Hodge's guess

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

The Clay Mathematics Institute has put the proof of this conjecture on its list of its Millennium Problems .

formulation

Let be a non-singular algebraic variety of the dimension over the complex numbers . Then can be viewed as a real manifold of the dimension and thus has De Rham cohomology groups , which are finite-dimensional complex vector spaces, indexed by a dimension with to . If you specify an even value , then two additional structures must be described on the -th cohomology group . ${\ displaystyle V}$ ${\ displaystyle n}$ ${\ displaystyle V}$ ${\ displaystyle 2n}$ ${\ displaystyle d}$ ${\ displaystyle d = 0}$ ${\ displaystyle 2n}$ ${\ displaystyle d = 2k}$ ${\ displaystyle d}$ ${\ displaystyle H}$

One is the Hodge decomposition of that into a direct sum of subspaces ${\ displaystyle H}$ ${\ displaystyle H}$ ${\ displaystyle 2k + 1}$

{\ displaystyle H (0.2k), H (1.2k-1), \ ldots, H (2k, 0)}

with the central summand relevant for the conjecture . ${\ displaystyle H (k, k)}$

The other is what is called a rational structure . The space was chosen as a cohomology group with complex coefficients (to which the Hodge decomposition refers). If we now start with the cohomology group with rational coefficients, we get an idea of a rational cohomology class in : for example a basis of the cohomology classes with rational coefficients can be used as the basis for , and one then considers the linear combination with rational coefficients of these basis vectors. ${\ displaystyle H}$ ${\ displaystyle H}$ ${\ displaystyle H}$ ${\ displaystyle H}$

Under these conditions one can define the vector space with which Hodge's conjecture is concerned. It consists of the vectors in , which are rational cohomology classes, and is a finite-dimensional vector space over the rational numbers. ${\ displaystyle H ^ {*}}$ ${\ displaystyle H (k, k)}$

statement

Hodge's conjecture says that the algebraic cycles of span the entire space , that is, that the specified conditions, which are necessary for a combination of algebraic cycles, are also sufficient . ${\ displaystyle V}$ ${\ displaystyle H ^ {*}}$

The concept of the algebraic cycle

Some standard techniques explain the relationship to the geometry of . If a sub-variety of dimension in is called codimension , founded an element of the cohomology . For example, in codimension 1, which is the most accessible case of geometrically used hyperplane sections, the associated class is in the second cohomology group and can be calculated using the means of the first Chern class of the straight line bundle . ${\ displaystyle V}$ ${\ displaystyle W}$ ${\ displaystyle nk}$ ${\ displaystyle V}$ ${\ displaystyle k}$ ${\ displaystyle W}$ ${\ displaystyle H}$

It is known that such classes, called algebraic cycles (at least if one does not become exact), satisfy the necessary conditions for the construction of . They are rational classes and are in the central summand . ${\ displaystyle H ^ {*}}$ ${\ displaystyle H (k, k)}$

Implications for geometry

The conjecture is known for and many other special cases. Access to codimensions greater than 1 is more difficult, since in general not everything can be found by repeated hyperplane cuts . ${\ displaystyle k = 1}$

The existence of non-empty spaces in these cases has predictive value for that part of the geometry of which is difficult to reach. In given examples is something that can be discussed much more easily. ${\ displaystyle H ^ {*}}$ ${\ displaystyle V}$ ${\ displaystyle H ^ {*}}$

This also applies, if has a large dimension, then the chosen one can be regarded as a special case, so that the conjecture deals with what could be called interesting cases, and the further one is from the general case, the harder it is to prove. ${\ displaystyle H ^ {*}}$ ${\ displaystyle V}$

Web links

Hodge's conjecture at the Clay Mathematics Institute (by P. Deligne, English) (PDF file; 147 kB)
Dan Freed on the Hodge Conjecture, English