×

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
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] D. Abramovich , Subvarieties of abelian varieties and of Jacobians of curves , P.H.D. Thesis, Harvard University, 1991.
[2] D. Abramovich and J. Harris , Abelian varieties and curves in W d(C) . Comp. Math. 78 (1991) 227-238. · Zbl 0748.14010
[3] D. Abramovich and J.F. Voloch , Toward a proof of the Mordell-lang conjecture in characteristic p . Duke I.M.R.N., 1992. · Zbl 0787.14026 · doi:10.1155/S1073792892000126
[4] F.A. Bogomolov , Points of finite order on an abelian variety . Math. of the U.S.S.R. Izvestya, vol. 17, 1 (1981) 55-72. · Zbl 0466.14015 · doi:10.1070/IM1981v017n01ABEH001329
[5] O. Debarre and R. Fahlaoui , Maximal abelian varieties in Wrd(C) and a counterexample to a conjecture of Abramovich and Harris . Preprint, 1991.
[6] G. Faltings , Diophantine approximation on abelian varieties . Annals of Math., to appear 1991. · Zbl 0734.14007 · doi:10.2307/2944319
[7] G. Faltings , The general case of S. Lang’s conjecture . Preprint, 1991. · Zbl 0823.14009
[8] P. Griffiths and J. Harris , Algebraic geometry and local differential geometry . Ann. Scient. Éc. Norm. Sup., Ser 4, 12 (1974) 355-432. · Zbl 0426.14019 · doi:10.24033/asens.1370
[9] A. Grothendieck et al., S.G.A. III Vol. 2, Expose VIII: Groups Diagonalisables . Lecture Notes in Mathematics 152.
[10] A. Grothendieck , Fondéments de la Géométrie Algébrique , Sec. Math. Paris, 1962. · Zbl 0239.14002
[11] Joe Harris , Letter to S. Lang , 1990.
[12] Joe Harris and J. Silverman , Bi-elliptic curves and symmetric products . A.M.S. Proceedings, to appear. · Zbl 0727.11023 · doi:10.2307/2048726
[13] M. Hindry , Autour d’une conjecture de Serge Lang . Inven. Math. 94 (1988) 575-603. · Zbl 0638.14026 · doi:10.1007/BF01394276
[14] Y. Kawamata , On Bloch’s conjecture . Inv. Math. 57 (1980) 97-100. · Zbl 0569.32012 · doi:10.1007/BF01389820
[15] S. Lang , Some theorems and conjectures in diophantine equations . Bul. Amer. Math. Soc. 66 (1960) 240-249. · Zbl 0095.26301 · doi:10.1090/S0002-9904-1960-10440-5
[16] S. Lang , Division points on curves . Annali di Mat. Pura ed Appl. LXX (1965) 229-234. · Zbl 0151.27401 · doi:10.1007/BF02410091
[17] S. Lang , Higher dimensional diophantine problems . Bull. A.M.S. 80 (1974) 779-787. · Zbl 0298.14014 · doi:10.1090/S0002-9904-1974-13516-0
[18] Z.-H. Luo , Kodaira dimension of algebraic function fields . Am. J. Math. vol. 109, 4, p. 669-693. · Zbl 0643.14020 · doi:10.2307/2374609
[19] Z.-H. Luo, An invariant approach to the theory of logarithmic Kodaira dimension of algebraic varieties . Bull. A.M.S. (N.S.)vol. 19, 1 (1988) 319-323. · Zbl 0692.14024 · doi:10.1090/S0273-0979-1988-15657-1
[20] S. Mori , Classification of higher dimensional varieties . In: Algebraic Geometry , Bowdoin 1985, (ed.) S. Bloch, Procedings of Symposia in Pure Mathematics , vol. 46, part 1, p. 269-331. A.M.S., Providence R.I. 1987. · Zbl 0656.14022
[21] A. Neeman , Weierstrass points in characteristic p . Inv. Math. 75 (1984) 359-376. · Zbl 0555.14009 · doi:10.1007/BF01388569
[22] J. Noguchi , Lemma on logarithmic derivatives and holomorphic curves in algebraic varieties . Nagoya Math. J. 83 (1981) 213-233. · Zbl 0429.32003 · doi:10.1017/S0027763000019504
[23] T. Ochiai , On holomorphic curves in algebraic varieties with ample irregularity . Invent. Math. 43 (1977) 83-89. · Zbl 0374.32006 · doi:10.1007/BF01390205
[24] Z. Ran , The structure of Gauss like maps . Comp. Math. 52(2) (1984) 171-177. · Zbl 0547.14004
[25] J.P. Serre , Exp. 10: Morphismes universels et variété d’Albanese ; Exp. 11: Morhpisms Universels et différentielles de troisiéme espèce . Seminair C. Chevalley, 3e année: 1958/59. · Zbl 0123.14001
[26] J. Silverman , Curves of low genus in the symmetric square of a curve , unpublished (included in [12]).
[27] P. Vojta , Integral points on subvarieties of semiabelian varieties , preprint 1991. · Zbl 1011.11040 · doi:10.1007/s002220050092
[28] J.F. Voloch , On the conjectures of Mordell and Lang in positive characteristic . Inv. Math. 104 (1991) 643-646. · Zbl 0735.14019 · doi:10.1007/BF01245094
[29] K. Ueno , Classification of algebraic varieties I . Comp. Math. 27 (1973) 277-342. · Zbl 0284.14015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.