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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Then the topological cohomology of X does not change, but the Hodge decomposition does change. Since X is a compact oriented manifold, X has a fundamental class.
It was formulated by the Scottish mathematician William Vallance Douglas Hodge as a result of a work in between and to enrich the description of de Conjftura cohomology to include extra structure that is present in the case of complex algebraic varieties.
With this notation, the Hodge conjecture becomes:. Say that a cohomology class on X is of co-level c conjetrua c if it is the pushforward of a cohomology class on a c -codimensional subvariety of X. This is too optimistic, because there are not enough subvarieties to make this work.
Birch and Swinnerton-Dyer conjecture
In Zucker showed that it is possible to construct a counterexample to the Hodge conjecture as complex tori with analytic rational cohomology of type ppwhich is not projective algebraic. By the strong and weak Lefschetz theoremthe only non-trivial part of the Hodge conjecture for hypersurfaces is the degree m part i.
Talk at the BSD 50th anniversary conference, May Grothendieck observed that this cannot be true, even with rational coefficients, because the right-hand side is not always a Hodge structure. As of [update]only special cases of the conjecture have been proved.
This is now known to be false. Graduate Texts in Mathematics. A projective complex manifold is a complex manifold which can be embedded in complex projective space.
This L -function is analogous to the Riemann zeta function and the Dirichlet L-series that is defined for a binary quadratic form. Bhargava, Manjul ; Shankar, Arul Weil generalized this example by showing that whenever the variety has complex multiplication by an imaginary quadratic fieldthen Hdg 2 X is not generated by products of divisor classes.
Lecture Notes in Mathematics. Another way of phrasing the Hodge conjecture involves the idea of an algebraic cycle.
In fact, it predates the conjecture and provided some of Hodge’s motivation. NB that the reciprocal of the L-function is from some points of view a more natural object of study; on occasion this means that one should consider poles rather than zeroes. The cohomology classes of co-level at least c filter the cohomology of Xand it is easy to see that the c th step of the filtration N c H k XZ satisfies.
Such a cohomology class is called algebraic. For example, the cohomology class of the above cycle would be:. Because [ Z ] is a cohomology class, it has a Hodge decomposition. Totaro reinterpreted their result in the framework of cobordism and found many examples of cpnjetura classes.
If the degree d is 2, i. However, for large primes it is computationally intensive. A Survey of the Hodge Conjecture, Example 7.
Hodge conjecture – Wikipedia
A possible substitute is to ask instead one of the two following questions:. Nothing has been proved for curves with rank greater than 1, although there is conjstura numerical evidence for the truth of the conjecture.
A very quick proof can be given using sheaf cohomology and the exponential exact sequence. From Wikipedia, the free encyclopedia.
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