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formal Picard group

Contents

Context

Formal geometry

Group Theory

Contents

Idea

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

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

moduli spaces of line n-bundles with connection on nn-dimensional XX

nnCalabi-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=0n = 0unit in structure sheafmultiplicative group/group of unitsformal multiplicative groupcomplex K-theory
n=1n = 1elliptic curveline bundlePicard group/Picard schemeJacobianformal Picard groupelliptic cohomology3d Chern-Simons theory/WZW model
n=2n = 2K3 surfaceline 2-bundleBrauer groupintermediate Jacobianformal Brauer groupK3 cohomology
n=3n = 3Calabi-Yau 3-foldline 3-bundleintermediate JacobianCY3 cohomology7d Chern-Simons theory/M5-brane
nnintermediate 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 November 16, 2020 at 11:53:14. See the history of this page for a list of all contributions to it.