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Chern-Weil theory for line bundles with the family Arakelov metric. (English) Zbl 1443.14011

To motivate the results of this paper, consider a smooth connected complex algebraic variety \(S\) together with a compactification \(S \hookrightarrow \mathcal{X}\) with \(\mathcal{X}\) smooth and normal crossings boundary divisor \(\mathcal{X} \setminus S\). Let \(\overline{L} = \left(L, \| \cdot \| \right)\) be a smooth Hermitian line bundle on \(S\). Assume that \(\overline{L}\) extends canonically to a (not necessarily smooth) Hermitian line bundle \(\overline{L}^{\mathcal{X}}\) on \(\mathcal{X}\) in such a way that local sections have at most logarithmic growth along \(\mathcal{X} \setminus S\). The metric \(\| \cdot \|\) on \(\mathcal{X}\) is then good in the sense of D. Mumford [Invent. Math. 42, 239–272 (1977; Zbl 0365.14012)]. As was observed by Mumford, this canonical extension is then compatible with modifications \(\mathcal{X}' \to \mathcal{X}\) over \(S\). Moreover, the following identity is satisfied in \(\mathbb{Z}\). \[ \left( \overline{L}^{\mathcal{X}} \right)^n = \int_S c_1\left(\overline{L}\right)^{\wedge n}. \hspace{5cm} (1) \] Here, the product on the left hand side is the top intersection theoretic product of the extended line bundle on the compact variety \(\mathcal{X}\), and the product on the right hand side refers to the wedge product of the first Chern form associated to the smooth metrized line bundle \(\overline{L}\) over \(S\). One refers to \((1)\) as a result of Chern-Weil type for \(\overline{L}\) on \(S\).
It follows also from the work of Mumford that if \(S\) is taken to be a pure noncompact Shimura variety together with an automorphic line bundle \(\overline{L}\) equipped with its canonical invariant metric, and given a toroidal compactification \(S \hookrightarrow \mathcal{X}\), then the metrized line bundle \(\overline{L}\) extends to \(\mathcal{X}\) in such a way that the metric is good on the compactification. Hence, in this case, Chern-Weil theory is satisfied for \(\overline{L}\). It thus seems natural to try to extend \((1)\) to a broader range of geometrically relevant situations, including cases where the metric on the open variety \(S\) is no longer necessarily good on some compactification \(\mathcal{X}\). A first example of such a situation is given in [J. I. Burgos Gil et al., London Math. Soc. Lecture Note Ser. 427, 45–77 (2016; Zbl 1378.14027)]. Here, the authors consider the simplest case of a mixed Shimura variety of noncompact type together with an automorphic line bundle equipped with its invariant metric, namely, they consider the universal elliptic curve \(E\) (with an appropriate level structure) together with the automorphic line bundle \(\overline{J}\) of Jacobi forms endowed with its invariant metric. It turns out that given a toroidal compactification \(E \hookrightarrow \mathcal{X}\), there is a canonical extension \(\overline{J}^{\mathcal{X}}\) of the line bundle together with the metric. However, the extension is such that the metric is no longer good along a codimension two subset of the boundary divisor \(\mathcal{X}\setminus E\). Moreover, the extension is no longer compatible with modifications \(\mathcal{X}' \to \mathcal{X}\) and \((1)\) is no longer satisfied. The main original idea in {loc. cit.} is to replace the top intersection product of the extended line bundle with a limit (in the sense of nets) of top intersection numbers, indexed over all possible toroidal modifications \(\mathcal{X}' \to \mathcal{X}\). This limit can be seen as the degree of the \(b\)-divisor \(\left(\overline{J}^{\mathcal{X}}\right)_{\mathcal{X'}}\), where the term \(b\)-divisor is in the sense of Shokurov [V. A. Iskovskikh, Proc. Steklov Inst. Math. 240, 4–15 (2003; Zbl 1081.14022)]. Chern-Weil theory in this setting then reads as follows. \[ \lim_{\mathcal{X}'} \left( \overline{J}^{\mathcal{X}'}\right)^2 = \int_E c_1\left(\overline{J}\right)^{\wedge 2} \hspace{5cm} (2) \] in \(\mathbb{Q}\).
The main goal of the present paper is to generalize \((2)\) to the context of families of smooth complex projective curves of arbitrary genus \(g \geq 1\) fibered over a smooth complex curve. The line bundles considered are equipped with the fiberwise Arakelov metric [S. Y. Arakelov, Math. USSR, Izv. 8, 1167–1180 (1974; Zbl 0355.14002)] and it is assumed that the minimal regular model of the family is semistable. It turns out that in this setting, a b-divisorial Chern-Weil theory is also satisfied. Moreover, \((2)\) is obtained as a particular case.
One of the main tools used by the present authors, and which does not appear in the work of Burgos-Kramer-Kühn, or, to the knowledge of the reviewer, in any other work related to Chern-Weil theory, is the concept of the Deligne pairing of metrized line bundles [P. Deligne, Contemp. Math. 67, 93–177 (1987; Zbl 0629.14008)]. The idea of bringing this pairing into play is original and could be very useful in order to prove other Chern-Weil type results.

MSC:

14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14D06 Fibrations, degenerations in algebraic geometry
14E05 Rational and birational maps
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
32C30 Integration on analytic sets and spaces, currents
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32A27 Residues for several complex variables

References:

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