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Why is the Hodge conjecture Conjecture so important?The Hodge conjecture Conjecture states that every Hodge class of a non singular projective variety over $\mathbf{C}$ is a rational linear combination of cohomology classes of algebraic cycles: Even though I'm able to understand what it says, and at first glance I do find it a very nice assertion, I cannot grasp yet why it is so relevant as to be considered one of the biggest open problems in algebraic geometry. Which are its implications? Going a bit further, what about the Tate conjecture? |
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Why is the Hodge conjecture so important?The Hodge conjecture states that every Hodge class of a non singular projective variety over $\mathbf{C}$ is a rational linear combination of cohomology classes of algebraic cycles: Even though I'm able to understand what it says, and at first glance I do find it a very nice assertion, I cannot grasp yet why it is so relevant as to be considered one of the biggest open problems in algebraic geometry. Which are its implications? Going a bit further, what about the Tate conjecture?
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