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Projective regular models for abelian varieties, semistable reduction, and the height pairing. (English) Zbl 0955.14017

From the introduction: Let \(K\) be a number field, and let \(X_\eta\), be a smooth, projective variety over \(\eta=\text{Spec} K\). In order to describe the behavior of motivic \(L\)-functions associated with the variety \(X_\eta\) near the central point, Beilinson and Bloch defined real-valued height pairings between Chow groups of homologically trivial cycles on \(X_\eta\) that extend the classical pairing of Néron and Tate. Therefore, they have to assume that \(X_\eta\) has a regular model \(X\), which is proper over the ring of integers in \(K\), and that homologically trivial cycles on \(X_\eta\) admit suitable extensions to this model. In this paper, these assumptions are investigates in the case of abelian varieties. The author constructs projective regular models for abelian varieties with semiabelian reduction and prove that they have potentially semi-stable reduction. Using a nonarchimedean analogue of the \(dd^c\)-lemma for differential forms, he show that there is a well-defined height pairing if an abelian variety has totally degenerate reduction at all places of bad reduction.
In the first part of the paper, he considers the question of finding proper regular models for abelian varieties. Besides the context mentioned above, this question is also raised by B. W. Jordan and D. R. Morrison [J. Reine Angew. Math. 447, 1-22 (1994; Zbl 0791.14021)]. The author gives a positive answer to this problem in the case of semiabelian reduction. More precisely, let \(R\) be a Dedekind domain with quotient field \(K\), and let \(A_\eta\) be an abelian variety over \(\eta=\text{Spec} K\) that admits semiabelian reduction. Methods developed by G. Faltings and C. L. Chai in their book: “Degeneration of abelian varieties” (1990; Zbl 0744.14031) and D. Mumford [Compos. Math. 24, 239-272 (1972; Zbl 0241.14020)] are used to construct a regular model \(P\) of \(A_\eta\) that is projective and flat over \(\text{Spec} R\). The model \(P\) is by no means unique. Its construction depends on the choice of certain admissible cone decompositions. The reduced special fibers of \(P\) are divisors with normal crossings. A power of a given ample symmetric invertible sheaf \({\mathcal L}_\eta\) on \(A_\eta\) extends to an ample invertible sheaf \({\mathcal L}_P\) on \(P\). The author shows also that certain automorphisms, like multiplication with \(-1\), lift from \((A_\eta,{\mathcal L}_\eta)\) to \((P,{\mathcal L}_P)\). – Using a combinatorial result of Knudsen and Mumford, he proves the following version of potentially semistable reduction for abelian varieties:
Let \(R\) be a discrete valuation ring, and let \(S=\text{Spec} R\). After a finite flat extension of \(R\), the abelian variety \(A_\eta\) admits a projective semistable model \(P\) over \(S\). That is, \(P\) is regular, and the special fiber is a reduced divisor on \(P\) with normal crossings. Each stratum of the canonical stratification of the special fiber is a semiabelian scheme, and its closure in \(P\) is given as a contraction product of this semiabelian scheme with a smooth projective torus embedding.
In the second part of the paper, the author considers the problem of extending cycles from a smooth projective variety \(X_\eta\) to a given proper regular model \(X\). In order to define the height pairing, one has to know that homologically trivial cycles on \(X_\eta\) admit extensions to \(X\) that are perpendicular to cycles supported on the special fibers with respect to the arithmetic intersection pairing defined by H. Gillet and G. Soulé [Publ. Math., Inst. Hautes Étud. Sci. 72, 93-174 (1990; Zbl 0741.14012)]. The discussion in the paper under review is based on a remark of S. Bloch, H. Gillet and C. Soulé in the context of nonarchimedean Arakelov theory [J. Algebr. Geom. 4, No. 3, 427-485 (1995; Zbl 0866.14011)] and on a nonarchimedan analogue of the \(dd^c\)-lemma [in: Arithmetic Geometry, Proc. Symp. Cortona 1994, Symp. Math. 37, 45-69 (1997; Zbl 0955.14007)]. It is shown that the desired extensions of cycles exist if certain well-known standard conjectures are assumed. For abelian varieties, the author verifies that these conjectures are satisfied for the models \(P\) constructed above if \(A_\eta\) has totally degenerate reduction at all places of bad reduction. Hence in this case, there is a well-defined height pairing \[ \langle .,. \rangle_{A_\eta}: \text{CH}^p (A_\eta)^0_\mathbb{Q} \times\text{CH}^{\text{dim} (A_\eta)+1-p} (A_\eta)^0_\mathbb{Q} \to\mathbb{R}. \] Let \(p\) be a prime number. The Jacobian \(J_0 (p)_\mathbb{Q}\) of the modular curve \(X_0(p)_\mathbb{Q}\) is an example of an abelian variety that satisfies the assumption made above.

MSC:

14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14K15 Arithmetic ground fields for abelian varieties
11G10 Abelian varieties of dimension \(> 1\)
14C25 Algebraic cycles
14G25 Global ground fields in algebraic geometry
11G50 Heights
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