Timeline for Intuition for polarized Hodge structures
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Oct 18, 2017 at 14:41 | vote | accept | Saal Hardali | ||
| Oct 9, 2017 at 20:59 | comment | added | Saal Hardali | @nfdc23 I was reading claire voisin's book about hodge theory and so far i felt everything was very useful and motivated but when the definition of polarized hodge structure appeared it made me a bit confused. I was wondering how I should interpret it, as it looked like a convoluted way to encode the cup product. That's what made me ask the question. | |
| Oct 9, 2017 at 20:52 | comment | added | nfdc23 | Maybe this is a naive question, but what is your reason for trying to learn about polarized Hodge structures if you don't have a purpose for using them to do or understand something of prior interest? And are you aware of the importance of the more geometric incarnation of polarizations in the algebro-geometric and arithmetic study of abelian varieties (such as for the analytic moduli spaces thereof, vs. without a polarization; e.g., see #1 in Donu Arapura's answer below)? | |
| Oct 9, 2017 at 17:40 | answer | added | Donu Arapura | timeline score: 4 | |
| Oct 9, 2017 at 17:30 | comment | added | abx | And forget the cup-product? It is fundamental in algebraic geometry, already for curves (giving the principal polarization of the Jacobian), for surfaces (look at any text on K3 surfaces), etc. | |
| Oct 9, 2017 at 17:03 | history | edited | Saal Hardali | CC BY-SA 3.0 |
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| Oct 9, 2017 at 16:58 | comment | added | Saal Hardali | @abx That's exactly the one I'm reading right now. Can you give a specific reference inside that discusses motivation for polarization? What is suggested there is that the kahler form itself provides a polarization, I can't find anywhere motivation for defining this thing however, why not just define a something which is hodge structure together with a leftchetz operator for instance? | |
| Oct 9, 2017 at 16:45 | review | Close votes | |||
| Oct 10, 2017 at 7:53 | |||||
| Oct 9, 2017 at 16:29 | comment | added | abx | You'll find all the answers to your question in Hodge theory and Complex algebraic geometry by C. Voisin. | |
| Oct 9, 2017 at 15:21 | comment | added | Saal Hardali | @nfdc23 What specifically? Do you have a helpful suggestion for a particular topic/construction which can motivate the definition of a polarization? | |
| Oct 9, 2017 at 15:07 | comment | added | nfdc23 | Learn about complex-analytic projective geometry, since that is what provides the examples that motivate the abstract definitions. Also see Deligne's papers on Hodge structures. | |
| Oct 9, 2017 at 14:39 | history | asked | Saal Hardali | CC BY-SA 3.0 |