Contents

group theory

# Contents

## Idea

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.

## Definition

### Deformations of higher line bundles (of $H^n(-,\mathbb{G}_m)$-cohomology)

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).

###### Definition

For $X$ an algebraic variety as above, write

$\Phi_X^n \;\colon\; ArtAlg_k \to Grp$
$\Phi_X^n(S) \coloneqq \mathrm{ker}(H^n_{et}(X\times_{Spec(k)} S, \mathbb{G}_m)\to H^n_{et}(X, \mathbb{G}_m)) \,.$

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:

###### Proposition

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)).

###### Remark

The dimension of $\Phi^k_X$ is

$dim(\Phi^k_X) = dim H^k(X,\mathcal{O}_X) \,.$

### Deformations of higher line bundles with connection (of Deligne cohomology)

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).

## Examples

### General

###### Remark
• 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 $\widehat{Br}$.

### Of Calabi-Yau varieties

###### Example

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. this means that the Artin-Mazur formal group $\Phi^n_X$ exists. Since moreover $h^{0,n} = 1$ it follows by remark 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 $X$

$n$Calabi-Yau n-foldline n-bundlemoduli of line n-bundlesmoduli of flat/degree-0 n-bundlesArtin-Mazur formal group of deformation moduli of line n-bundlescomplex oriented cohomology theorymodular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory
$n = 0$unit in structure sheafmultiplicative group/group of unitsformal multiplicative groupcomplex K-theory
$n = 1$elliptic curveline bundlePicard group/Picard schemeJacobianformal Picard groupelliptic cohomology3d Chern-Simons theory/WZW model
$n = 2$K3 surfaceline 2-bundleBrauer groupintermediate Jacobianformal Brauer groupK3 cohomology
$n = 3$Calabi-Yau 3-foldline 3-bundleintermediate JacobianCY3 cohomology7d Chern-Simons theory/M5-brane
$n$intermediate Jacobian

The original article is

Further developments are in

• Jan Stienstra, Formal group laws arising from algebraic varieties, American Journal of Mathematics, Vol. 109, No.5 (1987), 907-925 (pdf)

Lecture notes touching on the cases $n = 1$ and $n = 2$ include

• Christian Liedtke, example 6.13 in Lectures on Supersingular K3 Surfaces and the Crystalline Torelli Theorem (arXiv.1403.2538)

Discussion of Artin-Mazur formal groups for all $n$ and of Calabi-Yau varieties of positive characteristic in dimension $n$ is in