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we know that a complex torus,which is algebraic,is called abelian variety.Recently i see another definition.Let $\mathbb{C}^n/\Lambda$ be a complex torus,where $\Lambda$ is a lattice of $\mathbb{C}^n$.$\mathbb{C}^n/\Lambda$ is abelian variety when there is a real skew-symmetric bilinear $E$ form on $\mathbb{C}^n$,satisfying

1)$E(iX,Y)=E(iY,X)$ for $X,Y\in\mathbb{C}^n$

2)$E(iX,X)>0$ for nonzero $X\in\mathbb{C}^n$

3)$E(X,Y)\in\mathbb{Z}$ for $X,Y\in\Lambda$

How can i see the equivalence?

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These conditions are called the Riemann conditions, and the equivalence is a big theorem to which many famous mathematicians of the 19th century contributed. You should look at Mumford's book an abelian varieties, for example. –  François Brunault Jan 13 '11 at 12:35
    
Thank you for you comments. –  Quanting Zhao Jan 15 '11 at 2:18

1 Answer 1

up vote 4 down vote accepted

In a complex torus it is (in principle) simple to relate Dolbeaut and deRham (with values in $\Z$ ) cohomologies. Therefore, under your assumptions, one can write down explicitly a holomorphic bundle (the theta bundle) which is postive. This bundle can be written down if only condition 1) and 3) are satisfies, but, it (and its powers) has holomorphic sections only under condition 2). In fact, the third power of the theta bundle gives an projective embedding (Lefschetz theorem), and by CHow it is algebraic. A nice reference for that is Griffiths and Harris

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