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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Birch and Swinnerton-Dyer conjecture
Analytic class number formula Riemann—von Mangoldt formula Weil conjectures. This is also false. His corrected form of the Hodge conjecture is:. With this notation, the Hodge conjecture becomes:. It received little attention before Hodge presented it in an address during the International Congress of Mathematiciansheld in Cambridge, Massachusetts. By the Hard Lefschetz theoremone can prove:.
Views Read Edit View history. This page was last edited on 13 Novemberat Then the topological cohomology of X does not change, but the Hodge decomposition does change. 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. It was formulated by cojnetura Scottish mathematician William Vallance Douglas Hodge as a result of a work in between and to enrich the description of de Rham cohomology to include extra structure that is present in the case of complex algebraic varieties.
Hodge conjecture at Wikipedia’s sister projects.
Because [ Z ] conjeyura a cohomology class, it has a Hodge decomposition. Around 0, we can choose local coordinates z 1This page was last edited on 30 Decemberat Arithmetic Theory of Elliptic Curves. Hodge made an additional, stronger conjecture than the integral Hodge conjecture.
Hodge conjecture – Wikidata
Birch, Bryan ; Swinnerton-Dyer, Peter The rank of certain elliptic curves can be calculated using numerical methods but in the current state of knowledge it is unknown if these methods handle all curves. A Survey of the Hodge Conjecture, Example 7. The number of independent basis points with infinite order is called the rank of the curve, and is an important invariant property of an elliptic curve.
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The cohomology classes of co-level at least c filter the cohomology of Xand ed is easy to see that the c th step of the filtration N c H k XZ satisfies. In mathematicsthe Hodge conjecture is a major unsolved problem in the field of algebraic geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties.
We define the cohomology class of an algebraic cycle to be the sum of the cohomology classes of its components. This conjecture was first proved by Deuring for elliptic curves with complex multiplication. Taking wedge products of these harmonic representatives corresponds to the cup product in cohomology, so the cup product is compatible with the Hodge decomposition:.
Let X be a compact complex manifold of complex dimension n. See Hodge theory for more details. Nothing has been proved for curves with rank greater than 1, although there is extensive numerical evidence for the truth of the conjecture.
Hodge Conjecture — from Wolfram MathWorld
Say that a cohomology class on X is of co-level c coniveau c if it is the pushforward of a cohomology class on a conjeturs -codimensional subvariety of X. Views Read Edit View history. Proceedings of the American Mathematical Society.
However, Griffiths transversality theorem shows that this approach cannot prove the Hodge conjecture for higher codimensional subvarieties. Then X is an orientable smooth manifold of real dimension 2 nso its cohomology groups lie in degrees zero through 2 n. Introduction to Elliptic Curves and Modular Forms.
In fact, it predates the conjecture and provided some of Hodge’s cinjetura.