My question is quite naive, and my knowledge limited on the subject. I heard lot of talks about Hodge conjecture. I wanted to ask about an intuitive way to figure out why we should care about Hodge conjecture, what kind of "application" in mathematics can we expect once the conjecture is proved or disproved?
I'm not asking to write details because I think there is a lot of textbook on the subject. I can summarize my question as follows:
Why should we care about Hodge conjecture ?
I don't know if we can compare this conjecture to the Fermat Theorem or Poincare Theorem, since the letters are just the visible part a deep mathematics. In fact, behind Poincare Theorem there is Thurston geometrization theorem (G. Perelman) and behind Fermat's last Theorem there is a deep theory about modular forms (A. Wiles)
What is behind the Hodge conjecture nowadays (if there is)?