I checked my 2011 paper on the Hodge conjecture. I made some corrections to the paper, especially to Lemma 3.5, but basically the construction was correct and I present it now as a checked counterexample to the Hodge conjecture. In 2011 I did not state that it is a counterexample but asked for comments in arXiv. I received one comment from an expert on the field. At that time I did not find his comment correct but decided to to accept it and in a new version of the paper in arXiv stated that it is not a counterexample giving the explanation that the expert had offered to me. Now I have checked the paper and considered carefully what the expert said. His comment is wrong and it is a counterexample. The paper is here
https://www.researchgate.net/publication/48178200_A_note_on_the_Hodge_conjecture
If you have read any papers in algebraic geometry, or especially on Hodge theory, you can appreciate the paper I wrote. It is by far the simplest that you can imagine in that very complicated field. As all work on that field is very difficult to understand (let’s say compared to algebraic topology, differential geometry, or geometric topology of low-dimensional manifolds, fields that i know well), I am afraid I did not quite reach the targeted level that I have tried to follow in Clay Millennium Problems, that of a second year mathematics student. But certainly this my paper is readable for a 4th year university student of mathematics. You will not find many papers on algebraic geometry that a 4th year student of mathematics, not from the precise field of mathematics, could read. But my paper he can read and understand. As has been the case with many of these problems, those experts who answer to me (the majority do not) make wrong claims. This was true also with the Hodge conjecture paper.
Now I will make a break with mathematics and physics and write something nicer, easier and maybe a bit more interesting. But it was the time to check these proofs. The proof of the Riemann Hypothesis has already been 12 weeks in the Annals of Mathematics, in Princeton. Probably they have not read it. Probably they simply put a submitted (and acknowledged submission by them, meaning that in a finite time they should make a review) paper on a shelf. That would be typical. No answer from the physics book either. It seems like throwing papers in a well. They just disappear, no answers ever.