Proof. Since pullbacks commute with determinants we have

\begin{align*} f^*\delta (f_*L)=\delta (f^*f_*L). \end{align*}

The isogeny $f$ is a $\ker (f)$ -torsor. Therefore,

\begin{align*} f^*f_*L\cong \bigoplus _{\lambda \in \ker (f)} T_\lambda ^*L. \end{align*}

As translates of $L$ are algebraically equivalent, the assertion follows.

Proof. Being a locally free sheaf is an open condition by [Reference Huybrechts and LehnHL10, Lemma 2.1.8], as is the condition of having slope $H$ and rank $k\cdot n(H)$ . We may thus assume that $\mathcal{F}$ is a family of vector bundles of slope $H$ and rank $k\cdot n(H)$ .

By [Reference MukaiMuk78, Proposition 5.4], a vector bundle $E$ on $A$ is semi-homogeneous if and only if $\pi ^*E$ is a semi-homogeneous vector bundle on $A_H$ . If $E$ is of slope $H$ , then $\pi ^*E$ is of slope $\pi ^*H$ and by [Reference MukaiMuk78, Proposition 7.6] we have $\pi ^*H\in \textrm {NS}(A)$ . Replacing $A$ by $A_H$ we may thus assume, from the onset, that $H\in \textrm {NS}(A)$ .

Let ${\mathcal{P}}_H$ denote the universal line bundle on $A\times \textrm {Pic}^H(A)$ and let

\begin{align*} \begin{split} \Phi \colon D^b\big ({\textrm {Pic}}^H(A)\big )&\longrightarrow D^b(A) \\[5pt] {\mathcal{K}}^\bullet &\longmapsto \mathbf R p_{1*} (p_{2}^*{\mathcal{K}}^\bullet \otimes ^{\mathbf L} {\mathcal{P}}_H) \end{split} \end{align*}

denote the associated Fourier–Mukai transform. By Theorem2.4, semi-homogeneous bundles on $A$ of slope $H$ are precisely the successive extensions of elements of $\textrm {Pic}^H(A)$ . These correspond under $\Phi$ to the successive extension of skyscraper sheaves on $A$ , that is to zero-dimensional coherent sheaves on $A$ . Applying the inverse $\Phi ^{-1}$ to $\mathcal{F}$ (this can be done in families, see [Reference Bartocci, Bruzzo and RuipérezBBHR09, Section 1.2.1]), it suffices to prove the following: given an element ${\mathcal{K}}^\bullet \in D^b(\textrm {Pic}^H(A)\times _K S)$ , the set $U$ of $s\in S$ such that $\mathbf L i_s^*{\mathcal{K}}^\bullet$ is finite of length $k$ is open. Here, we denote by $i_s\colon \textrm {Pic}^H(A)\times _K \kappa (s) \to \textrm {Pic}^H(A)\times _K S$ the inclusion. To prove openness, it suffices to show that $U$ is both constructible and stable under generalization.

We first prove that $U$ is stable under generalization. Suppose that $s\in S$ specializes to $t\in U$ , let $R$ be the local ring of $t$ in $\overline {\{s\}}$ , and let $\iota \colon \textrm {Spec} R\to S$ denote the inclusion. Then after replacing $S$ by $\textrm {Spec} R$ and ${\mathcal{K}}^\bullet$ by $\mathbf L \iota ^*{\mathcal{K}}^\bullet$ , we may assume that $t$ is the unique closed point of $S$ and that $\mathbf L i_t^*{\mathcal{K}}^\bullet$ is of finite length $k$ . By [Reference HuybrechtsHuy06, Lemma 3.31], it follows that ${\mathcal{K}}^\bullet$ is isomorphic to a sheaf $\mathcal{L}$ that is flat over $S$ . Since Hilbert polynomials are locally constant in flat families, it follows that $i_s^*{\mathcal{L}}\cong \mathbf Li_s^*{\mathcal{K}}^\bullet$ is also of finite length $k$ .

To prove constructibility, by Noetherian induction it suffices to prove that $U\cap V$ is open for some non-empty open subset $V\subset S$ . By generic flatness, we may choose $V$ to be a non-empty open subset such that ${\mathcal{H}}^i({\mathcal{K}}^\bullet )\vert _V$ is flat over $V$ for all $i\in \mathbb{Z}$ . Then for all $s\in S$ we have ${\mathcal{H}}^i(\mathbf Li_s^*{\mathcal{K}}^\bullet )= i_s^*{\mathcal{H}}^i({\mathcal{K}}^\bullet )$ , and therefore the set $W$ consisting of all $s\in V$ such that $\mathbf Li_s^*{\mathcal{K}}^\bullet$ is a sheaf, is open. As ${\mathcal{K}}^\bullet$ is a flat sheaf over $W$ by construction, the set of all $s\in W$ such that $\mathbf L i_s^*{\mathcal{K}}^\bullet$ is a finite sheaf of length $k$ is open.

Proof. We have already seen that every bundle in $\mathcal{S}_H$ is isomorphic to $f_*(L)$ for some $L\in \textrm {Pic}^{f^*H}(A')(K)$ . If $f_*(L)\cong f_*(L')$ for two line bundles $L$ and $L'$ in $\textrm {Pic}^{f^*H}(A')(K)$ , then

\begin{align*} \bigoplus _{g\in {\textrm {Aut}}(A'/A)} g^*L\cong f^*f_*L \cong f^*f_* L'\cong \bigoplus _{g\in {\textrm {Aut}}(A'/A)}g^*L', \end{align*}

where the first and last isomorphisms exist because $f$ is an ${\textrm {Aut}}(A'/A)$ -torsor. It follows that $L\cong g^*L'$ for some $g\in {\textrm {Aut}}(A'/A)$ .

Proof. First note that since $\textrm {Pic}^{f^*H}(A')$ is projective and ${\textrm {Aut}}(A'/A)$ is finite, the geometric quotient $\textrm {Pic}^{f^*H}(A')\big /{\textrm {Aut}}(A'/A)$ exists by [SGA03, Exposé V, Section 1] or [Sta24, Tag 07S7]. Let $L\in \textrm {Pic}^{f^*H}(A')(K)$ . Then every other $K$ -point of $\textrm {Pic}^{f^*H}(A')$ is represented by $L\otimes f^*M$ for some $M\in \textrm {Pic}^0(A)$ . Suppose $g^*(L)\cong g^*(L\otimes f^*M)$ . Then $\pi ^*M\cong {\mathcal{O}}_{A_H}$ and, hence, $M\in \Sigma (H)$ . It follows that $f_*(L)\cong f_*(L\otimes f^*M)$ . Using Lemma 2.13, we conclude that $g^*$ induces a morphism $\textrm {Pic}^{f^*H}(A')/{\textrm {Aut}}(A'/A)\to \textrm {Pic}^{\pi ^*H}(A_H)$ that is bijective on $K$ -points. Since $\textrm {Pic}^{\pi ^*H}(A_H)$ is normal, this concludes the proof.

Proof. Let ${\mathcal{P}}_H$ denote the universal line bundle on $A'\times _K\textrm {Pic}^{f^*H}(A')$ (whose fiber over a point $[L]\in \textrm {Pic}^{f^*H}(A')(K)$ is $L$ ). Define

\begin{align*} {\mathcal{E}}_1= (f\times {\textrm {id}})_*{\mathcal{P}}_H. \end{align*}

This is a vector bundle on $ A\times _K\textrm {Pic}^{f^*H}(A')$ whose fiber over a point $[L]\in \textrm {Pic}^{f^*H}(A')(K)$ is the simple semi-homogeneous vector bundle $f_*L$ . Let

\begin{align*} \phi _1'\colon \textrm {Pic}^{f^*H}(A')\longrightarrow S_{H,1}(A) \end{align*}

be the associated morphism. For $\lambda \in {\textrm {Aut}}(A'/A)$ , the fiber of ${\mathcal{E}}_1$ over $T_\lambda ^*[L]=[T_\lambda ^*L]$ is isomorphic to the fiber over $[L]$ , because we have

\begin{align*} f_*(T_\lambda ^*L)\cong f_*(T_{-\lambda })_*L=f_*L. \end{align*}

It follows that $({\textrm {id}}\times T_{\lambda }^*)^* {\mathcal{E}}_1$ and ${\mathcal{E}}_1$ have isomorphic fibers over all points of $\textrm {Pic}^{f^*H}(A')$ . As $\textrm {Pic}^{f^*H}(A')$ is reduced, we can conclude from [Reference MukaiMuk78, Theorem 1.8] (see also [Reference Huybrechts and LehnHL10, Lemma 4.6.3] for the assumption on simplicity over the base) that there exists a line bundle ${\mathcal{L}}_\lambda$ on $\textrm {Pic}^{f^*H}(A')$ such that

\begin{align*} ({\textrm {id}}\times T_{\lambda }^*)^* {\mathcal{E}}_1\cong {\mathcal{E}}_1\otimes p_2^*{\mathcal{L}}_\lambda, \end{align*}

where $p_2$ denotes the projection to $\textrm {Pic}^{f^*H}(A')$ . This shows that $\phi _1'$ is ${\textrm {Aut}}(A'/A)$ -equivariant and thus descends, by Lemma 2.14, to a morphism

\begin{align*} \phi _1\colon \textrm {Pic}^{\pi ^*H}(A_H) \to S_{H,1}(A). \end{align*}

For $F\in {\mathcal{S}}_H$ , the morphism $\phi _1$ maps the point $\psi _H(F)\in \textrm {Pic}^{\pi ^*H}(A_H)(K)$ to the point of $\textrm {Pic}^{\pi ^*H}(A_H)$ corresponding to $F$ . We conclude that $\phi _1$ is bijective on $K$ -points. Moreover, the space $S_{H,1}(A)$ is the image of the proper space $\textrm {Pic}^{\pi ^*H}(A_H)$ and is therefore closed in the moduli space of semi-stable sheaves, as well as connected. By definition, it is also open, showing that it is a component. By [Reference GulbrandsenGul08, Proposition 3.1], the component $S_{H,1}(A)$ is smooth and, thus, automatically isomorphic to $M_{H,1}(A)$ . The bijectivity of $\phi _1$ on $K$ -points implies that $\phi _1$ is an isomorphism by Zariski’s main theorem.

For the case $k\gt 1$ , consider the vector bundle

\begin{align*} {\mathcal{E}}_k:= \bigoplus _{1=1}^k p_i^*{\mathcal{E}}_1 \end{align*}

on $(\textrm {Pic}^{f^*H}(A))^k\times _K A$ . The ${\textrm {Aut}}(A'/A)$ -action on $\textrm {Pic}^{f^*H}(A)$ and the $S_k$ -action on $(\textrm {Pic}^{f^*H}(A))^k$ combine to an $(S_k\ltimes {\textrm {Aut}}(A'/A)^k)$ -action and it follows from the $k=1$ case that the morphism

\begin{align*} \phi _k'\colon (\textrm {Pic}^{f^*H}(A))^k\longrightarrow S_{H,k}(A) \end{align*}

induced by ${\mathcal{E}}_k$ descends to a morphism

\begin{align*} \phi _k \colon \textrm {Sym}^k(M_{H,1}(A))=(\textrm {Pic}^{f^*H}(A))^k /(S_k\ltimes {\textrm {Aut}}(A'/A)^k) \longrightarrow S_{H,k}(A). \end{align*}

By Theorem2.4 and the $k=1$ case, the morphism $\phi _k$ is bijective on $K$ -points. The statement about being a component follows as in the $k=1$ case. That $\phi _k$ induces an isomorphism between $\textrm {Sym}^k(M_{H,1}(A))$ and the normalization $M_{H,k}(A)$ follows from Zariski’s main theorem.

Proof. The subgroup $\Lambda _{H}^{\mathbb{Z}}$ of $\Lambda$ is precisely the kernel of the composite

\begin{align*} \Lambda \xrightarrow {\,\,\,H\,\,} M_{\mathbb{Q}}\longrightarrow M_{\mathbb{Q}}/M \end{align*}

Since every element in $M_{\mathbb{Q}}/M$ has finite order, the quotient $\Lambda /\Lambda _{H}^{\mathbb{Z}}$ is a finitely generated abelian group, all of whose elements have finite order. Therefore, the quotient $\Lambda /\Lambda _{H}^{\mathbb{Z}}$ is finite.

Proof. Let $\gamma =l\theta \in l\Lambda$ and let $\lambda \in \Lambda _{H}^{\mathbb{Z}}$ . Then we have

\begin{align*} [\gamma ,\lambda ]_H= [\theta ,\lambda ]_{lH}=[\lambda ,\theta ]_{lH}=[\lambda ,\gamma ]_H. \end{align*}

It immediately follows that $l\Lambda \subseteq \Lambda _{H}^{c}$ .

Proof. Let $\mathcal{G}$ denote the category fibered in groupoids over the small site $O(X)$ of open subsets of $X$ consisting of pairs $(Y\xrightarrow {\pi }U ,L)$ , where $\pi$ is a free cover of degree $r$ of an open subset $U$ of $X$ and $L$ be a tropical line bundle on $Y$ . Note that this is equivalent to the category consisting of pairs $(Y\xrightarrow {\pi }U,L)$ , where $\pi$ is a cover of the underlying topological space $U$ and $L$ is a $\pi ^{-1}{\rm Aff}_U$ -torsor. Since covering spaces and torsors glue, the fibered category $\mathcal{G}$ is a stack over $O(X)$ . Let $I\in {\mathcal{G}}(X)$ denote the trivial degree- $r$ cover of $X$ equipped with the trivial torsor. Then every element of $\mathcal{G}$ is locally isomorphic to $I$ . In other words, the stack $\mathcal{G}$ is a neutral gerbe on $O(X)$ . It follows that there is an equivalence of gerbes

\begin{align*} {\mathcal{G}} \xrightarrow {\,\sim \,} {\rm Tors}\big (\underline{{\rm Isom}}(I,I)\big )\quad \textrm {given by}\ J\longmapsto \underline{{\rm Isom}}(I,J), \end{align*}

where ${\rm Tors}\big (\underline{{\rm Isom}}(I,I)\big )$ denotes the gerbe of $\underline{{\rm Isom}}(I,I)$ torsors. In particular, we obtain an equivalence of categories between ${\mathcal{G}}(X)$ and the category of torsors over $\underline{{\rm Isom}}(I,I)= S_r\ltimes {\rm Aff}_X^r$ , which are precisely the tropical vector bundles of rank $r$ on $X$ .

Proof. By definition of sums, the cover associated to a sum $E_1\oplus E_2$ of tropical vector bundles is never connected. Therefore, if $Y$ is connected, then $E(Y\to X,L)$ is indecomposable. To complete the proof, it suffices to show that every tropical vector bundle can be expressed uniquely as a sum of tropical vector bundles corresponding to connected covers. But this is clear because the connected components of a cover are uniquely determined by the cover.

Proof. The projection $N_{\mathbb{R}}\to \Sigma$ exhibits $N_{\mathbb{R}}$ as the universal covering space of $\Sigma$ and induces an isomorphism between $\Lambda$ and the fundamental group $\pi _1(\Sigma ,0)$ of $\Sigma$ . By the classification of covering spaces, the connected finite cover $f$ defines a subgroup $\Lambda '$ of $\pi _1(\Sigma ,0)=\Lambda$ of finite index and $\Sigma '$ can be recovered from $\Lambda '$ as the quotient $N_{\mathbb{R}}/\Lambda '$ .

Proof. The equality

\begin{align*} T_x^{-1}(f_*L) =(T_{-x})_*(f_*L) =f_*\big ((T_{-y})_*L\big )= f_*(T_y^{-1} L), \end{align*}

is a consequence of $T_{-x}\circ f=f\circ T_{-y}$ .

Proof. Tropical vector bundles are equivalent to isomorphism classes of tropical line bundles on finite free covers. Therefore, the pushforward $f_*L(H,l)$ is isomorphic to $f_*L(H',l')$ if and only if there exists an automorphism of $f$ along which $L(H,l)$ pulls back to $L(H',l')$ . The automorphisms of $f$ are precisely the translations by the classes of elements $\lambda \in \Lambda$ and the pullback of $L(H,l)$ via a translation by $\lambda$ is given by $L\big (H, l- H(\lambda )\big )$ . This is isomorphic to $L(H',l')$ if and only if $H=H'$ and $l-l'-H(\lambda )\in M$ .

Proof. For every $i\in \{1,\ldots , k\}$ there exists a finite-index sublattice $\Lambda _i$ of $\Lambda$ , a symmetric morphism $H_i\colon \Lambda _i\to M$ , and a linear map $l_i\colon \Lambda _i\to {\mathbb{R}}$ such that $E_i =(\pi _i)_*L(H_i, l_i)$ , where $\pi _i\colon N_{\mathbb{R}}/\Lambda _i\to N_{\mathbb{R}}/\Lambda$ is the quotient map. For $x\in N_{\mathbb{R}}$ , we have

\begin{align*} T_{\overline x}^*E \cong \bigoplus _{i=1}^k L\big (H_i, l_i-H_i(x)\big ) \end{align*}

by Lemma 4.14. By Lemma 4.15, this is isomorphic to $E\otimes L(0, l)$ for some $l\colon \Lambda \to {\mathbb{R}}$ if and only if there exists a permutation $\sigma$ on $\{1,\ldots , k\}$ with $\Lambda _i=\Lambda _{\sigma (i)}$ and $H_i=H_{\sigma (i)}$ as well as for each $1\leqslant i\leqslant k$ elements $\lambda _i\in \Lambda$ and $m_i\in M$ with

(3) \begin{equation} l_i-H_i(x)= l_{\sigma (i)}+H_i(\lambda _i)+m_i+l. \end{equation}

If $E$ is homogeneous, we can take $l=0$ and see that there are only countably many choices for $l_i-H_i(x)$ as $i$ varies over $\{1,\ldots , k\}$ and $x$ over $N_{\mathbb{R}}$ . As every one-dimensional subspace of $N_{\mathbb{R}}$ is uncountable, this implies that $H_i(x)=0$ for all $x\in N_{\mathbb{R}}$ and $i\in \{1,\ldots , k\}$ , and, hence, $\delta (E_i)= H_i=0$ for all $i\in \{1,\ldots , k\}$ . Conversely, if $H_i=0$ for all $1\leqslant i\leqslant k$ , then we can take $\sigma ={\textrm {id}}$ as well as $l$ and all $\lambda _i$ and $m_i$ to be zero in (3), showing that $E$ is homogeneous.

If $E$ is semi-homogeneous, we take differences of the Equations (3) for given $1\leqslant i,j\leqslant n$ . For every $x \in N_{\mathbb{R}}$ there exists a permutation $\sigma$ of $\{1,\ldots , k\}$ and elements $\lambda \in \Lambda$ and $m\in M$ with

\begin{align*} l_i-l_j +\big (H_j(x)-H_i(x)\big )= l_{\sigma (i)}-l_{\sigma (j)}+ H(\lambda )+m. \end{align*}

As there is only a countable set of choices for the triple $(\sigma , \lambda , m)$ , there exists a set $S\subseteq N_{\mathbb{R}}$ whose affine span is $N_{\mathbb{R}}$ such that $H_j(s)-H_i(s)$ is independent of the choice of $s\in S$ . But this implies that

\begin{align*} \delta (E_i)=H_i=H_j=\delta (E_j). \end{align*}

Conversely, if all $H_i$ coincide, then we can take $\sigma ={\textrm {id}}$ as well as $l=-H_1(x)$ , and all $\lambda _i$ and $m_i$ equal to zero in (3) and conclude that $E$ is semi-homogeneous.

Proof. We have $\textrm {Pic}^0(N_{\mathbb{R}}/\Lambda _i)= {\textrm {Hom}}(\Lambda _i,{\mathbb{R}})/M$ and the pullback is induced by the inclusion $\Lambda _1\to \Lambda _2$ . Since its dual ${\textrm {Hom}}(\Lambda _2,{\mathbb{R}})\to {\textrm {Hom}}(\Lambda _1,{\mathbb{R}})$ is a bijection that is the identity on $M$ , the assertion follows.

Proof. We have $(f\circ g)_*g^*L\cong (f\circ g)_*g^*M$ if and only if there exists an automorphism $\phi$ of $f\circ g$ such that $\phi ^*g^* L\cong g^* M$ . Similarly, we have $f_*L\cong f_*M$ if and only if there exists an automorphism $\psi$ of $f$ such that $\psi ^*L\cong M$ . Any automorphism on a connected cover $h$ of $\Sigma$ is induced by an element $\lambda \in \pi _1(\Sigma ,0)$ ; we denote the automorphism induced by $\lambda$ by $\phi _{h,\lambda }$ . Thus, we have $(f\circ g)_*g^*L\cong (f\circ g)_*g^*M$ if and only if there exists $\lambda \in \pi _1(A,0)$ with $\phi _{f\circ g,\lambda }^*g^L\cong g^*M$ . Since $g\circ \phi _{f\circ g,\lambda }= \phi _{f,\lambda }\circ g$ , this is equivalent to the existence of $\lambda$ with $\phi _{f,\lambda }^*L\cong M$ by Lemma 4.17. This, in turn, is equivalent to $f_*L\cong f_*M$ .

Proof. All underlying covers of indecomposable $\Gamma$ -compatible tropical vector bundles can be dominated by the cover $\pi \colon N_{\mathbb{R}}/\Gamma \to N_{\mathbb{R}}/\Lambda$ , so by Lemma 4.18 (see also Remark 4.20) the set of equivalence classes of $\Gamma$ -compatible indecomposable tropical vector bundles on $N_{\mathbb{R}}/\Lambda$ of slope $H$ is in natural bijection with the set of isomorphism classes of tropical vector bundles that appear as pushforwards of tropical line bundles of Néron–Severi class $H$ on $N_{\mathbb{R}}/\Gamma$ along $\pi$ . This, in turn, is in bijection with the dual abelian variety of $N_{\mathbb{R}}/\Gamma$ modulo the automorphism group $\Lambda /\Gamma$ of the cover, that is with

\begin{align*} \big ({\textrm {Hom}}(\Gamma ,{\mathbb{R}})/M\big )\big /(\Lambda /\Gamma ). \end{align*}

A class $\overline \lambda \in \Lambda /\Gamma$ acts on $N_{\mathbb{R}}/\Gamma$ as translation by $\lambda$ . On the level of factors of automorphy we have described the action of $\lambda$ on tropical line bundles of Néron–Severi class $H$ by

\begin{align*} (H,l)\longmapsto \big (H, l-H(\lambda )\big ). \end{align*}

This allows us to identify the quotient $\big ({\textrm {Hom}}(\Gamma ,{\mathbb{R}})/M\big )\big /(\Lambda /\Gamma )$ with ${\textrm {Hom}}(\Gamma , {\mathbb{R}})/M'$ .

Proof.

(4) \begin{equation} \langle \lambda ' , m\rangle =\big \langle x, H(\lambda ')\big \rangle \end{equation}

for all $\lambda '\in \Lambda '$ . The kernel of $f$ is given by $\Lambda /\Lambda '$ . So if $\lambda \in \Lambda _{H}^{\mathbb{Z}}$ with $[\lambda ,\lambda ']_H=[\lambda ',\lambda ]_H$ for all $\lambda '\in \Lambda$ , then we can take $x=\lambda$ and $m=H(\lambda )$ in (4) and see that the class $\overline \lambda$ in $T^{{\rm an}}/\Lambda '$ satisfies $\overline \lambda \in \ker (f)\cap K(L)$ .

Conversely, let $\lambda \in \Lambda$ with $\overline \lambda \in \ker (f)\cap K(L)$ . Then by virtue of (4) there exists $m\in M$ such that

\begin{align*} \langle \lambda ' , m\rangle =\big \langle \lambda , H(\lambda ')\big \rangle \end{align*}

for all $\lambda '\in \Lambda '$ . Because $H\in \textrm {NS}(A)_{\mathbb{Q}}$ , for a sufficiently large integer $k$ we have

\begin{align*} \langle \lambda ', km\rangle = \big \langle \lambda , kH(\lambda ')\big \rangle =\big \langle \lambda ', kH(\lambda )\big \rangle \end{align*}

for all $\lambda '\in \Lambda '$ . Applying the valuation on both sides and using that $\mathbb{R}$ is torsion-free allows us to conclude that $H(\lambda )= m\in M$ , from which we conclude that $\lambda \in \Lambda _{H}^{\mathbb{Z}}$ . Plugging in $H(\lambda )$ for $m$ above, we see that

\begin{align*} [\lambda ',\lambda ]_H=\big \langle \lambda ', H(\lambda )\big \rangle =\big \langle \lambda , H(\lambda ')\big \rangle =[\lambda ,\lambda ']_H \end{align*}

for all $\lambda '\in \Lambda '$ .

By our description of $\ker (f)\cap K(L)$ , we have $\ker (f)\cap K(L)=0$ if and only if there exists no sublattice of $\Lambda _{H}^{\mathbb{Z}}$ that is strictly larger than $\Lambda '$ and on which $H$ defines a symmetric bilinear form. On the other hand, by [Reference MukaiMuk78, Proposition 5.6], the vector bundle $f_*L$ is simple if and only if $\ker (f)\cap K(L)=0$ , completing the proof.

Proof. If $\Lambda _{H}^{c} = \Lambda _{H}^{\mathbb{Z}}$ , we can take $\Lambda '= \Lambda _{H}^{\mathbb{Z}}$ . Otherwise, by definition of $\Lambda _{H}^{c}$ , there exists $\delta \in \Lambda _{H}^{\mathbb{Z}}$ such that $[\delta ,\theta ]_H\neq [\theta ,\delta ]_H$ . Moreover, the restriction of $H$ to $\Lambda _{H}^{c}+\mathbb{Z}\delta$ is symmetric and by Lemma 3.6, $\Lambda _{H}^{c}+\mathbb{Z}\delta$ has finite index in $\Lambda$ . Let $\Lambda '$ be maximal among the sublattices of $\Lambda$ that contain $\Lambda _{H}^{c}+\mathbb{Z}\delta$ and on which $H$ is symmetric. Let $L$ be a line bundle on $B:= N\otimes _{\mathbb{Z}}{\mathbb{G}}_m^{{\rm an}}/\Lambda '$ of Néron–Severi class $H$ , and let $f\colon B\to A$ denote the quotient map. By construction, we have $\delta (f_*L)=H$ , and by Proposition 5.2 it follows that $\Lambda '$ is $H$ -admissible. Since $H$ is symmetric on $\Lambda '$ and $\delta \in \Lambda '$ , we must have $\theta \notin \Lambda '$ .

Proof. By Proposition 5.2, the lattice $\Lambda _{H}^{c}$ is contained in the intersection, and by Proposition 5.3 the opposite inclusion holds as well.

Proof. Since $f_i$ is a $\Lambda /\Lambda _i$ -torsor, we have

\begin{align*} f_i^*f_{i*}L_i\cong \bigoplus _{\overline \lambda \in \Lambda /\Lambda _i} T_{\overline \lambda }^*L_i. \end{align*}

Let $\lambda _1,\ldots ,\lambda _k\in \Lambda$ be a set of representatives for $\Lambda /\Lambda _2$ , and let $h\colon T^{{\rm an}}/\Lambda _{H}^{c} \to A$ denote the projection. Then we have

\begin{align*} \bigoplus _{\overline \lambda \in \Lambda /\Lambda _1} g_1^*T_{\overline \lambda }^*L_1 \cong h^*f_{1*}L_1 \cong h^*f_{2*}L_2 \cong \bigoplus _{\overline \lambda \in \Lambda /\Lambda _2} g_2^*T_{\overline \lambda }^*L_2. \cong \bigoplus _{i=1}^k T_{\overline \lambda _i}^*g_2^*L_2. \end{align*}

Therefore, we have $T_{\overline \lambda _i}^*g_2^*L_2 \cong g_1^*L_1$ for some $1\leqslant i\leqslant k$ by the Krull–Schmidt theorem.