In recent years, considerable progress has been made in studying
algebraic cycles using infinitesimal methods. These methods have usually
been applied to Hodge-theoretic constructions such as the cycle class
and the Abel-Jacobi map. Substantial advances have also occurred in the
infinitesimal theory for subvarieties of a given smooth variety,
centered around the normal bundle and the obstructions coming from the
normal bundle's first cohomology group. Here, Mark Green and Phillip
Griffiths set forth the initial stages of an infinitesimal theory for
algebraic cycles.
The book aims in part to understand the geometric basis and the
limitations of Spencer Bloch's beautiful formula for the tangent space
to Chow groups. Bloch's formula is motivated by algebraic K-theory and
involves differentials over Q. The theory developed here is
characterized by the appearance of arithmetic considerations even in the
local infinitesimal theory of algebraic cycles. The map from the tangent
space to the Hilbert scheme to the tangent space to algebraic cycles
passes through a variant of an interesting construction in commutative
algebra due to Angéniol and Lejeune-Jalabert. The link between the
theory given here and Bloch's formula arises from an interpretation of
the Cousin flasque resolution of differentials over Q as the tangent
sequence to the Gersten resolution in algebraic K-theory. The case of
0-cycles on a surface is used for illustrative purposes to avoid undue
technical complications.