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\title{Towards a proof of the Hodge conjecture,\\
and cycle spaces in positive characteristic}
\titlemark{Towards a proof of the Hodge conjecture}
\author{Alex Grothenstruck \&\ Paul-Henri Troiligne}
\authormark{A.\ Grothenstruck \&\ P.-A.\ Troiligne}
\date{} % Empty date or tweak it according to your needs
\journal{\'Epijournal de G\'eom\'etrie Alg\'ebrique -- (yyyy), ---} % Epijournal name
\acceptation{submitted dd/mm/yyyy, accepted dd/mm/yyyy, revised dd/mm/yyyy}
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\begin{document}
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\dedication{in the memory of William V.D.\ Hodge}
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\thanks{We thank \href{http://episciences.org}{episciences.org} for providing open access hosting of \emph{\'Epijournal de G\'eom\'etrie Alg\'ebrique}.}
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\def\abstractname{Abstract}
\abstract{An English abstract is suggested and should be
descriptive enough by itself. Please do not include citations,
footnotes or references to numbered equations, theorems, figures
or tables in your abstract. Avoid complicated formulae or
displayed equations.}
\keywords{arithmetic functions, primitive roots, cyclotomic fields, Dirichlet series, multiplicative functions.}
\MSCclass{11R18, 11R32, 11M41, 11L07, 11N37, 11N45}
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\tableofcontents
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\section{Introduction}
First of all, we want to stress that this paper has been typeset with
the wonderful system \TeX\ created by Don Knuth \cite{Knuth}, in the
form later adapted by Leslie Lamport \cite{Lamport}, the current
version of which is \LaTeXe (older versions won't work).
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style works (using the class file \verb|epiarticle.cls| and the
more specific style \verb|epimath.sty|).
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sheets without much change. When Letter paper format is used, the text
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at the beginning. The name of the particular epijournal (supposedly, from the
Episciences project) is set by using
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\verb|\journal{Mathematica Universalis}| here.
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navigation back and forth.
\smallskip
As far as mathematics are concerned, our main reference is
\cite{Grothendieck} in what follows.
\section{Preliminary results}
The following basic property will be used systematically.
\begin{property} ...
\end{property}
\section{Fundamental results about motives}
% Use this if the title is too long to fit on the page header.
% \sectionmark{\thesection~Abbreviation of title}
\subsection{A new fonctor}
\forceindent
We start with a few basic facts that will be useful in the sequel.
The first paragraph in a section, subsection or subsubsection is
normally not indented (as you may guess, these are introduced by
typesetting \verb|\section{...}|, \verb|\subsection{...}|,
\verb|\subsubsection{...}| respectively). You can force indentation
by putting \verb|\forceindent| before. We did so here.
However, if you start a new paragraph in the course of a continued
exposition, The new paragraph will appear to be indented, unless you
specify \verb|\noindent| before.
By default, all statement environments are set so as to use the same
counter for theorems, propositions, lemmas, to avoid the possibly
confusing situation where different types of statements get the same
numbering. We recommend to respect this rule for any new environment
to be created. You can e.g.\ specify
\verb|\newtheorem{assertion}[theorem]{Assertion}| to define the new
environment ``assertion'', that will use the same counter as
``theorem''.
You may wish to use a vertical skip to indicate a new logical step,
otherwise there will be only a new paragraph without any vertical
skip. Here we use \verb|\smallskip|.
\smallskip
Our main mathematical statement is
\begin{theorem}
\label{thm-prelim}
Let ${\cal F}$ be a fonctor from the category of algebraic schemes to the
derived category of $A$-modules. $($...$)$
\end{theorem}
\begin{proof} The proof is a bit long, so we first present a short sketch.
Surprisingly, the cohomology calculations require the famous Rogers-Ramanujan
identity
\begin{equation}
\label{eq-ramanujan}
G(q) = \sum_{n=0}^\infty \frac {q^{n^2}} {(q;q)_n} =
\frac {1}{(q;q^5)_\infty (q^4; q^5)_\infty}
=1+ q +q^2 +q^3 +2q^4+2q^5 +3q^6+\cdots \,,
\end{equation}
where $({\scriptscriptstyle\bullet}\,;{\scriptscriptstyle\bullet})_n$
denotes as usual the $q$-Pochhammer symbol
$$(a;q)_n =\prod_{k=0}^{n-1} (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^{n-1}),
\qquad (a;q)_0 = 1.$$
Another step uses the dilogarithm function
\begin{equation}
\label{eq-dilog}
\mathop{\rm Li}\nolimits_2(z)=\sum_{k=1}^\infty\frac{z^k}{k^2}.
\end{equation}
Our arguments involve some preliminary steps which we call ``first case''
and ``sporadic cases'', as detailed below.
\end{proof}
\begin{remark} {\rm We recommend Theorems, Propositions and Lemmas to be
typeset in italics, Remarks and Examples in roman characters.}
\end{remark}
\subsubsection{First case}
We proceed here by induction on dimension.
\paragraph{Initialization.} Assume $n=1$.
\paragraph{Inductive step.} Assume the result known up to dimension
$n-1$ with $n\ge 2$.
\subsubsection{Sporadic cases}
We settle the sporadic cases of Theorem~\ref{thm-prelim} by means of
Formulae~\eqref{eq-ramanujan} and ~\eqref{eq-dilog}.
\subsection{More about \emph{l}-adic technology}
We give here some technical facts about $l$-adic cohomology, along
the lines of standard conjectures stated in \cite{Grothendieck}.
\begin{proposition}
Let ${\cal R}$ be a representable fonctor ...
\end{proposition}
\begin{proof} Lorem ipsum dolor sit amet, consectetur adipiscing
elit. Sed non risus. Suspendisse lectus tortor, dignissim sit amet,
adipiscing nec, ultricies sed, dolor. Cras elementum ultrices
diam. Maecenas ligula massa, varius a, semper congue, euismod non,
mi. Proin porttitor, orci nec nonummy molestie, enim est eleifend
mi, non fermentum diam nisl sit amet erat. Duis semper. Duis arcu
massa, scelerisque vitae, consequat in, pretium a,
enim. Pellentesque congue. Ut in risus volutpat libero pharetra
tempor. Cras vestibulum bibendum augue. Praesent egestas leo in
pede. Praesent blandit odio eu enim. Pellentesque sed dui ut augue
blandit sodales. Vestibulum ante ipsum primis in faucibus orci
luctus et ultrices posuere cubilia Curae; Aliquam nibh. Mauris ac
mauris sed pede pellentesque fermentum. Maecenas adipiscing ante non
diam sodales hendrerit.
\smallskip
Ut velit mauris, egestas sed, gravida nec, ornare ut, mi. Aenean ut orci vel massa suscipit pulvinar. Nulla sollicitudin. Fusce varius, ligula non tempus aliquam, nunc turpis ullamcorper nibh, in tempus sapien eros vitae ligula. Pellentesque rhoncus nunc et augue. Integer id felis. Curabitur aliquet pellentesque diam. Integer quis metus vitae elit lobortis egestas. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Morbi vel erat non mauris convallis vehicula. Nulla et sapien. Integer tortor tellus, aliquam faucibus, convallis id, congue eu, quam. Mauris ullamcorper felis vitae erat. Proin feugiat, augue non elementum posuere, metus purus iaculis lectus, et tristique ligula justo vitae magna.\qed
\end{proof}
\noindent
(Notice here again the use of \verb|\smallskip| to split two logically distinct
parts of the explanations in the ``proof''!)
\subsection{Diagrammatic picture of the proof}
The reader will understand better our strategy by looking at the
following diagram. At the same time, this provides an example of how
to include an EPS or PDF figure. With \texttt{dvips}, one must include
encapsulated postscript files (EPS), while \texttt{pdflatex} needs
JPG, PNG or (preferably) PDF format (search for \texttt{Flowchart.pdf}
in the present source file).
\begin{figure}[ht!]
% \hspace{4.8cm}\includegraphics[width=7.5cm]{Flowchart.pdf} % with pdflatex
\hspace{4.8cm}\includegraphics[width=7.5cm]{Flowchart.pdf} % with dvips
\caption{\emph{Flow chart of proof}}
\end{figure}
\section{Main arguments}
The idea is to use a double induction on the weight and coniveau of
the involved Hodge structures.
\begin{lemma} Let $X$ be a nonsingular projective scheme over an
algebraically closed field $k$ of characteristic~$0$. $($...$)$
\end{lemma}
\noindent ...
\noindent ...
\noindent ...
\section{Further comments}
In case the manuscript title is very long (or the list of authors is
very long), the header line of odd pages might not have enough space
to include this information. You can adjust this by specifying
\verb|\titlemark| and \verb|\authormark|. Similarly, for titles of
sections that would be too long to fit on headers of even pages, use
\verb|\sectionmark|.
The \verb|epimath| latex style is otherwise pretty standard. The font
size and line spacing of the list of references can be adjusted: it is
recommended to use \verb|\bibliographysize{small}| for a more compact
presentation. The default heading ``References'' can be changed, e.g.\ to
``Bibliography'', by setting before
\verb|\renewcommand{\refname}{Bibliography}|.
Also, a command like \verb|\bibliographymark{References}| set
before the bibliography list will let ''References'' appear in the right
page headers instead if the last section -- this can be useful if the
bibliography list is very long.
Finally, \verb|epimath.sty| provides special macros \verb|\DOIstring{...}|
to set the DOI on top of the first page, as well as \verb|\ARXIV{...}|,
\verb|\HAL{...}| and \verb|\MR{...}| to specify references to the
preprint archives \emph{arXiv} and \emph{HAL}, and to
\emph{Math Reviews}, respectively, with appropriate hyperlinks.
See e.g.\ \cite{Voisin1} and \cite{Voisin2} below.
% use this if the bibliography list is very long and you want "References"
% to appear in the headers of even pages.
% \bibliographymark{References}
\bibliographystyle{amsalpha}
% This would change the heading "References" by "Bibliography"
% \renewcommand{\refname}{Bibliography}
\begin{thebibliography}{ABC93}
% \bibliographysize{normal}
\bibliographysize{small} % recommended size is small
\bibitem[Gro68]{Grothendieck} A. Grothendieck, \emph{Standard Conjectures
on Algebraic Cycles}, Algebraic Geometry (Internat.\ Colloq., Tata
Inst.\ Fund.\ Res., Bombay, 1968), Oxford University Press, 193-–199.
\MR{0268189}
\bibitem[Knu84]{Knuth} D.E. Knuth, \emph{The \TeX book},
(Addison--Wesley, New York, 1984).
\bibitem[Lam94]{Lamport}
L. Lamport, \emph{\LaTeX: A document preparation system \textup{(}%
updated for \LaTeXe\textup{)}} (Addison--Wesley, New York, 1994), this is
the fundamental book.
\bibitem[Voi11]{Voisin1}
C. Voisin, \emph{The generalized Hodge and Bloch conjectures are equivalent
for general complete intersections}, e-print 13~July 2011, \ARXIV{1107.2600}
\bibitem[Voi13]{Voisin2}
C. Voisin, \emph{Unirational threefolds with no universal codimension
$2$ cycle}, e-print 7 Dec.\ 2013, \HAL{hal-01002966}
\end{thebibliography}
\authoraddresses{
Alex Grothenstruck\\
Las Dopicos University, Department of Mathematics,
144 Plekszy-Gladz Street\\
ST-05772 San Theorodos\\
\email alexgroth@univ.las-dopicos.st
Paul-Henri Troiligne\\
Sbrodj Research Center, Space Technology Institute,
Mathematics Department\\
SY-27182 Syldavia\\
\email ph.troiligne@center.sbrodj.sy
}
\end{document}