Hodge conjecture

The conjecture of Hodge is one of the great unsolved problems in algebraic geometry. It is the representation of a putative binding element between the algebraic topology of non- singular complex algebraic varieties and their geometry, which is described by defining sub-varieties of polynomial equations. The assumption is the result of the work of William Vallance Douglas Hodge (1903-1975), who extended the representation of the De Rham cohomology 1930-1940 to special structures, which are present in algebraic varieties (although not limited to them ) to include.

The Clay Mathematics Institute has set the proof of this conjecture on the list of his Millennium problems.

Formulation

Be a non-singular algebraic variety of dimension over the complex numbers. Then can be considered as a real manifold of dimension, and thus has de Rham cohomology, the finite complex vector spaces are indexed by a dimension up. If we specify an even value, then two additional structures on the -th cohomology group should be described.

One is the Hodge decomposition of which in a direct sum of subspaces

Divided by the relevant central assumption for the summands.

The other is a so-called rational structure. The room was chosen as the cohomology with complex coefficients ( to which the Hodge decomposition applies ). You will start the cohomology with rational coefficients, we get an idea of a rational cohomology class in: for example, a base of Kohomologieklassen with rational coefficients are used as the basis for, and we then consider the linear combination with rational coefficients of these basis vectors.

Under these conditions, it is the vector space, at issue in the conjecture of Hodge define. It consists of the vectors that are rational Kohomologieklassen, and is a finite dimensional vector space over the rational numbers.

Statement

The conjecture of Hodge states that the algebraic cycles of V the whole space H span *, which means that the specified conditions that are necessary for a combination of algebraic cycles, and are sufficient.

The concept of algebraic Zykels

Some standard methods explain the relationship with the geometry of V. If W is a subvariety of dimension n - k in V, called the codimension k, founded W an element of the cohomology group H. For example, in codimension 1, which is the most accessible case geometrically unused hyperplane sections, is the corresponding class in the second cohomology group and can be calculated by means of the first Chern class of the line bundle.

It is known that such classes, algebraic cycles called ( at least if one is not exact), * satisfy the necessary conditions for the construction of H. They are rational and classes located in the central summands H (k, k).

Implications for the geometry

The conjecture is known for k = 1 and many other special events. To Kodimensionen greater than 1, the access is more difficult because in general not everything can be found by repeated hyperplane sections.

The existence of non-empty spaces H * in these cases has a predictive value for the part of the geometry of V, which can be difficult to reach. Examples given in H * is something which can be considered much simpler.

This is true even if H * a large dimension, the selected V can be considered as a special case, so that the presumption treated what could be called interesting cases, and is more difficult to prove, the farther away you are from the general case.