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Given two complex tori $X_1$ and $X_2$, there is always a canonical isomorphism

$\widehat{X_1 \times X_2} \cong \widehat{X}_1 \times \widehat{X}_2$,

see for instance Birkenhake-Lange's book Complex Abelian Varieties, Exercise 11 page 43.

Indeed let us write $X_i=\mathbb{C}^{g_i}/\Gamma_i$, where $\Gamma_i$ is a lattice. Then

$X_1 \times X_2 \cong \mathbb{C}^{g_1+g_2}/ \Gamma_1 \times \Gamma_2$

and, by standard representation theory of abelian groups, the character group of $\Gamma_1 \times \Gamma_2$ coincides with the direct product of the character groups of $\Gamma_1$ and $\Gamma_2$.

Given two complex tori $X_1$ and $X_2$, there is always a canonical isomorphism

$\widehat{X_1 \times X_2} \cong \widehat{X}_1 \times \widehat{X}_2$,

see for instance Birkenhake-Lange's book Complex Abelian Varieties, Exercise 11 page 43.

Indeed let us write $X_i=V_i/\Gamma_i$, where $\Gamma_i$ is a lattice in the complex vector space $V_i$. Then

$X_1 \times X_2 \cong V_1 \times V_2/ \Gamma_1 \times \Gamma_2$

and, by standard representation theory of abelian groups, the character group of $\Gamma_1 \times \Gamma_2$ coincides with the direct product of the character groups of $\Gamma_1$ and $\Gamma_2$.

Given two complex tori $X_1$ and $X_2$, there is always a canonical isomorphism

$\widehat{X_1 \times X_2} \cong \widehat{X}_1 \times \widehat{X}_2$,

see for instance Birkenhake-Lange's book Complex Abelian Varieties, Exercise 11 page 43.

In fact, write $X_i=\mathbb{C}^{g_i}/\Gamma_i$, where $\Gamma_i$ is a lattice. Then

$X_1 \times X_2 \cong \mathbb{C}^{g_1+g_2}/ \Gamma_1 \times \Gamma_2$,

and, by standard representation theory of abelian groups, the character group of $\Gamma_1 \times \Gamma_2$ coincides with the direct product of the character groups of $\Gamma_1$ and $\Gamma_2$.

Given two complex tori $X_1$ and $X_2$, there is always a canonical isomorphism

$\widehat{X_1 \times X_2} \cong \widehat{X}_1 \times \widehat{X}_2$,

see for instance Birkenhake-Lange's book Complex Abelian Varieties, Exercise 11 page 43.

Indeed let us write $X_i=\mathbb{C}^{g_i}/\Gamma_i$, where $\Gamma_i$ is a lattice. Then

$X_1 \times X_2 \cong \mathbb{C}^{g_1+g_2}/ \Gamma_1 \times \Gamma_2$

and, by standard representation theory of abelian groups, the character group of $\Gamma_1 \times \Gamma_2$ coincides with the direct product of the character groups of $\Gamma_1$ and $\Gamma_2$.

Given two complex tori $X_1$ and $X_2$, there is always a canonical isomorphism

$\widehat{X_1 \times X_2} \cong \widehat{X}_1 \times \widehat{X}_2$,

see Birkenhake-Lange's book Complex Abelian Varieties, Exercise 11 page 43.

Given two complex tori $X_1$ and $X_2$, there is always a canonical isomorphism

$\widehat{X_1 \times X_2} \cong \widehat{X}_1 \times \widehat{X}_2$,

see for instance Birkenhake-Lange's book Complex Abelian Varieties, Exercise 11 page 43.

In fact, write $X_i=\mathbb{C}^{g_i}/\Gamma_i$, where $\Gamma_i$ is a lattice. Then

$X_1 \times X_2 \cong \mathbb{C}^{g_1+g_2}/ \Gamma_1 \times \Gamma_2$,

and, by standard representation theory of abelian groups, the character group of $\Gamma_1 \times \Gamma_2$ coincides with the direct product of the character groups of $\Gamma_1$ and $\Gamma_2$.