I have trouble with the following problem
Let $V$ be a complex vector space, of real dimension $2d$, and let $L$ be a lattice in $V$, that is the abelian group generated by a basis of $V$ as a real vector space. It is a free abelian group of rank $2d$, by definition.
Assume that I've been given $\alpha$ an integral alternated $2$-form, that is $\alpha:L\times L\to \mathbb{Z}$, with $\alpha(u,v)=−\alpha(v,u)$, that is non degenerate.
I'd like to find $L′$ a sub-lattice of $L$ of rank $d$ such that $\alpha(L'×L')=0$ and $(\mathbb{R}.L′)∩L=L′$.
On the book I'm reading they just say "choose such an $L'$".
Thank you!