The Hodge conjecture is tough since sophisticated submanifolds are extremely rigid objects (because They're described by polynomials). In actual fact, it is very hard to build elaborate submanifolds, and you'll find only a few of them. Proving that submanifolds exist without developing them (a very handy kind of oblique reasoning) https://extrabookmarking.com/story19342250/the-basic-principles-of-bos138