# Complex torus and integral of forms

Start with a complex torus $X$ of dimension $n$ and a basis $e_1,...,e_{2n}$ for its first integral cohomology.

Let $f_1,...f_{2n}$ be the dual basis vectors, so they are in $H^{1}(X,\mathbb Z)^*$. But this is isomorphic to $H^{1}(X^{*},\mathbb Z)$, with $X^{*}$ the dual torus.

Let $I$ denote a subset of {1,...,2n} and denote with $e_I$ the wedge product of the basis vectors $e_i$ with respect to $I$ (where you order them also succedingly by size). Let $K$ be the complement of $I$ in {1,...,2n}.

Now my question: is the integral $\int_X e_I\wedge e_K$ the same as $\int_{X^*} f_K \wedge f_I$ (observe the order)?

Of course one considers the integrand as a $2n-$form on the torus resp. it's dual by regarding the cohomology with coefficients in $\mathbb C$.

-