Gysin maps and cycle classes for Hodge cohomology (Q1329243)

Gysin maps \(f_ *\) of Hodge cohomology groups are associated to projective morphisms \(f\) of regular varieties over a field. They lead to cycle class maps \(\text{CH}^ i (X, \text{Sing} X) \to H^ i (X, \Omega_{X/Z})\) for any quasi-projective variety \(X\) over a field. As an application one obtains the infinite dimensionality of the Chow group of zero cycles of a normal complex projective variety \(X\) of dimension \(n\) with \(H^ n (X, {\mathcal O}_ X) \neq 0\); this is an extension of results of Mumford (who proved this for smooth surfaces) and Roitman (who proved this for smooth \(X\) of arbitrary dimension).