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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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