nLab formal Picard group

Idea

Given an algebraic variety with Picard scheme $Pic_X$, if the connected component $Pic_X^0$ is a smooth scheme then the completion of $Pic_X$ at its neutral global point is a formal group. This is called the formal Picard group of $X$. (ArtinMazur 77, Liedtke 14, example 6.13)

This construction is the special case of the general construction of Artin-Mazur formal groups for $n = 1$ (see also this Remark at elliptic spectrum). The next case is called the formal Brauer group.

Related concepts

• Picard stack

moduli spaces of line n-bundles with connection on $n$-dimensional $X$

$n$	Calabi-Yau 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
$n = 0$		unit in structure sheaf	multiplicative group/group of units		formal multiplicative group	complex K-theory
$n = 1$	elliptic curve	line bundle	Picard group/Picard scheme	Jacobian	formal Picard group	elliptic cohomology	3d Chern-Simons theory/WZW model
$n = 2$	K3 surface	line 2-bundle	Brauer group	intermediate Jacobian	formal Brauer group	K3 cohomology
$n = 3$	Calabi-Yau 3-fold	line 3-bundle		intermediate Jacobian		CY3 cohomology	7d Chern-Simons theory/M5-brane
$n$				intermediate Jacobian

References

The original account of the construction of formal Picard groups 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, MR56:15663

Modern reviews include

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