English Spanish online dictionary Term Bank, translate words and terms with different pronunciation options. Products of it with itself give candidates for counterexamples to the Hodge conjecture which may be of interest. We also study the Kuga-Satake. The Hodge conjecture asserts that, for particularly nice types of spaces called projective algebraic varieties, the pieces called Hodge cycles are actually rational.

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It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems. A possible substitute is to ask instead one of the two following questions:. From Wikipedia, the free encyclopedia.

Hodge Conjecture

This is too optimistic, because there are not enough subvarieties to make this work. Proceedings of the American Mathematical Society. From Wikipedia, the free encyclopedia. The Lefschetz theorem on 1,1 -classes also implies that if all Hodge classes are generated by the Hodge classes of divisors, then the Hodge conjecture is true:.

The first result on the Hodge conjecture is due to Lefschetz In fact, it predates the conjecture and provided some of Hodge’s motivation. The Millennium prize problems.


[math/] Counterexample to the Hodge Conjecture

Suppose that we vary the complex structure of X over a simply connected base. A Survey of the Hodge Conjecture, Example 7. Journal of the American Mathematical Society.

It was subsequently shown to be true conjehura all elliptic curves over Qas a consequence of the modularity theorem. Such a cohomology class is called algebraic.

A very quick proof can be given using sheaf cohomology and the exponential exact sequence. CS1 French-language sources fr.

This means that for any elliptic curve there is a finite sub-set of df rational points on the curve, from which all further rational points may be generated. As of [update]only special cases of the conjecture have been proved. That is, these are the cohomology classes represented by differential forms which, in some choice of local coordinates z 1Articles containing potentially dated statements from All articles containing potentially dated statements.

Graduate Texts in Mathematics.

Term Bank – conjetura de hodge – Spanish English Dictionary

Totaro reinterpreted their result in the framework of cobordism and found many examples of such classes. Dokchitser, Tim ; Dokchitser, Vladimir Arithmetic Theory of Elliptic Curves.

His corrected form of the Hodge conjecture is:. In the latter case, the Hodge conjecture is only known in special cases. In mathematicsthe Hodge conjecture is a major unsolved problem in the field of algebraic re that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties. 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.


If the degree d is 2, i.

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Retrieved from ” https: It is conjecturally given by. In other projects Wikiquote. An algebraic cycle on X is a formal combination of subvarieties of X ; that is, it is something of the form:. We define the cohomology class of an algebraic cycle to be the sum of the cohomology classes of conjrtura components.