Prove or disprove the Hodge conjecture

Understanding
A complete description of the problem can be found here: Claymath Millennium prize problems - Hodge conjecture

Everything else
The description of the problem given in the link will be rewritten in the simplest and most commonly understood words that we can think of so that people who are not familiar with mathematical jargon can still understand and play with the mathematical concepts. All jargon will be clearly defined. Then we can move on from the problem definition with our own deductions and manipulations to develop a deeper understanding of the problem space.

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