The arithmetic Hodge index theorem for adelic line bundles
The main theorem (Theorem 3.2) of this paper is the arithmetic Hodge index theorem for integrable adelic line bundles. This generalizes the index theorem for line bundles [\textit{A. Moriwaki}, Math. Res. Lett. 3, No. 2, 173--183 (1996; Zbl 0873.14005)], which in turn is a higher-dimensional generalization of Faltings-Hriljac. More precisely, let \(X\) be a normal and geometrically integral projective variety over \(\overline{\mathbb Q}\) of dimension \(n\geq 1\). Let \(\overline M\) be an integral adelic \(\mathbb Q\)-line bundle on \(X\) as in [\textit{S. Zhang}, J. Algebr. Geom. 4, No. 2, 281--300 (1995; Zbl 0861.14019)], and let \(\overline L_1, \ldots, \overline L_{n-1}\) be nef adelic \(\mathbb Q\)-line bundles. Suppose also that each \(L_i\) is big and \(M\cdot L_1\cdots L_{n-1} = 0\). Then \(\overline M^2\cdot \overline L_1 \cdots \overline L_{n-1} \leq 0\), and if in addition each \(\overline L_i\) is arithmetically positive and \(\overline M\) is \(\overline L_i\)-bounded for each \(i\), the equality holds if and only if \(\overline M\in \pi^*\widehat{\mathrm{Pic}(\overline{\mathbb Q})}_{\mathbb Q}\), where \(\pi\) is the structure morphism. The authors first prove the local index theorem for integrable metrized line bundles, using the induction formula for integration on the Berkovich analytic space \(X^{\mathrm{an}}\) [\textit{A. Chambert-Loir} and \textit{A. Thuillier}, Ann. Inst. Fourier 59, No. 3, 977--1014 (2009; Zbl 1192.14020)]. Then the authors use the vanishing result of curvatures of flat metrics [\textit{W. Gubler}, Invent. Math. 169, No. 2, 321--376 (2007; Zbl 1153.14036)] and a Lefschetz-type theorem to obtain Theorem 3.2. An appendix discusses many of these ingredients in detail, including local and arithmetic intersections, arithmetic positivity, flat metrics, and the Lefschetz theorems. As a consequence of the local index theorem, the authors derive a non-archimedean analog of [\textit{E. Calabi}, The space of Kähler Metrics, Proc. Internat. Congress Math. Amsterdam II, 206--207 (1954)]: given an integral projective variety \(X\) over an algebraically closed field endowed with a complete and nontrivial absolute value, an ample line bundle \(L\) on \(X\), and two semipositive metrics \(\|\cdot \|_1\) and \(\|\cdot \|_2\) on \(L\), \[ c_1(L, \|\cdot \|_1)^{\mathrm{dim} X} = c_1(L, \|\cdot \|_2)^{\mathrm{dim} X} \] if and only if \(\|\cdot \|_1 / \|\cdot \|_2\) is a constant function on \(X^{\mathrm{an}}\). As a consequence of the global index theorem, the authors derive a rigidity theorem for preperiodic points in algebraic dynamical systems. More precisely, let \(X\) be a projective variety over \(\overline{\mathbb Q}\), and let \(f_i:X\rightarrow X\) be a morphism such that there exists an ample \(\mathbb{Q}\)-line bundle \(L_i\) satisfying \(f_i^*(L_i) \simeq q_i L_i\) for some \(q_i \in \mathbb{Q}^{> 1}\) (\(i=1,2\)). Let \(\mathrm{Prep}(f_i)\) denote the subset of \(X(\overline{\mathbb Q})\) consisting of points with finite forward orbits under \(f_i\). Then Theorem 4.1 proves that \[ \mathrm{Prep}(f_1) \cap Z(\overline{\mathbb Q}) = \mathrm{Prep}(f_2) \cap Z(\overline{\mathbb Q}), \] where \(Z\) is the Zariski closure of \(\mathrm{Prep}(f_1) \cap \mathrm{Prep}(f_2)\).
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