Construction of the isomorphism $V^*\simeq H^0(V/L,\Omega^1_{V/L})$ for a complex vector space $V$ with lattice $L$ Le $V$ be a complex vector space and $L\subset V$ be a lattice in $V$. Then $T=V/L$ is by definition a complex torus with universal covering space $V$. In Beauville ch 5, it is stated that there is an isomorphism
$\delta:V^* \simeq H^0(T,\Omega_T^1)$
where $V^*$ is the dual, given explicitly as follows: let $x^*\in V^*$, then $x^*(v+\gamma)=x^*(v)+\text{constant}$, for all $v\in V,\gamma\in L$. From here we can see that $dx^*$ defines a global one form on the torus, $\delta x^*$.
Question: Why does $x^*(v+\gamma)=x^*(v)+\text{constant}$? Indeed this seems like a contradiction, as it seems to imply $x^*(\gamma)$ is constant as $\gamma$ varies, which seems to contradict linearity of $x^*$. Is this statement from Beauville correct? And if not, how does one construct the isomorphism?
 A: Let me denote tangent spaces by $\mathcal T$, with suitable decorations, so that $\Omega^1=\mathcal T^*$ with the same decorations.
What is  $\mathcal T(V)$ ?
The tangent space to the  vector space $V$ is  the product $\mathcal T(V)=V\times V$, i.e. at the point $a\in V$ we have $\mathcal T_a(V)=\{a\}\times V$.
And, given a vector $\gamma \in V$, the translation $$\theta_\gamma: V\to V:x\mapsto x+\gamma$$ has as tangent map at $a$:  $$\mathcal T_a(\theta_\gamma):\mathcal T_a(V)=\{a\}\times V \to \mathcal T_{a+\gamma}(V)=\{a+\gamma \}\times V:(a,v)\mapsto (a+\gamma,v)$$
The fact that $v$ does not change in this description of the tangent map allows us to answer the question:
What is  $\mathcal T(T)$ ?
It is simply $\mathcal T=T\times V$: the tangent space of the torus $T$ is trivial.
In other words, for any point $[a] \in T$ and any $\gamma \in L$ we have $$\mathcal T_{[a]}T=\mathcal T_{[a+\gamma]}T=\{[a]\}\times V=\{[a+\gamma ]\}\times V$$
And finally the required isomorphism is...
...the isomorphism $\delta:V^* \simeq H^0(T,\Omega_T^1)$ sending $\omega \in V^*$ to the section $s_\omega \in H^0(T,\Omega_T^1)= H^0(T,\mathcal T_T^*)$ defined by $s_{\omega} [a]([a],v)=\omega(v)$.
Et voilà!
