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I am trying to understand the theory of cubical structures and am interested in knowing if a disconnected commutative group variety whose identity component is a semi-abelian variety satisfies the theorem of the cube.

Recall that if $X$ is an abelian variety and $L$ is a line bundle on $X$ that is rigidified along the identity sectionn, then the Theorem of the Cube implies that the line bundle

$$\Theta(L) := m_{123}^{*}(L) \otimes m_{12}^{*}(L^{-1}) \otimes m_{13}^{*}(L^{-1}) m_{23}^{*}(L^{-1}) \otimes m_{1}^{*}(L) \otimes m_{2}^{*}(L) \otimes m_{3}^{*}(L)$$

admits a unique trivialization that makes

$$\Lambda(L) := m^{*}(L) \otimes p_1^{*}(L^{-1}) \otimes p_{2}^{*}(L^{-1})$$

into a symmetric biextension of of $X \times X$ by $\mathbb{G}_{m}$.

Here $m$ denotes the multiplication map, $p_i$ the projection maps, and $m_{\underline{i}}$ the morphism $X \times X \times X \to X$ given by summing the coordinates whose indices are in $\underline{i}$.

More generally, Breen proved this fact remains true when $X$ is a semi-abelian variety.

Does this statement remain true if we allow $X$ to have non-trivial component group?

If not, what is a example of a rigidified line bundle that does not have canonical cubical structure.

Does the theorem remain valid you rigidify along $1$ fixed point on every component (rather along the identity element)?

Added As BCnrd notes, over a more general base the formulas should be modified by adding the term $0^{*}(L^{\otimes \pm 1})$, which should be thought of as $m_{\emptyset}^{*}(L^{\otimes \pm 1})$. This (rigidified) line bundle is trivial when the base if a field, but not in general.

3 added 292 characters in body

I am trying to understand the theory of cubical structures and am interested in knowing if a disconnected commutative group variety whose identity component is a semi-abelian variety satisfies the theorem of the cube.

Recall that if $X$ is an abelian variety and $L$ is a line bundle on $X$ that is rigidified along the identity sectionn, then the Theorem of the Cube implies that the line bundle

$$\Theta(L) := m_{123}^{*}(L) \otimes m_{12}^{*}(L^{-1}) \otimes m_{13}^{*}(L^{-1}) m_{23}^{*}(L^{-1}) \otimes m_{1}^{*}(L) \otimes m_{2}^{*}(L) \otimes m_{3}^{*}(L)$$

admits a unique trivialization that makes

$$\Lambda(L) := m^{*}(L) \otimes p_1^{*}(L^{-1}) \otimes p_{2}^{*}(L^{-1})$$

into a symmetric biextension of of $X \times X$ by $\mathbb{G}_{m}$.

Here $m$ denotes the multiplication map, $p_i$ the projection maps, and $m_{\underline{i}}$ the morphism $X \times X \times X \to X$ given by summing the coordinates whose indices are in $\underline{i}$.

More generally, Breen proved this fact remains true when $X$ is a semi-abelian variety.

Does this statement remain true if we allow $X$ to have non-trivial component group?

If not, what is a example of a rigidified line bundle that does not have canonical cubical structure.

Does the problem theorem remain valid you rigidify along $1$ fixed point on every component (rather than just along the identity component)element)?

Added As BCnrd notes, over a more general base the formulas should be modified by adding the term $0^{*}(L^{\otimes \pm 1})$, which should be thought of as $m_{\emptyset}^{*}(L^{\otimes \pm 1})$. This (rigidified) line bundle is trivial when the base if a field, but not in general.

2 added 124 characters in body

I am trying to understand the theory of cubical structures and am interested in knowing if a disconnected commutative group variety whose identity component is a semi-abelian variety satisfies the theorem of the cube.

Recall that if $X$ is an abelian variety and $L$ is a line bundle on $X$ that is rigidified along the identity sectionn, then the Theorem of the Cube implies that the line bundle

$$\Theta(L) := m_{123}^{*}(L) \otimes m_{12}^{*}(L^{-1}) \otimes m_{13}^{*}(L^{-1}) m_{23}^{*}(L^{-1}) \otimes m_{1}^{*}(L) \otimes m_{2}^{*}(L) \otimes m_{3}^{*}(L)$$

admits a unique trivialization that makes

$$\Lambda(L) := m^{*}(L) \otimes p_1^{*}(L^{-1}) \otimes p_{2}^{*}(L^{-1})$$

into a symmetric biextension of of $G X \times G$ X$by$\mathbb{G}_{m}$. Here$m$denotes the multiplication map,$p_i$the projection maps, and$m_{\underline{i}}$the morphism$G X \times G X \times G X \to G$X$ given by summing the coordinates whose indices are in $\underline{i}$.

More generally, Breen proved this fact remains true when $X$ is a semi-abelian variety.

Does this statement remain true if we allow $X$ to have non-trivial component group?

If not, what is a example of a rigidified line bundle that does not have canonical cubical structure.

Does the problem remain valid you rigidify along $1$ point on every component (rather than just the identity component)?

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