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Cambio de parrafo sobre extrategia del artículo; uso de complejo B para calcular cohomologia de B_\infty

NonisoMHS.tex

@@ -32,7 +32,7 @@
 %\begin{enumerate}[noitemsep]
 
 % THEOREM Environments ---------------------------------------------------
-
+\theoremstyle{theorem} 
 \newtheorem{theorem}{Theorem}[section]
 \newtheorem*{MainTheorem}{Main Theorem}
 \newtheorem{hypothesis}[theorem]{Hypothesis}
@@ -339,12 +339,22 @@ Consider also the universal covering of the punctured disk $S^\ast := S \setminu
 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 $A^\bullet / A^\bullet(Y_0)$.
+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 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 $D_k := i_k(\check{D}_k)$.
 \end{itemize}
+Using the above hypothesis and the spectral sequence associated ith the complex $B^\bullet$ we will show that $H^q(B_\infty, \mathbb{Q})=\mathbb{H}^p(D, B^\bullet_\mathbb{Q})$.
+
+
+
+%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}
 
 
 
@@ -352,4 +362,4 @@ Our aim is to compute $H^p(B_\infty)$ and construct a mixed Hodge structure on i
 
 
 
-\end{document}
+\end{document}