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Artin-Mazur formal group

Context

Formal geometry

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 nn is a formal group version of the Picard n-group of flat/holomorphic circle n-bundles on the given variety. Therefore for n=1n = 1 one also speaks of the formal Picard group and for n=2n = 2 of the formal Brauer group.

Definition

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

Let XX be a smooth proper nn dimensional variety over an algebraically closed field kk of positive characteristic pp.

Writing 𝔾 m\mathbb{G}_m for the multiplicative group and H et (,)H_{et}^\bullet(-,-) for etale cohomology, then H et n(X,𝔾 m)H_{et}^n(X,\mathbb{G}_m) classifies 𝔾 m\mathbb{G}_m-principal n-bundles (line n-bundles, bundle (n-1)-gerbes) on XX. Notice that, by the discussion at Brauer group – relation to étale cohomology, for n=1n = 1 this is the Picard group while for n=2n = 2 this contains (as a torsion subgroup) the Brauer group of XX.

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

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

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

As SS varies, these groups of deformations naturally form a presheaf on “infinitesimally thickened points” (formal duals to Artin algebras).

Definition

For XX an algebraic variety as above, write

Φ X n:ArtAlg kGrp \Phi_X^n \;\colon\; ArtAlg_k \to Grp
Φ X n(S)ker(H et n(X× Spec(k)S,𝔾 m)H et n(X,𝔾 m)). \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)) \,.

(Artin-Mazur 77, II.1 “Main examples”)

The fundamental result of (Artin-Mazur 77, II) is that under the above hypotheses this presheaf if pro-representable by formal group, which we may hence also denote by Φ X n\Phi_X^n. This is called the Artin-Mazur formal group of XX in degree nn.

More in detail:

Proposition

Let XX be an algebraic variety proper over an algebraically closed field kk of positive characteristic.

A sufficient condition for Φ X k\Phi_X^k to be pro-representable by a formal group is that Φ X k1\Phi_X^{k-1} is formally smooth.

In particular if dimH k1(X,𝒪 X)=0dim H^{k-1}(X,\mathcal{O}_X) = 0 then Φ k1(X)\Phi^{k-1}(X) vanishes, hence is trivially formally smooth, hence Φ k(X)\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 Φ X k\Phi^k_X is

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

(Artin-Mazur 77, II.4).

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

Of Calabi-Yau varieties

Example

Let XX be a strict Calabi-Yau variety of dimension nn (strict meaning that the Hodge numbers h 0,r=0h^{0,r} = 0 vanish for 0<r<n0 \lt r \lt n, i.e. over the complex numbers that the holonomy group exhausts SU(n)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 Φ X n\Phi^n_X exists. Since moreover h 0,n=1h^{0,n} = 1 it follows by remark 1 that it is of dimension 1

For discussion of Φ X n\Phi_X^n for Calabi-Yau varieties XX of dimension nn and in positive characteristic see (Geer-Katsura 03).

moduli spaces of line n-bundles with connection on nn-dimensional XX

nnCalabi-Cau 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=0n = 0unit in structure sheafmultiplicative group/group of unitsformal multiplicative groupcomplex K-theory
n=1n = 1elliptic curveline bundlePicard group/Picard schemeJacobianformal Picard groupelliptic cohomology3d Chern-Simons theory/WZW model
n=2n = 2K3 surfaceline 2-bundleBrauer groupintermediate Jacobianformal Brauer groupK3 cohomology
n=3n = 3Calabi-Yau 3-foldline 3-bundleintermediate JacobianCY3 cohomology7d Chern-Simons theory/M5-brane
nnintermediate Jacobian

References

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=1n = 1 and n=2n = 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 nn and of Calabi-Yau varieties of positive characteristic in dimension nn is in

Revised on June 5, 2014 12:53:25 by Urs Schreiber (82.136.246.44)