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Subvarieties of semiabelian varieties. (English) Zbl 0814.14041
Recall that a group variety $$A$$ over a field $$K$$ is called a semiabelian variety if it fits in an exact sequence $$0 \to T \to A \to B \to 0$$, where $$B$$ is an abelian variety and $$T$$ is a torus, that is, after extending to the algebraic closure $$T$$ becomes a product of multiplicative groups: $$T \otimes_ K \overline{K} \simeq G^ d_ m$$. Analogously, we say that a commutative complex group $$A$$ is a semitorus if it is an extension of a compact complex torus by $$(\mathbb{C}^*)^ n$$, or equivalently, a quotient of $$(\mathbb{C}^*)^ g$$ by a discrete subgroup. The geometric results in this paper can be summarized by the following theorem 1.
Let $$X \subset G$$ be a reduced, irreducible, closed subvariety of $$G$$, where $$G$$ is either a complex group or an algebraic group over an algebraically closed field. Let $$Z(X) = \{x \in X \mid \exists B$$, $$\dim B > 0$$, $$B$$ a subgroup, $$xB \subset X\}$$ be the Mordell exceptional locus on $$X$$. Then $$Z(X)$$ is a closed subvariety of $$X$$.
Theorem 2. Let $$X \subset A$$ be a reduced, irreducible, closed subvariety of $$A$$, where $$A$$ is either a complex semitorus or a semiabelian variety over an algebraically closed field. Assume $$Z(X) = X$$. Then there is a positive dimensional subgroup $$B$$ of $$A$$ such that $$B + X = X$$, that is, $$\dim(\text{Stab}(X)) > 0$$.
We denote by $$\overline{\kappa}(X)$$ the logarithmic Kodaira dimension of $$X$$. In the complex case we need to assume that $$X$$ is meromorphic, that is, the closure of $$X$$ in a compactification of $$A$$ is a complex space.
Theorem 3. Let $$X \subset A$$ be as in theorem 2. In the analytic case assume that $$X$$ is meromorphic, that is, it extends to a complex subspace of a compactification of $$A$$. Let $$B = \text{Stab}(X)$$, that is, $$B$$ is the maximal closed subgroup $$B$$ of $$A$$ such that $$B + X = X$$. Then $$\overline{\kappa}(X) = \dim(X/B)$$.
The above theorem is proved using a generalized Gauss map defined using jets. The usual Gauss map for $$X \subset A$$ is defined by sending a smooth point $$x$$ of $$X$$ to the point on the Grassmann variety representing the tangent space of $$X$$ at $$x$$, inside the tangent space to $$A$$. Here we have:
Theorem 4. Let $$X \subset A$$ be a subvariety of an abelian variety. Assume that $$X$$ is nonsingular and has a finite stabilizer in $$A$$. Then the Gauss map $$X \to \text{Gr}(\dim X,\dim A)$$ is finite.

##### MSC:
 14K05 Algebraic theory of abelian varieties 14J10 Families, moduli, classification: algebraic theory
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##### References:
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