Artin-Mazur formal group in nLab

Contents

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 S∈ArtAlg 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).

(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 Φ X n\Phi_X^n. This is called the Artin-Mazur formal group of XX in degree nn.

More in detail:

The first statement appears as (Artin-Mazur 77, corollary (2.12)). The second as (Artin-Mazur 77, corollary (4.2)).

(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

Of Calabi-Yau varieties

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).

• formal group

• height of a variety

References

The original article is

• Michael Artin, Barry Mazur, Formal Groups Arising from Algebraic Varieties, Annales scientifiques de l’École Normale Supérieure, Sér. 4, 10 no. 1 (1977), p. 87-131 (numdam:ASENS_1977_4_10_1_87_0, MR56:15663)

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

• Gerard van der Geer, T. Katsura, On the height of Calabi-Yau varieties in positive characteristic, Documenta Math. 8. 97-113 (2003) (arXiv:math/0302023)