Reference for Complex Abelian Varieties - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T10:05:23Z http://mathoverflow.net/feeds/question/121555 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/121555/reference-for-complex-abelian-varieties Reference for Complex Abelian Varieties mkreisel 2013-02-12T04:42:42Z 2013-02-12T05:09:24Z <p>I am looking for a reference which explains how theta functions, algebraically independent meromorphic functions, and line bundles all fit together in the context of complex tori. More explicitly, given the right kind of Hermitian form on \$\mathbb{C}^g\$ with respect to some lattice \$L\$ I'd like an explicit construction of the g independent meromorphic functions on \$\mathbb{C}^g/L\$, which is something that Mumford, for example, does not give.</p> http://mathoverflow.net/questions/121555/reference-for-complex-abelian-varieties/121558#121558 Answer by Abhinav Kumar for Reference for Complex Abelian Varieties Abhinav Kumar 2013-02-12T05:09:24Z 2013-02-12T05:09:24Z <p>Perhaps not in his "Abelian varieties" book, but certainly in his "Tata lectures on Theta" Mumford describes this setup. See Chapter II of book 1. Another reference is Birkenhake and Lange's book "Complex abelian varieties" (see Chapters 3 and 8, for instance). By the way, I don't think you generally construct just \$g\$ independent meromorphic functions; you construct a whole bunch more of them - enough to embed the abelian variety into projective space.</p>