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?



A equivalent definition of abelian variety
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


