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stratifications 2 and biblio

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36
NonisoMHS.tex
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@ -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 $k<leq r+1$, if $r$ is the dimension of the singular locus.
\end{remark}
%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}
@ -377,7 +386,8 @@ Using the above hypothesis and the spectral sequence associated with the complex
\bibliographystyle{amsplain}
\bibliography{biblio}

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biblio.bib Normal file
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@ -0,0 +1,38 @@
@inproceedings {st:77,
AUTHOR = {Steenbrink, J.H.M.},
TITLE = {Mixed {H}odge structure on the vanishing cohomology},
BOOKTITLE = {Real and complex singularities ({P}roc. {N}inth {N}ordic
{S}ummer {S}chool/{NAVF} {S}ympos. {M}ath., {O}slo, 1976)},
PAGES = {525--563},
PUBLISHER = {Sijthoff and Noordhoff, Alphen aan den Rijn},
YEAR = {1977},
}
@article {st:75,
AUTHOR = {Steenbrink, J.H.M.},
TITLE = {Limits of {H}odge structures},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {31},
YEAR = {1975/76},
NUMBER = {3},
PAGES = {229--257},
ISSN = {0020-9910},
DOI = {10.1007/BF01403146},
URL = {https://doi.org/10.1007/BF01403146},
}
@article {sch:73,
AUTHOR = {Schmid, W.},
TITLE = {Variation of {H}odge structure: the singularities of the
period mapping},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {22},
YEAR = {1973},
PAGES = {211--319},
ISSN = {0020-9910},
DOI = {10.1007/BF01389674},
URL = {https://doi.org/10.1007/BF01389674},
}