strata

NonisoMHS.tex

@@ -32,7 +32,6 @@
 %\begin{enumerate}[noitemsep]
 
 % THEOREM Environments ---------------------------------------------------
-\theoremstyle{theorem} 
 \newtheorem{theorem}{Theorem}[section]
 \newtheorem*{MainTheorem}{Main Theorem}
 \newtheorem{hypothesis}[theorem]{Hypothesis}
@@ -223,7 +222,7 @@
 \renewcommand{\b}{{J}}
 \newcommand{\OO}{\mathcal{O}}
 \newcommand{\JJ}{\mathcal{J}}
-\renewcommand{\ll}{\operatorname{\ell\ell}}
+%\renewcommand{\ll}{\operatorname{\ell\ell}}
 \newcommand{\ctr}{N} % For a generic number of items
 
 %%%%%%%%%%%%%%%%%%%%%
@@ -309,9 +308,13 @@ Steenbrink constructed a MHS on the vanishing cohomology of an isolated singular
 
 \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 $t \in S$.
+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 $\rho: W \rightarrow \mathbb{P}^{n+1}$ 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 $\bar o \in W$. Let $S$ a small disk about $0 \in \mathbb{C} \cup \infty$ and let 
+$Y:=(\rho \circ \bar P)^{-1}(S)$ and let $f_1:=(\rho \circ \bar P)_{|Y}$. 
+The disk has be chosen small enought such that the value $0 \in  \mathbb{C} \cup \infty$ is the unique critical value of the map $f_1$; as it is proper
+the fibers $Y_t= f_1^{-1}(t)$ are smooth and projective for $t\neq 0$.
 
-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 $X$. 
+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 isomorphic to $Y_t$, for $t \neq 0$. The central fiber $X_0=f^{-1}(0)$ is a simple normal crossing divisor on $X$
+while $Y_0$ had more complicated singularities. 
 
 
 	\begin{figure}[ht]
@@ -332,20 +335,36 @@ The irreducible components of $X_0$ can be classified into three types. Let us d
 
 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 $\tilde{X}$ 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 $\pi^{-1}(X_0)$. A decomposition
+into $D',D'',D'''$ is done as for $E$.
 
-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 $B \times_S H$. 
+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 $X_\infty$ the fibered product $X \times_S H$. 
 
-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 $o$. 
+Let  $A^\bullet$ be the complex of sheaves on the divisor $D$ that defines, according to  Schmid, in the unipotent case,  and Steenbrink, a limit Hodge structure on the cohomology groups $H^p(X_\infty)$ associated with the fibration $f_{|{X \setminus X_0}}:X \setminus X_0 \rightarrow S \setminus \{0\}$. 
+
+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 $\eta\ll\epsilon$ 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 first aim is to compute $H^p(B_\infty)$ by means of a complex of sheaves $B^\bullet$ on the divisor $D$ with support on $D'$. This complex should satisfy the following additional properties:
+
+Denote by $B_\infty$ the fibered product $B \times_S H$.  Our first aim is to compute $H^p(B_\infty)$ by means of a complex of sheaves $B^\bullet$ on the divisor $D$ with support on $D'$. This complex should satisfy the following additional properties:
 \begin{itemize} 
-\item It will be defined as a locally free on the abstract strata $\check{D}_k$ consisting on disjoint unions of intersections of $k$ irreducible components of $D$, i.e., $B^\bullet_{|_{\check{D}_k}}$ is locally free on $\check{D}_k = \sqcup_{I, |I|=k} \cap_{i \in I} D_i$ for each $k$.
+\item The complex $B^\bullet$ will be defined as a locally free on the abstract strata 
+\[
+\check{D}_k:=\bigsqcup_{I, |I|=k} \bigcap_{i \in I} D_i
+\]
+consisting on disjoint unions of intersections of exactly $k$ irreducible components of $D$. 
+
 \item Given the natural map $i_k : \check{D}_k \rightarrow D$, the direct image $(i_k)_\ast B^\bullet$ is locally free too.
-\item Let  $A^\bullet$ be the complex of sheaves on the divisor $D$ tat defines, according to  Schmid, in the unipotent case,  and Steenbrink, a limit Hodge structure on the cohomology groups $H^p(X_\infty)$ associated with the fibration $f_{|_{X \setminus X_0}}:X \setminus X_0 \rightarrow S \setminus \{0\}$. The direct images $(i_k)_\ast B^\bullet$ coincide with $A^\bullet_{|_D_k}$ where $D_k := i_k(\check{D}_k)$.
+
+
+\item These properties are also satisfied by $A^\bullet$. 
+%Actually, if 
+% $\tilde{f}(D_I)=\bar{o}$ then $B^\bullet_{|D_I}=A^\bullet_{|D_I}$; otherwise $B^\bullet_{|D_I}=0$.
+% % $I$ satisfies that at least one $D_i$ is in $D'$, then 
+% % $B^\bullet_{|D_I}=A^\bullet_{|D_I}$; otherwise $B^\bullet_{|D_I}=0$.
+
 \end{itemize}
-Using the above hypothesis and the spectral sequence associated ith the complex $B^\bullet$ we will show that $H^q(B_\infty, \mathbb{Q})=\mathbb{H}^p(D, B^\bullet_\mathbb{Q})$.
+Using the above hypothesis and the spectral sequence associated with the complex $B^\bullet$ we will show that $H^q(B_\infty, \mathbb{Q})=\mathbb{H}^p(D, B^\bullet_\mathbb{Q})$.
 
 
 
@@ -362,4 +381,4 @@ Using the above hypothesis and the spectral sequence associated ith the complex
 
 
 
-\end{document}
+\end{document}