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The following question was motivated by one of the earliest exercises of Complex Abelian Variaties by Birkenhake and Lange during my presentation last year.

It can be shown that any complex torus $X$ $(=V/\Lambda$, where $V$ is a complex vector space and $\Lambda$ is lattice of maximal real dimension in $V)$ admits at most countably many complex subtori.

My question:

Is there sort of algorithm s.t. one could find simple (not admitting any non-trivial complex subtorus) complex tori of dimension $\geq 2?$ how about simple abelian varieties of dimension $\geq 2?$

Note that, $X$ admits a complex subtorus of dimension $g'$ if and only if there exists a subgroup $\Lambda' \subset \Lambda$ of rank $2g'$ s.t. the image of the canonical map $\Lambda' \otimes \mathbb{R} \to V$ is a complex subvector space of $V.$

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2 Answers 2

up vote 8 down vote accepted

Recall that the Néron-Severi group of a complex manifold $X$ is the subgroup of $NS(X)\subset H^2(X, \mathbb Z) $ consisting of first Chern classes of holomorphic line bundles on $X$.
More algebraically, it is the quotient group $PicX/Pic_0X$, as results from the exact sequence

$$ 0\to Pic_0X \to PicX \stackrel {c_1}{\to }NS(X) \to 0 $$

Since a torus is Kählerian, it will have no divisor at all if its Néron-Severi group is zero.
You can find an explicit calculation of the Picard number $\rho (X)=rank_ {\mathbb Z}NS(X)$ of 2-dimensional tori here, in the Appendix

You will see there for example a calculation of the Picard number of the torus determined by the lattice in $\mathbb C^2$ of matrix

$$\begin{pmatrix} 1&0&ip&ir\\ 0&1&iq & is \end{pmatrix} \quad \quad (p,q,r,s \in \mathbb R) $$

To give a completely explicit example, if $ \; p=1,r=\sqrt 2, q=\sqrt 3, s=\sqrt 5 \;$ then the corresponding (highly non algebraic!) torus has no holomorphic divisor (=curve) whatsoever.

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That's great! so how can we construct the period matrices of simple complex tori for any dimension $g \geq 2?$ –  Ehsan M. Kermani Dec 18 '11 at 21:42
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You could try to generalize the easy calculations done for dimension $2$ in the reference to higher dimensions . You could also have a look at Chapter VIII of Debarre's [Complex Tori and Abelian Varieties][1] and the references given there. [1]:ams.org/bookstore-getitem/item=SMFAMS-11 –  Georges Elencwajg Dec 18 '11 at 23:14
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I don't really know what "some sort of algorithm" means, but here is a source of examples of simple abelian varieties. As you probably know, if $L$ is a lattice in $\mathbf{C}$ then $\mathbf{C}/L$ is an abelian variety (of dimension 1). Here are some examples of $L$ coming from arithmetic: take an imaginary quadratic field $K$ living in the complexes, and let $L$ be the ring of integers of $K$.

This construction generalises. Let $K$ be a "CM field", i.e. a totally imaginary quadratic extension of a totally real field $F$. Let $L$ be the integers of $K$. The embeddings $F\to\mathbf{R}$ extend to embeddings $K\to\mathbf{C}$ and the resulting map $L\to\mathbf{C}^n$, $n=\frac{1}{2}[K:\mathbf{Q}]$ has a Riemann form by some basic results in arithmetic. The resulting abelian varieties $\mathbf{C}^n/L$ are typically simple (and have endomorphism ring containing $L$, so quite big). Google for "CM abelian variety" for more information.

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Thanks for your response. That does not really answer my question. I think, I should have been more precise. By algorithm, I mean how to construct the period matrices associated with simple complex tori/abelian varieties. Moreover, the method you described was explained in the above given reference. –  Ehsan M. Kermani Dec 18 '11 at 21:39
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