Mathematics

Dr. Souvik Goswami
Texas A&M University

Let X be a regular scheme, which is flat and quasi-projective over an arithmetic ring (typically the ring of integers of a number field, or the number field itself). For such an X , Gillet and Soul ́e defined a theory of arithmetic Chow groups, denoted by ̂ CH p (X), whose elements are classes of pairs ( Z, g Z), where Z is a codimension p subvariety of X , and g Z is a Green current for Z. As one can see by projecting to the first component,̂CH p(X) is related to the usual Chow group

CHp(X)via a surjective map.

For a smooth and quasi-projective variety X defined over a field,Spencer Bloch developed a theory for higher Chow groups, denoted by CHp(X, n). These are bigraded abelian groups (with bigrading (p, n)),and has an associative intersection theory which is graded commuta-tive on the index given by n.

In this talk, we will explore the possibility to put a hat on

CHp(X, n).