Statement of The Hodge Conjecture
Let:
We call this the group of Hodge classes of degree 2k on X.
The modern statement of the Hodge conjecture is:
-
- Hodge conjecture. Let X be a projective complex manifold. Then every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X.
A projective complex manifold is a complex manifold which can be embedded in complex projective space. Because projective space carries a Kähler metric, the Fubini–Study metric, such a manifold is always a Kähler manifold. By Chow's theorem, a projective complex manifold is also a smooth projective algebraic variety, that is, it is the zero set of a collection of homogenous polynomials.
Read more about this topic: Hodge Conjecture
Famous quotes containing the words statement of the, statement of, statement and/or conjecture:
“Eroticism has its own moral justification because it says that pleasure is enough for me; it is a statement of the individual’s sovereignty.”
—Mario Vargas Llosa (b. 1936)
“The honor my country shall never be stained by an apology from me for the statement of truth and the performance of duty; nor can I give any explanation of my official acts except such as is due to integrity and justice and consistent with the principles on which our institutions have been framed.”
—Andrew Jackson (1767–1845)
“After the first powerful plain manifesto
The black statement of pistons, without more fuss
But gliding like a queen, she leaves the station.”
—Stephen Spender (1909–1995)
“What these perplexities of my uncle Toby were,—’tis impossible for you to guess;Mif you could,—I should blush ... as an author; inasmuch as I set no small store by myself upon this very account, that my reader has never yet been able to guess at any thing. And ... if I thought you was able to form the least ... conjecture to yourself, of what was to come in the next page,—I would tear it out of my book.”
—Laurence Sterne (1713–1768)