The following question was motivated by one of the earliest exercises of [Complex Abelian Variaties by Birkenhake and Lange](http://books.google.ca/books/about/Complex_Abelian_varieties.html?id=MOW2gEP7HIkC) 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 $V.$



Explicit way to construct simple complex tori/abelian varieties of dimension at least 2
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](http://archive.numdam.org/ARCHIVE/AIF/AIF_1982__32_4/AIF_1982__32_4_25_0/AIF_1982__32_4_25_0.pdf) 


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.


