differential form with logarithmic singularities in nLab

Contents

Contents

Idea

A differential form with logarithmic singularities is a meromorphic differential form on some space XX which is a holomorphic differential form on a suitably dense open subspace with at most logarithmic singularities at the boundary.

These are the differential forms on spaces in logarithmic geometry. They form the logarithmic generalization of the holomorphic de Rham complex.

Examples

• The moduli spaces of connections with logarithmic singularity at prescribed points are for instance the object of interest in “ramified” geometric Langlands duality. (e.g. Witten 08, section 4)

• A 't Hooft operator is a gauge field configuration with certain logarithmic differential form singularities.

• logarithmic geometry

• logarithmic connection

• differential algebraic K-theory

References

The brief idea is well described in

• Pottharst, Logarithmic structures on schemes (pdf)

Further details are discussed in

• Arthur Ogus, Chapter IV of Lectures on logarithmic algebraic geometry, TeXed notes, 2001, pdf

• Arthur Ogus, slides 31 ff in Logarithmic geometry, talk slides 2009 (pdf)

• José Ignacio Burgos, A C ∞C^\infty-logarithmic Dolbeault complex, Compositio Math. 92 (1994), no. 1, 61-86. MR 1275721 (95g:32056)

• Claire Voisin, section 8.2.2 of Hodge theory and Complex algebraic geometry I,II, Cambridge Stud. in Adv. Math. 76, 77, 2002/3

Discussion in the context of geometric Langlands duality includes

• Edward Witten, Mirror Symmetry, Hitchin’s Equations, And Langlands Duality (arXiv:0802.0999)

In the context of differential algebraic K-theory

• Ulrich Bunke, Georg Tamme, section 3.1 of Regulators and cycle maps in higher-dimensional differential algebraic K-theory (arXiv:1209.6451)