Timeline for Example illustrating necessity of considering birational equivalence and not biholomorphic equivalence in MMP
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Jul 7, 2022 at 21:45 | vote | accept | ABBC | ||
| Jul 7, 2022 at 21:45 | comment | added | ABBC | @TabesBridges and Dori: Thank you both, this is a point that obviously never clicked for me. Although I had all the pieces: I understood the birational picture and then understood that one could then consider moduli spaces, and so forth. My question now seems completely trivial, so thank you again for entertaining my ignorance and giving great answers :-) | |
| Jul 7, 2022 at 15:26 | comment | added | Dori Bejleri | I'm not an expert in the non-algebraic setting but my impression is that the MMP is conjectured to hold for Kähler manifolds as well albeit with different methods. I have no clue what could happen for non-Kähler compact manifolds though. | |
| Jul 7, 2022 at 15:20 | comment | added | Dori Bejleri | @ABBC I second what Tabes said. The point of my very long comment was to challenge the premise of the question at least in the algebraic setting. The classification question is fruitful and the MMP is a strategy to achieve this classification using the birational classification as the first step as Tabes illustrated. | |
| Jul 6, 2022 at 23:31 | comment | added | Tabes Bridges | @ABBC it's not that the question is not fruitful, it's really that we break the question into two pieces: classify up to birational equivalence, and then classify isomorphism classes within a birational equivalence class. At least for complex surfaces, we can basically answer this: every birational class contains a unique minimal model except for rational surfaces, for which there are infinitely many minimal models (the Hirzebruch surfaces $F_k$ for $k\ne 1$, and $\mathbb P^2$). Within a given birational class, any two models can be related by blowing curves up and down. | |
| Jul 6, 2022 at 21:49 | comment | added | ABBC | Thank you for this very detailed answer. I really appreciate it, and Serre's example that you link to is very interesting. I have to say that most of what you have written here, I was aware of previously, but I feel that my main question was not answered: My question is effectively asking for an illustrative example on why a classification of compact complex manifolds up to biholomorphism is not fruitful, while classification up to birational isomorphism (or bimeromorphism) is worth pursuing. Perhaps you have answered this, but explicit examples are what I'm mainly after. Thanks again :) | |
| Jul 6, 2022 at 4:29 | history | answered | Dori Bejleri | CC BY-SA 4.0 |