##
**Complete moduli in the presence of semiabelian group action.**
*(English)*
Zbl 1052.14017

Section \(1\) of the paper is devoted to summarize the results. We are going to extract some of them. A semiabelian variety is a group variety, \(G\), with an exact sequence of groups
\[
1\to T\to G \to A \to 1
\]
being \(T\) a torus and \(A\) an abelian variety.

Definition. A (semi)abelic pair \((P, \Theta)\) of degree \(d\) is a variety \(P\) together with an action of a (semi)abelian variety \(G\), satifying: (1) \(P\) is normal. (2) There are only finitely many orbits. (3) The stabilizer of the generic point is connected, reduced and lies in the toric part \(T\) of \(G\). \(\Theta\) is an effective ample Cartier divisor on \(P\) which does not contain any \(G\)-orbit \(d=h^0({\mathcal O}(\Theta))\).

\(X\) is a lattice \(X\simeq {\mathbb Z}^r\), \(i:Y\subset X\) a sublattice, \(\overline X\) a trivial \(X\)-torsor and \(\overline{ X}_{\mathbb R}:=\overline X\otimes_{\mathbb Z}{\mathbb R}\).

Theorem: 1. The moduli stack \({\mathcal A P}_{g,d}\) of abelic pairs, \((P, \Theta)\), of degree \(d\) is a separated Artin stack with finite stabilizers and it comes with a natural map of relative dimension \(d-1\) to the stack \({\mathcal A }_{g,d}\) of polarized abelian varieties.

2. \({\mathcal A P}_{g,d}\) has a coarse moduli space \({ A P}_{g,d}\) which is a separated scheme and comes with a natural projective map of relative dimension \(d-1\) to \({ A }_{g,d}\).

3. Over \({\mathbb Z}[1/d]\) the schemes \({ A P}_{g,d}\) and \({ A }_{g,d}\) are disjoint unions of components naturally labeled by \(g\)-dimensional \(1\)-cell complexes \(\overline{ X}_{\mathbb R}/Y\) with \(| X/Y| =d\).

The numerical type of a polarized toric variety is a lattice polytope \(Q\subset \overline{ X}_{\mathbb R}\simeq {\mathbb R}^r\). Let \((A,P,L)\) be a polarized abelic variety of degree \(d\), \(\operatorname{char} k\neq d\). Let \(\lambda :A\to A^t\) be the induced polarization on \(A\) with \(\text{ker} (\lambda)\simeq H\times \widehat H\), and say \(H=X/Y\). In this case the numerical type of \((P,L)\) is \(\overline{ X}_{\mathbb R}/Y\simeq {\mathbb R}^g/{\mathbb Z}^g\).

Theorem: 1. The component \(\overline {\mathcal A P}_{g,d}\) of the moduli stack of semiabelic pairs containing \({\mathcal A P}_{g,d}\) and pairs of the same numerical type is a proper Artin stack with finite stabilizers;

2. \({\mathcal A P}_{g,d}\) has a coarse moduli space \(\overline {A P}_{g,d} \) as a proper algebraic space;

3. The space \(\overline { A P}_{g,d}\) is naturally stratified according to the toric part of \((P, \Theta)\), and every stratum corresponds 1-to-1 to a cell decomposition of \(\overline{X}_{\mathbb R}/Y\) with \(\dim X=g\) and \(| X/Y| =d\), modulo symmetries.

The definition of semiabelic pairs is extended to stable semiabelic pairs.

Definition. A stable semiabelic pair, \((P, \Theta)\), of degree \(d\) is a variety \(P\) together with an action of a semiabelian variety, \(G\), of the same dimension and satifying the same conditions as in the definition above but now \(P\) is seminormal.

A classification of stable (toric) semiabelic pairs is made over closed fields and \({\mathbb C}\) in terms of sublattices, \(X\)-torsors, complexes of lattice polytopes….

Definition. A (semi)abelic pair \((P, \Theta)\) of degree \(d\) is a variety \(P\) together with an action of a (semi)abelian variety \(G\), satifying: (1) \(P\) is normal. (2) There are only finitely many orbits. (3) The stabilizer of the generic point is connected, reduced and lies in the toric part \(T\) of \(G\). \(\Theta\) is an effective ample Cartier divisor on \(P\) which does not contain any \(G\)-orbit \(d=h^0({\mathcal O}(\Theta))\).

\(X\) is a lattice \(X\simeq {\mathbb Z}^r\), \(i:Y\subset X\) a sublattice, \(\overline X\) a trivial \(X\)-torsor and \(\overline{ X}_{\mathbb R}:=\overline X\otimes_{\mathbb Z}{\mathbb R}\).

Theorem: 1. The moduli stack \({\mathcal A P}_{g,d}\) of abelic pairs, \((P, \Theta)\), of degree \(d\) is a separated Artin stack with finite stabilizers and it comes with a natural map of relative dimension \(d-1\) to the stack \({\mathcal A }_{g,d}\) of polarized abelian varieties.

2. \({\mathcal A P}_{g,d}\) has a coarse moduli space \({ A P}_{g,d}\) which is a separated scheme and comes with a natural projective map of relative dimension \(d-1\) to \({ A }_{g,d}\).

3. Over \({\mathbb Z}[1/d]\) the schemes \({ A P}_{g,d}\) and \({ A }_{g,d}\) are disjoint unions of components naturally labeled by \(g\)-dimensional \(1\)-cell complexes \(\overline{ X}_{\mathbb R}/Y\) with \(| X/Y| =d\).

The numerical type of a polarized toric variety is a lattice polytope \(Q\subset \overline{ X}_{\mathbb R}\simeq {\mathbb R}^r\). Let \((A,P,L)\) be a polarized abelic variety of degree \(d\), \(\operatorname{char} k\neq d\). Let \(\lambda :A\to A^t\) be the induced polarization on \(A\) with \(\text{ker} (\lambda)\simeq H\times \widehat H\), and say \(H=X/Y\). In this case the numerical type of \((P,L)\) is \(\overline{ X}_{\mathbb R}/Y\simeq {\mathbb R}^g/{\mathbb Z}^g\).

Theorem: 1. The component \(\overline {\mathcal A P}_{g,d}\) of the moduli stack of semiabelic pairs containing \({\mathcal A P}_{g,d}\) and pairs of the same numerical type is a proper Artin stack with finite stabilizers;

2. \({\mathcal A P}_{g,d}\) has a coarse moduli space \(\overline {A P}_{g,d} \) as a proper algebraic space;

3. The space \(\overline { A P}_{g,d}\) is naturally stratified according to the toric part of \((P, \Theta)\), and every stratum corresponds 1-to-1 to a cell decomposition of \(\overline{X}_{\mathbb R}/Y\) with \(\dim X=g\) and \(| X/Y| =d\), modulo symmetries.

The definition of semiabelic pairs is extended to stable semiabelic pairs.

Definition. A stable semiabelic pair, \((P, \Theta)\), of degree \(d\) is a variety \(P\) together with an action of a semiabelian variety, \(G\), of the same dimension and satifying the same conditions as in the definition above but now \(P\) is seminormal.

A classification of stable (toric) semiabelic pairs is made over closed fields and \({\mathbb C}\) in terms of sublattices, \(X\)-torsors, complexes of lattice polytopes….

Reviewer: Arturo Alvarez (Salamanca)

### MSC:

14D20 | Algebraic moduli problems, moduli of vector bundles |

14L30 | Group actions on varieties or schemes (quotients) |

14A20 | Generalizations (algebraic spaces, stacks) |