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.
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 .
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 an algebraic variety as above, write
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:
In particular if then vanishes, hence is trivially formally smooth, hence is representable
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).
Let be a strict Calabi-Yau variety 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).
|Calabi-Cau n-fold||line n-bundle||moduli of line n-bundles||moduli of flat/degree-0 n-bundles||Artin-Mazur formal group of deformation moduli of line n-bundles||complex oriented cohomology theory||modular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory|
|unit in structure sheaf||multiplicative group/group of units||formal multiplicative group||complex K-theory|
|elliptic curve||line bundle||Picard group/Picard scheme||Jacobian||formal Picard group||elliptic cohomology||3d Chern-Simons theory/WZW model|
|K3 surface||line 2-bundle||Brauer group||intermediate Jacobian||formal Brauer group||K3 cohomology|
|Calabi-Yau 3-fold||line 3-bundle||intermediate Jacobian||CY3 cohomology||7d Chern-Simons theory/M5-brane|
The original article is
Further developments are in
Lecture notes touching on the cases and include