Hodge cohomology in nLab

Contents

Contents

Idea

(Absolute) Hodge cohomology is a variant of de Rham cohomology for complex varieties induced by a canonical Hodge filtration on differential forms.

• holomorphic de Rham complex

• Hodge-filtered differential cohomology

References

The definition of absolute Hodge cohomology originates around

• Alexander Beilinson, Notes on absolute Hodge cohomology, Applications of algebraic K-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), Contemp. Math., vol. 55, Amer. Math. Soc., Providence, RI, 1986, pp. 35-68. MR 862628 (87m:14019)

see also the references at Hodge theory for background.

• Francois Charles, Christian Schnell, Notes on absolute Hodge classes, lecture notes 2010 (pdf)

Application to Beilinson regulators appears in

• Jose Ignacio Burgos and Steve Wang, Higher Bott-Chern forms and Beilinson’s regulator, Invent. Math. 132 (1998), no. 2, 261{305. MR 1621424 (99j:14008)

and then with application to differential algebraic K-theory and in terms of differential forms with logarithmic singularities is in

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