I'm reading about Complex Torus (H. Lange and C. Birkenhake, Complex Abelian Varieties - Chapter 1).
In the book it is written:
"In order to describe a complex totus $X=V/\Lambda$, choose bases $e_{1},...,e_{g}$ of $V$ and $\lambda_{1},...,\lambda_{2g}$ of the lattice $\Lambda$"
The author walks to set the period matrix for $X=V/\Lambda$.
I'm confused about the cardinality of generators of the lattice. Why $2g$ ??
I understand that for example:
Take $n$ vectors $v_{1},...,v_{g} \in \mathbb{C}^{g}$ that are linearly
independent over $\mathbb{C}$. Then the lattice $\Lambda$ generated by these vectors consists of all linear combinations
$a_{1}v_{1}+...+a_{g}v_{g}$, where $a_{1},...,a_{g} \in \mathbb{Z}$.
And here is my question: How can I say that the cardinality of generator set of lattice is $2g$?