Hodge conjecture

In simple terms, the Hodge conjecture asserts that the basic topological information like the number of holes in certain geometric spaces, complex algebraic varieties, can be understood by studying the possible nice shapes sitting inside those spaces, which look like zero sets of polynomial equations.

The latter objects can be studied using algebra and the calculus of analytic functions, and this allows one to indirectly understand the broad shape and structure of often higher-dimensional spaces which can not be otherwise easily visualized.

It was formulated by the Scottish mathematician William Vallance Douglas Hodge as a result of a work in between 1930 and 1940 to enrich the description of de Rham cohomology to include extra structure that is present in the case of complex algebraic varieties.

It received little attention before Hodge presented it in an address during the 1950 International Congress of Mathematicians, held in Cambridge, Massachusetts.

Assume X is a Kähler manifold, so that there is a decomposition on its cohomology with complex coefficients where

is the subgroup of cohomology classes which are represented by harmonic forms of type

That is, these are the cohomology classes represented by differential forms which, in some choice of local coordinates

, can be written as a harmonic function times Since X is a compact oriented manifold, X has a fundamental class, and so X can be integrated over.

Another way of phrasing the Hodge conjecture involves the idea of an algebraic cycle.

An algebraic cycle on X is a formal combination of subvarieties of X; that is, it is something of the form The coefficients are usually taken to be integral or rational.

In 1977, Steven Zucker showed that it is possible to construct a counterexample to the Hodge conjecture as complex tori with analytic rational cohomology of type

In fact, it predates the conjecture and provided some of Hodge's motivation.

A very quick proof can be given using sheaf cohomology and the exponential exact sequence.

Lefschetz's original proof proceeded by normal functions, which were introduced by Henri Poincaré.

However, the Griffiths transversality theorem shows that this approach cannot prove the Hodge conjecture for higher codimensional subvarieties.

[3] For most abelian varieties, the algebra Hdg*(X) is generated in degree one, so the Hodge conjecture holds.

In particular, the Hodge conjecture holds for sufficiently general abelian varieties, for products of elliptic curves, and for simple abelian varieties of prime dimension.

[4][5][6] However, Mumford (1969) constructed an example of an abelian variety where Hdg2(X) is not generated by products of divisor classes.

Weil (1977) generalized this example by showing that whenever the variety has complex multiplication by an imaginary quadratic field, then Hdg2(X) is not generated by products of divisor classes.

Moonen & Zarhin (1999) proved that in dimension less than 5, either Hdg*(X) is generated in degree one, or the variety has complex multiplication by an imaginary quadratic field.

Totaro (1997) reinterpreted their result in the framework of cobordism and found many examples of such classes.

The simplest adjustment of the integral Hodge conjecture is: Equivalently, after dividing

Rosenschon & Srinivas (2016) have shown that in order to obtain a correct integral Hodge conjecture, one needs to replace Chow groups, which can also be expressed as motivic cohomology groups, by a variant known as étale (or Lichtenbaum) motivic cohomology.

A natural generalization of the Hodge conjecture would ask: This is too optimistic, because there are not enough subvarieties to make this work.

Consequently, the only known formulations of the Hodge conjecture for Kähler varieties are false.

The cohomology classes of co-level at least c filter the cohomology of X, and it is easy to see that the cth step of the filtration NcHk(X, Z) satisfies Hodge's original statement was: Grothendieck (1969) observed that this cannot be true, even with rational coefficients, because the right-hand side is not always a Hodge structure.

The strongest evidence in favor of the Hodge conjecture is the algebraicity result of Cattani, Deligne & Kaplan (1995).

Suppose that we vary the complex structure of X over a simply connected base.

It is known that if the Hodge conjecture is true, then the locus of all points on the base where the cohomology of a fiber is a Hodge class is in fact an algebraic subset, that is, it is cut out by polynomial equations.

Cattani, Deligne & Kaplan (1995) proved that this is always true, without assuming the Hodge conjecture.

Topological features of a space , such as a hole (labelled by ) are usually detected using singular (co)homology , where the presence of a non-zero class indicates the space has a (dimension ) hole. Such a class is represented by a (co)chain of simplices , depicted by the red polygon built out of 1-simplices (line segments) on the left. This class detects the hole by looping around it. In this case, there is in fact a polynomial equation whose zero set, depicted in green on the right, also detects the hole by looping around it. The Hodge conjecture generalises this statement to higher dimensions.