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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 is a formal group version of the Picard n-group of flat/holomorphic circle n-bundles on the given variety. Therefore for one also speaks of the formal Picard group and for of the formal Brauer group.
Let be a smooth proper dimensional variety over an algebraically closed field of positive characteristic .
Writing for the multiplicative group and for etale cohomology, then classifies -principal n-bundles (line n-bundles, bundle (n-1)-gerbes) on . Notice that, by the discussion at Brauer group – relation to étale cohomology, for this is the Picard group while for this contains (as a torsion subgroup) the Brauer group of .
Accordingly, for each Artin algebra regarded as an infinitesimally thickened point the cohomology group is that of equivalence classes of -principal n-bundles on a formal thickening of .
The defining inclusion of the unique global point induces a restriction map which restricts an -bundle on the formal thickening to just itself. The kernel of this map hence may be thought of as the group of -parameterized infinitesimal deformations of the trivial --bundle on .
(For this is an infinitesimal neighbourhood of the neutral element in the Picard scheme , for higher one will need to genuinely speak about Picard stacks and higher stacks.)
As varies, these groups of deformations naturally form a presheaf on “infinitesimally thickened points” (formal duals to Artin algebras).
For 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 . This is called the Artin-Mazur formal group of in degree .
More in detail:
Let be an algebraic variety proper over an algebraically closed field of positive characteristic.
A sufficient condition for to be pro-representable by a formal group is that is formally smooth.
In particular if then vanishes, hence is trivially formally smooth, hence is representable
The first statement appears as (Artin-Mazur 77, corollary (2.12)). The second as (Artin-Mazur 77, corollary (4.2)).
The dimension of is
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 (i.e. ), the Artin-Mazur group is often called the formal Picard group .
For a surface (i.e. ), the Artin-Mazur group is called the formal Brauer group .
Let be a strict Calabi-Yau variety in positive characteristic of dimension (strict meaning that the Hodge numbers vanish for , i.e. over the complex numbers that the holonomy group exhausts , 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 exists. Since moreover it follows by remark that it is of dimension 1
For discussion of for Calabi-Yau varieties of dimension and in positive characteristic see (Geer-Katsura 03).
moduli spaces of line n-bundles with connection on -dimensional
The original article is
Further developments are in
Lecture notes touching on the cases and include
Discussion of Artin-Mazur formal groups for all and of Calabi-Yau varieties of positive characteristic in dimension is in
Last revised on November 16, 2020 at 18:31:28. See the history of this page for a list of all contributions to it.