Hodge cycle
Encyclopedia
In differential geometry, a Hodge cycle or Hodge class is a particular kind of homology class defined on a complex
algebraic variety
V, or more generally on a Kähler manifold
. A homology class x in a homology group
where V is a non-singular complex algebraic variety or Kähler manifold is a Hodge cycle, provided it satisfies two conditions. Firstly, k is an even integer 2p, and in the direct sum
decomposition of H shown to exist in Hodge theory
, x is purely of type (p,p). Secondly, x is a rational class, in the sense that it lies in the image of the abelian group homomorphism
defined in algebraic topology
(as a special case of the universal coefficient theorem
). The conventional term Hodge cycle therefore is slightly inaccurate, in that x is considered as a class (modulo
boundaries); but this is normal usage.
The importance of Hodge cycles lies primarily in the Hodge conjecture
, to the effect that Hodge cycles should always be algebraic cycle
s, for V a complete algebraic variety
. This is an unsolved problem, ; it is known that being a Hodge cycle is a necessary condition to be an algebraic cycle that is rational, and numerous particular cases of the conjecture are known.
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
V, or more generally on a Kähler manifold
Kähler manifold
In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...
. A homology class x in a homology group
- Hk(V, C) = H
where V is a non-singular complex algebraic variety or Kähler manifold is a Hodge cycle, provided it satisfies two conditions. Firstly, k is an even integer 2p, and in the direct sum
Direct sum of modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The result of the direct summation of modules is the "smallest general" module which contains the given modules as submodules...
decomposition of H shown to exist in Hodge theory
Hodge theory
In mathematics, Hodge theory, named after W. V. D. Hodge, is one aspect of the study of the algebraic topology of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised...
, x is purely of type (p,p). Secondly, x is a rational class, in the sense that it lies in the image of the abelian group homomorphism
- Hk(V, Q) → H
defined in algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
(as a special case of the universal coefficient theorem
Universal coefficient theorem
In mathematics, the universal coefficient theorem in algebraic topology establishes the relationship in homology theory between the integral homology of a topological space X, and its homology with coefficients in any abelian group A...
). The conventional term Hodge cycle therefore is slightly inaccurate, in that x is considered as a class (modulo
Modulo (jargon)
The word modulo is the Latin ablative of modulus which itself means "a small measure."It was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801...
boundaries); but this is normal usage.
The importance of Hodge cycles lies primarily in the Hodge conjecture
Hodge conjecture
The Hodge conjecture is a major unsolved problem in algebraic geometry which relates the algebraic topology of a non-singular complex algebraic variety and the subvarieties of that variety. More specifically, the conjecture says that certain de Rham cohomology classes are algebraic, that is, they...
, to the effect that Hodge cycles should always be algebraic cycle
Algebraic cycle
In mathematics, an algebraic cycle on an algebraic variety V is, roughly speaking, a homology class on V that is represented by a linear combination of subvarieties of V. Therefore the algebraic cycles on V are the part of the algebraic topology of V that is directly accessible in algebraic geometry...
s, for V a complete algebraic variety
Complete algebraic variety
In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety X, such that for any variety Y the projection morphismis a closed map, i.e. maps closed sets onto closed sets....
. This is an unsolved problem, ; it is known that being a Hodge cycle is a necessary condition to be an algebraic cycle that is rational, and numerous particular cases of the conjecture are known.