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group theory

Contents

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 groupof $X$. (ArtinMazur 77, Liedtke 14, example 6.13)

This construction is the special case of the general construction of Artin-Mazur formal groups for $n = 1$. The next case is called the formal Brauer group.

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

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

References

The original account of the construction of formal Picard groups is

Modern reviews include

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

Last revised on May 21, 2014 at 21:22:39. See the history of this page for a list of all contributions to it.