{"id":"openai-math/The-rational-Hodge-conjecture-for-CM-abelian-varieties","pdf":"https://github.com/openai/math/blob/main/preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-October-6-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-October-6-2026","tags":["algebraic-geometry","hodge-theory","abelian-varieties"],"score":0,"title":"The rational Hodge conjecture for CM abelian varieties","domain":"Algebraic and complex geometry","family":"032","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.AG","summary":"We prove the rational Hodge conjecture for complex abelian varieties with complex multiplication: every rational Hodge class on such a variety is a rational linear combination of algebraic cycle classes. As consequences, we obtain the generalized Hodge conjecture for CM abelian varieties, the Tate conjecture for abelian varieties over finite fields, and the Hodge standard conjecture for abelian varieties in arbitrary characteristic.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":4,"published":"2026-10-06","reactions":{},"discussion":[{"url":"https://x.com/ElliotGlazer/status/2107602154756489340","date":"2026-10-06","kind":"x","added":"2026-11-03","author":"Elliot Glazer","handle":"ElliotGlazer"},{"url":"https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/#:~:text=Currently%2C%20I%20am%20reading%20the%20paper%20on%20the%20Hodge%20conjecture%20for%20CM%20abelian%20varieties%3B%20I%20find%20it%20visionary","date":"2026-10-08","kind":"web","site":"Proofs and Prompts","text":"Currently, I am reading the paper on the Hodge conjecture for CM abelian varieties; I find it visionary. No point in continuing to move the goalposts. (2) Bad behavior from private companies is to be expected. Mathematicians have, in the past, been insulated from certain behaviors which plague the…","added":"2026-10-24","title":"100+ reactions to 100+ solutions","author":"Philip Engel"},{"url":"https://x.com/ElliotGlazer/status/2108125478469111983","date":"2026-10-08","kind":"x","added":"2026-11-03","author":"Elliot Glazer","handle":"ElliotGlazer"},{"url":"https://mathoverflow.net/a/516477","date":"2026-10-28","kind":"web","site":"mathoverflow.net","text":"The answer depends to some extent on what you think the purpose of formalization is. If the sole purpose of formalizing a theorem is to get a rubber stamp that lets you declare that the proof is correct, then 13 million lines of unreadable Lean code is probably just fine. (That's of course…","added":"2026-11-04","title":"Answer to: Are the main libraries of proof assistants becoming unnecessary?","author":"Timothy Chow"}],"formalized":true,"family_title":"The rational Hodge conjecture for CM abelian varieties","formalizations":["https://github.com/openai/math/blob/main/lean/docs/032.md"]}