stratifications 2 and biblio

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 
+Actually, if 
-% $\tilde{f}(D_I)=\bar{o}$ then $B^\bullet_{|D_I}=A^\bullet_{|D_I}$; otherwise $B^\bullet_{|D_I}=0$.
+$\tilde{f}(D_I)=\bar{o}$ 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 
+% $I$ satisfies that at least one $D_i$ is in $D'$, then 
-% % $B^\bullet_{|D_I}=A^\bullet_{|D_I}$; otherwise $B^\bullet_{|D_I}=0$.
+% $B^\bullet_{|D_I}=A^\bullet_{|D_I}$; 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}
 
 
 

biblio.bib

@@ -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},
+}