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\begin{document}

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\section*{Artin-Mazur formal group}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{formal_geometry}{}\paragraph*{{Formal geometry}}\label{formal_geometry}

[[!include formal geometry -- contents]]

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{deformations_of_higher_line_bundles_of_cohomology}{Deformations of higher line bundles (of $H^n(-,\mathbb{G}_m)$-cohomology)}\dotfill \pageref*{deformations_of_higher_line_bundles_of_cohomology} \linebreak
\noindent\hyperlink{DeformationsOfDeligneCohomology}{Deformations of higher line bundles with connection (of Deligne cohomology)}\dotfill \pageref*{DeformationsOfDeligneCohomology} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{OfCalabiYauVarieties}{Of Calabi-Yau varieties}\dotfill \pageref*{OfCalabiYauVarieties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Every [[variety]] in [[positive characteristic]] has a [[formal group]] attached to it, called the \emph{Artin-Mazur formal group}. This group is often related to arithmetic properties of the variety such as being ordinary or supersingular.

The Artin-Mazur formal group in dimension $n$ is a [[formal group]] version of the [[Picard infinity-group|Picard n-group]] of flat/holomorphic [[circle n-bundles]] on the given variety. Therefore for $n = 1$ one also speaks of the \emph{[[formal Picard group]]} and for $n = 2$ of the \emph{[[formal Brauer group]]}.

\hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition}

\hypertarget{deformations_of_higher_line_bundles_of_cohomology}{}\subsubsection*{{Deformations of higher line bundles (of $H^n(-,\mathbb{G}_m)$-cohomology)}}\label{deformations_of_higher_line_bundles_of_cohomology}

Let $X$ be a [[smooth scheme|smooth]] [[proper scheme|proper]] $n$ [[dimension|dimensional]] [[variety]] over an [[algebraically closed field]] $k$ of [[positive number|positive]] [[characteristic]] $p$.

Writing $\mathbb{G}_m$ for the [[multiplicative group]] and $H_{et}^\bullet(-,-)$ for [[etale cohomology]], then $H_{et}^n(X,\mathbb{G}_m)$ classifies $\mathbb{G}_m$-[[principal infinity-bundle|principal n-bundles]] ([[line n-bundles]], [[bundle 2-gerbe|bundle (n-1)-gerbes]]) on $X$. Notice that, by the discussion at \emph{\href{Brauer%20group#RelationToEtaleCohomology}{Brauer group -- relation to \'e{}tale cohomology}}, for $n = 1$ this is the [[Picard group]] while for $n = 2$ this contains (as a [[torsion subgroup]]) the [[Brauer group]] of $X$.

Accordingly, for each [[Artin algebra]] regarded as an [[infinitesimally thickened point]] $S \in ArtAlg_k^{op}$ the [[cohomology group]] $H_{et}^n(X\times_{Spec(k)} S,\mathbb{G}_m)$ is that of [[equivalence classes]] of $\mathbb{G}_n$-[[principal infinity-bundle|principal n-bundles]] on a formal thickening of $X$.

The defining inclusion $\ast \to S$ of the unique global point induces a restriction map $H^n_{et}(X\times_{Spec(k)} S, \mathbb{G}_m)\to H^n_{et}(X, \mathbb{G}_m))$ which restricts an $n$-bundle on the formal thickening to just $X$ itself. The [[kernel]] of this map hence may be thought of as the group of $S$-parameterized infinitesimal [[deformations]] of the trivial $\mathbb{G}_m$-$n$-bundle on $X$.

(For $n = 1$ this is an [[infinitesimal neighbourhood]] of the neutral element in the [[Picard scheme]] $Pic_X$, for higher $n$ one will need to genuinely speak about [[Picard stacks]] and higher stacks.)

As $S$ varies, these groups of deformations naturally form a [[presheaf]] on ``[[infinitesimally thickened points]]'' ([[Isbell duality|formal duals]] to [[Artin algebras]]).

\begin{defn}
\label{}\hypertarget{}{}
For $X$ an [[algebraic variety]] as above, write

\begin{displaymath}
\Phi_X^n \;\colon\; ArtAlg_k \to Grp
\end{displaymath}
\begin{displaymath}
\Phi_X^n(S) \coloneqq \mathrm{ker}(H^n_{et}(X\times_{Spec(k)} S, \mathbb{G}_m)\to H^n_{et}(X, \mathbb{G}_m))
  \,.
\end{displaymath}
\end{defn}
(\hyperlink{ArtinMazur77}{Artin-Mazur 77, II.1 ``Main examples''})

The fundamental result of (\hyperlink{ArtinMazur77}{Artin-Mazur 77, II}) is that under the above hypotheses this presheaf is [[prorepresentable functor|pro-representable]] by a [[formal group]], which we may hence also denote by $\Phi_X^n$. This is called the \textbf{Artin-Mazur formal group} of $X$ in degree $n$.

More in detail:

\begin{prop}
\label{SufficientConditionsForRepresentability}\hypertarget{SufficientConditionsForRepresentability}{}
Let $X$ be an [[algebraic variety]] [[proper morphism|proper]] over an [[algebraically closed field]] $k$ of [[positive number|positive]] [[characteristic]].

A sufficient condition for $\Phi_X^k$ to be [[prorepresentable functor|pro-representable]] by a [[formal group]] is that $\Phi_X^{k-1}$ is [[formally smooth morphism|formally smooth]].

In particular if $dim H^{k-1}(X,\mathcal{O}_X) = 0$ then $\Phi^{k-1}(X)$ vanishes, hence is trivially formally smooth, hence $\Phi^k(X)$ is representable

\end{prop}
The first statement appears as (\hyperlink{ArtinMazur77}{Artin-Mazur 77, corollary (2.12)}). The second as (\hyperlink{ArtinMazur77}{Artin-Mazur 77, corollary (4.2)}).

\begin{remark}
\label{Dimension}\hypertarget{Dimension}{}
The [[dimension]] of $\Phi^k_X$ is

\begin{displaymath}
dim(\Phi^k_X) = dim H^k(X,\mathcal{O}_X)
  \,.
\end{displaymath}
\end{remark}
(\hyperlink{ArtinMazur77}{Artin-Mazur 77, II.4}).

\hypertarget{DeformationsOfDeligneCohomology}{}\subsubsection*{{Deformations of higher line bundles with connection (of Deligne cohomology)}}\label{DeformationsOfDeligneCohomology}

In (\hyperlink{ArtinMazur77}{Artin-Mazur 77, section III}) is also discussed the formal [[deformation theory]] of [[line n-bundles with connection]] (classified by [[ordinary differential cohomology]], being [[hypercohomology]] with [[coefficients]] in the [[Deligne complex]]). Under suitable conditions this yields a [[formal group]], too.

Notice that by the discussion at \emph{\href{http://ncatlab.org/nlab/show/intermediate%20Jacobian#CharacterizationAsHodgeTrivialDeligneCohomology}{intermediate Jacobian -- Characterization as Hodge-trivial Deligne cohomology}} the formal deformation theory of Deligne cohomology yields the formal completion of [[intermediate Jacobians]] (all in suitable degree).

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{remark}
\label{}\hypertarget{}{}
\begin{itemize}%
\item For a [[curve]] $X$ (i.e. $dim(X)= 1$), the Artin-Mazur group is often called the \textbf{[[formal Picard group]]} $\widehat{\mathrm{Pic}}$.


\item For a [[surface]] $X$ (i.e. $dim(X) =2$), the Artin-Mazur group is called the \textbf{[[formal Brauer group]]} $\widehat{Br}$.



\end{itemize}
\end{remark}
\hypertarget{OfCalabiYauVarieties}{}\subsubsection*{{Of Calabi-Yau varieties}}\label{OfCalabiYauVarieties}

\begin{example}
\label{AMfgOfStrictCalabiYau}\hypertarget{AMfgOfStrictCalabiYau}{}
Let $X$ be a strict [[Calabi-Yau variety]] in [[positive characteristic]] of [[dimension]] $n$ (strict meaning that the [[Hodge numbers]] $h^{0,r} = 0$ vanish for $0 \lt r \lt n$, i.e. over the [[complex numbers]] that the [[holonomy group]] exhausts $SU(n)$, this is for instance the case of relevance for [[supersymmetry]], see at \emph{[[supersymmetry and Calabi-Yau manifolds]]}).

By prop. \ref{SufficientConditionsForRepresentability} this means that the Artin-Mazur formal group $\Phi^n_X$ exists. Since moreover $h^{0,n} = 1$ it follows by remark \ref{Dimension} that it is of [[dimension]] 1

\end{example}
For discussion of $\Phi_X^n$ for [[Calabi-Yau varieties]] $X$ of [[dimension]] $n$ and in [[positive number|positive]] [[characteristic]] see (\hyperlink{GeerKatsura03}{Geer-Katsura 03}).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[formal group]]


\item [[height of a variety]]



\end{itemize}
[[!include moduli of higher lines -- table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

The original article is

\begin{itemize}%
\item [[Michael Artin]], [[Barry Mazur]], \emph{Formal Groups Arising from Algebraic Varieties}, Annales scientifiques de l'\'E{}cole Normale Sup\'e{}rieure, S\'e{}r. 4, 10 no. 1 (1977), p. 87-131 \href{http://www.numdam.org/item?id=ASENS_1977_4_10_1_87_0}{numdam}, \href{http://www.ams.org/mathscinet-getitem?mr=56:15663}{MR56:15663}

\end{itemize}
Further developments are in

\begin{itemize}%
\item [[Jan Stienstra]], \emph{Formal group laws arising from algebraic varieties}, American Journal of Mathematics, Vol. 109, No.5 (1987), 907-925 (\href{http://www.math.rochester.edu/people/faculty/doug/otherpapers/stienstra1.pdf}{pdf})

\end{itemize}
Lecture notes touching on the cases $n = 1$ and $n = 2$ include

\begin{itemize}%
\item Christian Liedtke, example 6.13 in \emph{Lectures on Supersingular K3 Surfaces and the Crystalline Torelli Theorem} (\href{http://arxiv.org/abs/1403.2538}{arXiv.1403.2538})

\end{itemize}
Discussion of Artin-Mazur formal groups for all $n$ and of [[Calabi-Yau varieties]] of [[positive number|positive]] [[characteristic]] in [[dimension]] $n$ is in

\begin{itemize}%
\item [[Gerard van der Geer]], T. Katsura, \emph{On the height of Calabi-Yau varieties in positive characteristic}, Documenta Math. 8. 97-113 (2003) (\href{http://arxiv.org/abs/math/0302023}{arXiv:math/0302023})

\end{itemize}
[[!redirects Artin-Mazur formal group]] [[!redirects Artin-Mazur formal groups]] [[!redirects Artin–Mazur formal group]] [[!redirects Artin–Mazur formal groups]] [[!redirects Artin--Mazur formal group]] [[!redirects Artin--Mazur formal groups]]



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