Hodge conjecture
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The Hodge conjecture is a major unsolved problem of algebraic geometry. It is a conjectural description of the link between the algebraic topology of a non-singular complex algebraic variety, and its geometry as captured by polynomial equations that define sub-varieties. It arose as a result of the work of W. V. D. Hodge, who between 1930 and 1940 enriched the description of De Rham cohomology to include extra structure which is present in the case of algebraic varieties (though not restricted to that case).
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Formulating the conjecture
Suppose V is a non-singular algebraic variety of dimension n over the complex numbers. Then we can think of V as a real manifold of dimension 2n. As such it has cohomology groups that are finite-dimensional complex vector spaces indexed by a dimension d, for d = 0 to 2n. Fixing an even value d = 2k, there are two additional structures to describe on H, the d-th cohomology group.
Firstly, there is the Hodge decomposition of H. This is known to split H up as a direct sum of 2k+1 subspaces known as
- H(0,2k), H(1, 2k-1), and so on up to H(2k,0).
The summand relevant to the conjecture is the 'central' one,
- H(k,k).
See Hodge theory for the background.
Secondly, there is a so-called rational structure on H. We have taken H to be the cohomology group with complex coefficients (to which the Hodge decomposition applies). Starting with the cohomology group with rational coefficients, we have a notion of a rational cohomology class in H: for example, a basis for the cohomology classes with rational coefficients can be used as a basis for H and we look at the linear combinations with rational coefficients of those basis vectors.
In terms of those structures, we can define the vector space H* of interest for the Hodge conjecture. It consists of the vectors in H(k,k) that are rational cohomology classes. It is a finite-dimensional vector space over the rational numbers.
The notion of algebraic cycle
Some standard machinery explains the relationship with the geometry of V. If W is a subvariety of dimension n - k in V, which we call co-dimension k, it gives rise to an element of the cohomology group H. For example in codimension 1, which is the most accessible case geometrically using hyperplane sections, the corresponding class is in the second cohomology group and can be computed by means of the first Chern class of the line bundle.
What is known is that such classes, traditionally called algebraic cycles (at least if we talk loosely), satisfy the necessary conditions suggested by the construction of H*. They are rational classes, and also lie in the central H(k,k) summand.
What the Hodge conjecture states
The conjecture says that the algebraic cycles of V span the whole space H*. From what has been said, this means that the stated conditions, necessary to be a combination of algebraic cycles, are also sufficient.
The implications for geometry
The conjecture is known, for k = 1 and for many special cases. Codimension greater than 1 is more difficult to access, because not everything can be 'found' by repeated hyperplane sections, in general.
The existence of non-zero spaces H* in those cases has a predictive value for the part of the geometry of V which is hard to get at. In given examples H* is something that can be discussed much more easily.
It is also the case that when H* is large in dimension, the example chosen as V can be regarded as somewhat special: so the conjecture discusses what you could call the interesting cases and is harder to prove, the further away we are from a generic case.