Hodge conjecture

"The twin conjectures of Hodge and Tate have a status in algebraic and arithmetic geometry similar to that of the Riemann hypothesis in analytic number theory."

"The Hodge conjecture postulates a deep and powerful connection between three of the pillars of modern mathematics: algebra, topology, and analysis. Take any variety. To understand its shape (topology, leading to cohomology classes) pick out special instances of these (analysis, leading to Hodge classes by way of differential equations). These special types of cohomology class can be realised using subvarieties (algebra: throw in some extra equations and look at algebraic cycles). That is, to solve the topology problem 'what shape is this thing?' for a variety, turn the question into analysis and then solve that using algebra. Why is that important? The Hodge conjecture is a proposal to add two new tools to the algebraic geometer's toolbox: topological invariants and Laplace's equation. It's not really a conjecture about a mathematical theorem; it's a conjecture about new kinds of tools."

"In his 1950 Congress address, Hodge reported on the topological and differential-geometric methods in studying algebraic varieties and complex manifolds which had been initiated by Lefschetz and developed by Hodge himself. He raised there many problems, and most of them were settled in 1950's by extensive works due to Kodaira and others. One notable exception to this is the so-called Hodge Conjecture which, if true, will give a characterization of cohomology classes of algebraic cycles on a nonsingular projective variety, generalizing the Lefschetz criterion for the case of divisors. This conjecture has an arithmetic flavour, as is common to most problems concerning algebraic cycles, which makes the problem interesting and difficult at the same time."