In mathematics, more specifically in
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, the Griffiths group of a
projective complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a com ...
''X'' measures the difference between homological equivalence and algebraic equivalence, which are two important
equivalence relations
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relati ...
of
algebraic cycle In mathematics, an algebraic cycle on an algebraic variety ''V'' is a formal linear combination of subvarieties of ''V''. These are the part of the algebraic topology of ''V'' that is directly accessible by algebraic methods. Understanding the al ...
s.
More precisely, it is defined as
:
where
denotes the group of algebraic cycles of some fixed
codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.
For affine and projective algebraic varieties, the codimension equals the ...
''k'' and the subscripts indicate the groups that are homologically trivial, respectively algebraically equivalent to zero.
[Voisin, C., ''Hodge Theory and Complex Algebraic Geometry II'', Cambridge University Press, 2003. See Chapter 8]
This group was introduced by
Phillip Griffiths
Phillip Augustus Griffiths IV (born October 18, 1938) is an American mathematician, known for his work in the field of geometry, and in particular for the complex manifold approach to algebraic geometry. He was a major developer in particul ...
who showed that for a general
quintic
In algebra, a quintic function is a function of the form
:g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\,
where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, a q ...
in
(projective 4-space), the group
is not a torsion group.
References
{{Reflist
Algebraic geometry