There exist many beautifull texts on that, on the basics about and the startling arithmetic application of the Picard-Fuch eq., I recommend to look into Clemens' "Scrap book of complex curve theory" and Katz'articlesKatz' articles, esp. "On the differential equations satisfied by period matrices" (probably in numdam). General infos are in Brieskorn's "plane algebraic curves" and in Griffiths-Harris. I don#t remember in the moment don't know about a connection to Gromov-Witten theory. The curiosity in Periods of algebraic varieties comes probably from that they are special numbers like pi or e coming from a natural process, satisfying some set of relations and having transcentality properties. Then, Manin showed that the diff.-eq. they satisfy encodes arithmetic infos too (via "Gaus-Manin connection"). Further one wonders if one can produce with periods like with e and pi interesting algebraic numbers. There is a very beautifull article by Zagier "Periods" on the web somewhere.
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There exist many beautifull texts on that, on the basics about and the startling arithmetic application of the Picard-Fuch eq., I recommend to look into Clemens' "Scrap book of complex curve theory" and Katz'articles, esp. "On the differential equations satisfied by period matrices" (probably in numdam). General infos are in Brieskorn's "plane algebraic curves" and in Griffiths-Harris. I don#t remember in the moment a connection to Gromov-Witten theory. |
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