isolated) singular point of $V(P)$.
Let $d$ be the degree of the polynomial $P$, and denote by $P_h$ the homogeneous polynomial determined by $P$, i.e., $P_h(z_0 : \dots : z_{n+1}) = z_{n+1}^dP (z_0/z_{n+1}, \dots , z_n/z_{n+1})$. Denote by $\bar P$ the map from $\mathbb{P}^{n+1} \dashrightarrow \mathbb{P}^1= \mathbb{C} \cup \infty$ given by $P_h/z_{n+1}^d$. Let $\rho: Y \rightarrow \mathbb{P}^{n+1}$ be the resolution of the indeterminacies of $\bar P$ that are obtained by means of blow ups with center at the hyperplane at infinity. Notice that $\bar {P}$ is well defined at the point $\bar o := [o:1] \in \mathbb{P}^{n+1}$. Therefore we can consider $\bar o \in Y$. Let $S$ a small disk about $0 \in \mathbb{C} \cup \infty$ and let $f_1$ the restriction of the composition $\rho \circ \tilde P$ to $(\rho \circ \tilde P)^{-1}(S)$. The value $0 \in \mathbb{C} \cup \infty$ is the unique critical value of the map $f_1$. Denote the fibers of $f_1$ by $Y_t$, i.e., $Y_f= f_1^{-1}(t)$ for $t \in S$.
Finally, denote by $f: X \rightarrow S$ the embedded resolution of the map $f_1$. It is obtained by a modification $\psi : X \rightarrow Y$, i.e., $f = f_1 \circ \psi$. Denote the fibers of $f$ by $X_t$, i.e., $X_t = f^{-1}(t)$ for $t \in S$. The fibers $X_t$ are smooth and projective for $t \neq 0$. The central fiber $X_0=f^{-1}(0)$ is a simple normal crossing divisor on $X$.
\begin{figure}[ht]
\begin{tikzcd}[/tikz/column 4/.append style={anchor=base west}]
\tilde X \ar[r, "\pi"]\ar[d, "\tilde f"]& X\ar[r, "\psi"]\ar[d,"f"]&Y\ar[r, "\rho"]\ar[d,"f_1"]& \mathbb{P}^{n+1}\ar[d,"\bar{P}", start anchor=-4.629cm]\\
\tilde S \ar[r]& S\ar[r,"="]&S\ar[r, "i"] & \mathbb{P}^1\\[-20pt]
%&t\ar[r,mapsto]&t^e
\end{tikzcd}
\caption{}
\end{figure}
The irreducible components of $X_0$ can be classified into three types. Let us denote by $E'$ the set of irreducible components $E_i$ of $X_0$ that are exceptional and are mapped to $\bar o \in \mathbb{P}^{n+1}$ under $\rho \circ \psi$ or, in other words, $supp E_i \subset (\rho \circ \psi)^{-1}(\bar o)$. Let us denote by $E''$ the set of irreducible components $E_j$ of $X_0$ such that such that $\bar o$ is strictly contained in $(\rho \circ \psi)(E_j)$. In particular, the irreducible components of the strict transform of $V(P)$ correspond to irreducible components in $E''$. Finally, let us denote by $E'''$ the set of irreducible components $E_k$ of $X_0$ such that $\bar o \not \in (\rho \circ \psi)(E_k)$. %In particular, the exceptional and dicritical divisors that appear when resolving the indeterminacies of $\bar{P}$ correspond to irreducible components in $E'''$.
\begin{remark} Steenbrink asumes that $o$ is an isolated singular point of $V(P)$. Under this hypothesis Brieskorn constructed an irreducible homogeneous polynomial $P'$ such that $V(P)$ has a unique singular point which is analitically equivalent to $(V(P), o)$. Using $P'$, Steenbrink can assume that $E''$ is a prime divisor (the strict transform of $V(P'))$, and that $E'''$ is empty.
\end{remark}
Let $e$ be a common multiple of the multiplicities of the divisor $E$. Consider the covering $\sigma: \tilde{S}
\rightarrow S$ given by $\sigma(t)=t^e$ with $\tilde{S}$ the disk centered in $0 \in \mathbb{C}$ and radius $\eta^{1/e}$.
Consider $\tilde{X}$ the normalization of the fibered product $X \times_S \tilde{S}$ and let $\tilde{f}: \tilde{X} \rightarrow \tilde{S}$ and $\pi : \tilde{X} \rightarrow X$ the natural maps. Denote by $D$ the divisor $\pi^{-1}(X_0)$.
Consider also the universal covering of the punctured disk $S^\ast := S \setminus \{0\}$ given the map $H := \{ z \in \mathbb{C} \, : \, im(z) > 0\} \rightarrow S^\ast$ defined as $z \mapsto exp(2 \pi i z)$ Denote by $X_\infty$ and $B_\infty$ the fibered products $X \times_S H$ and $B \times_S H$.
Let $B$ a small ball in $Y$ with center $\tilde o$ and radius $\epsilon$. Assume that all balls centered at $\tilde o$ and with radius less than $\epsilon$ intersect $Y_0$ transversally. Choose $\eta$ such that all fibers with $Y_t$ with $|t| < \eta$ intersect $B$ transversally. Denote the intersection $Y_t \cap B$ by $B_t$. Under the above hypothesis for $\epsilon$ and $\eta$, the complex manifold $B_t$ with $t \not = 0$ is diffeomorphic to the Milnor fiber of $P$ at $o$.
Our aim is to compute $H^p(B_\infty)$ and construct a mixed Hodge structure on it by means of a complex $A^\bullet (B_\infty)$ and according to the following strategy:
\begin{itemize}
\item Deligne constructed a Hodge structure on $H^p_c (Y_0 \setminus \{o\})$ by means of a complex $A^\bullet (Y_0)$ on the divisor $C = D' \cap D''$ with support on the clousure of $C \setminus (C \cap E''')$. Dualizing ...
\item Schmid, in the unipotent case, and Steenbrink constructed a limit Hodge structure on the cohomology groups $H^p(X_\infty)$ associated with $X \setminus X_0 \rightarrow S \setminus \{0\}$ by means of a complex $A^\bullet$ on the divisor $D$ with support on the clousure of $D \setminus D'''$.
\item Since the complex $A^\bullet(Y_0)$ can be considered as a subcomplex of $A^\bullet$. Hence, the complex $A^\bullet (B_\infty)$ is defined as the quotient $A^\bullet / A^\bullet(Y_0)$.
\end{itemize}
\end{document}