From d205984483365a1ca79f5c4a9ab2f7414928fb34 Mon Sep 17 00:00:00 2001 From: Enrique Artal Date: Fri, 26 May 2023 18:35:33 +0200 Subject: [PATCH] stratifications 2 and biblio --- NonisoMHS.tex | 36 +++++++++++++++++++++++------------- biblio.bib | 38 ++++++++++++++++++++++++++++++++++++++ 2 files changed, 61 insertions(+), 13 deletions(-) create mode 100644 biblio.bib diff --git a/NonisoMHS.tex b/NonisoMHS.tex index b7042b0..e878928 100644 --- a/NonisoMHS.tex +++ b/NonisoMHS.tex @@ -31,6 +31,9 @@ %\begin{enumerate}[noitemsep] + +\newcommand{\Cech}{ \check{\rm{C}}} +\DeclareMathOperator{\supp}{Supp} % THEOREM Environments --------------------------------------------------- \newtheorem{theorem}{Theorem}[section] \newtheorem*{MainTheorem}{Main Theorem} @@ -42,7 +45,6 @@ \newcommand{\Fdif}[1]{{ \llbracket #1 \rrbracket }} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{claim}[theorem]{Claim} -\newcommand{\Cech}{ \check{\rm{C}}} \newtheorem{notation}[theorem]{Notation} \newtheorem*{convention}{Convention} \newtheorem*{notationCounter}{Notation for Section \ref{counterex}} @@ -59,6 +61,7 @@ \newtheorem{definition}[theorem]{Definition} \newtheorem{question}[theorem]{Question} \newtheorem{example}[theorem]{Example} +\theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{subsection} @@ -288,7 +291,6 @@ \newcommand{\lc}{\operatorname{lc}} \newcommand{\lt}{\operatorname{lt}} \newcommand{\init}{\operatorname{in}} -\newcommand{\supp}{\operatorname{supp}} \newcommand{\diff}{\operatorname{diff}} @@ -303,7 +305,7 @@ \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. +Steenbrink~\cite{st:77} 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 $V(P)$. @@ -319,7 +321,7 @@ while $Y_0$ had more complicated singularities. \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 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, dashrightarrow, "\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} @@ -327,9 +329,10 @@ while $Y_0$ had more complicated singularities. \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 $E'''$. +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 $E'''$. +Since $\bar{o}$ plays a special role, we will assume an extra hypothesis, namely that the preimage of $\bar{o}$ is contained 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. +\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. The extra hypothesis is always fullfilled in the isolated singularity case. \end{remark} @@ -338,9 +341,9 @@ Let $e$ be a common multiple of the multiplicities of the divisor $E$. Consider 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 $X_\infty$ the fibered product $X \times_S H$. +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 $A^\bullet$ 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 $A^\bullet$ be the complex of sheaves on the divisor $D$ that defines, according to Schmid~\cite{sch:73}, in the unipotent case, and Steenbrink~\cite{st:75,st:77}, 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 $o$. @@ -358,15 +361,21 @@ consisting on disjoint unions of intersections of exactly $k$ irreducible compon \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 $B^\bullet_{|D_I}=0$. +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 with the complex $B^\bullet$ we will show that $H^q(B_\infty, \mathbb{Q})=\mathbb{H}^p(D, B^\bullet_\mathbb{Q})$. +\begin{remark} +If I am not wrong the weight filtration in the spectral sequence is obtained gluing several weight filtrations for the logarithm +of the unipotent part of the monodromy centered at $n,n+1,\dots,n+k$, where $k