Timeline for is the Hodge conjecture birationally invariant?
Current License: CC BY-SA 4.0
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| Aug 4, 2021 at 5:47 | history | edited | abx | CC BY-SA 4.0 |
added 36 characters in body
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| Jan 27, 2014 at 4:43 | comment | added | abz | In fact, the Hodge conjecture, the Tate conjecture, and the various standard conjectures hold for all varieties if they hold for all rational varieties (and they do all hold for $\mathbb{P}^n$). See: Tankeev, S. G. Monoidal transformations and conjectures on algebraic cycles. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 71 (2007), no. 3, 197--224; translation in Izv. Math. 71 (2007), no. 3, 629--655 | |
| Jan 27, 2014 at 0:50 | comment | added | Sándor Kovács | @abx: I don't want to be nitpicking, but I had to reread what you wrote to understand what you're saying. Perhaps, saying "blow up $\mathbb P^n$ along a smooth subvariety $X$" would be a clearer way to put it. Then again, it might be just me... Cheers! (and +1) | |
| Jan 26, 2014 at 23:19 | comment | added | Olivier Benoist | On the other hand, the weak factorization theorem implies that the Hodge conjecture is a birational invariant of smooth projective varieties of dimension $\leq 4$. | |
| Jan 26, 2014 at 21:04 | history | answered | abx | CC BY-SA 3.0 |