synthetic differential geometry, deformation theory
infinitesimally thickened point
Every variety in positive characteristic has a formal group attached to it, called the 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 n-group of flat/holomorphic circle n-bundles on the given variety. Therefore for $n = 1$ one also speaks of the formal Picard group and for $n = 2$ of the formal Brauer group.
Let $X$ be a smooth proper $n$ dimensional variety over an algebraically closed field $k$ of 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 n-bundles (line n-bundles, bundle (n-1)-gerbes) on $X$. Notice that, by the discussion at Brauer group – relation to étale 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 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” (formal duals to Artin algebras).
For $X$ an algebraic variety as above, write
(Artin-Mazur 77, II.1 “Main examples”)
The fundamental result of (Artin-Mazur 77, II) is that under the above hypotheses this presheaf is pro-representable by a formal group, which we may hence also denote by $\Phi_X^n$. This is called the Artin-Mazur formal group of $X$ in degree $n$.
More in detail:
Let $X$ be an algebraic variety proper over an algebraically closed field $k$ of positive characteristic.
A sufficient condition for $\Phi_X^k$ to be pro-representable by a formal group is that $\Phi_X^{k-1}$ is 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
The first statement appears as (Artin-Mazur 77, corollary (2.12)). The second as (Artin-Mazur 77, corollary (4.2)).
In (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 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).
For a curve $X$ (i.e. $dim(X)= 1$), the Artin-Mazur group is often called the formal Picard group $\widehat{\mathrm{Pic}}$.
For a surface $X$ (i.e. $dim(X) =2$), the Artin-Mazur group is called the formal Brauer group $\widehat{Br}$.
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 supersymmetry and Calabi-Yau manifolds).
By prop. 1 this means that the Artin-Mazur formal group $\Phi^n_X$ exists. Since moreover $h^{0,n} = 1$ it follows by remark 1 that it is of dimension 1
For discussion of $\Phi_X^n$ for Calabi-Yau varieties $X$ of dimension $n$ and in positive characteristic see (Geer-Katsura 03).
moduli spaces of line n-bundles with connection on $n$-dimensional $X$
The original article is
Further developments are in
Lecture notes touching on the cases $n = 1$ and $n = 2$ include
Discussion of Artin-Mazur formal groups for all $n$ and of Calabi-Yau varieties of positive characteristic in dimension $n$ is in
Last revised on December 18, 2015 at 11:46:43. See the history of this page for a list of all contributions to it.