Welcome to MathOverflow

Question

2

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?

flag	asked Jan 13 2011 at 12:09 Quanting Zhao 73●5

1

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 2011 at 12:35

Thank you for you comments. – Quanting Zhao Jan 15 2011 at 2:18

Sebastian · Accepted Answer · 2011-01-13 13:29:29Z UTC

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

A equivalent definition of abelian variety

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.