The Hodge conjecture is hard because sophisticated submanifolds are really rigid objects (considering that These are described by polynomials). Actually, it is very hard to build elaborate submanifolds, and there are hardly any of them. Proving that submanifolds exist without having constructing them (an exceedingly handy kind of indirect reasoning) https://top10bookmark.com/story19196780/considerations-to-know-about-bos138