Semi-stability and heights of cycles. (English) Zbl 0873.14028

Let \(K\) be a number field, \({\mathcal O}_K\) the ring of integers of \(K\), and \(\overline E\) a non-zero hermitian vector bundle over \(S:= \text{Spec} {\mathcal O}_K\). Denote \(\mathbb{P}(E)= \text{Proj(Sym}^\bullet E^\vee)\), and by \(\pi: \mathbb{P}(E)\to S\) the canonical morphism. The canonical quotient line bundle \({\mathcal O}_E(1)\) on \(\mathbb{P}(E)\) will be equipped with the hermitian metric; the hermitian line bundle thus defined will be denoted \(\overline {{\mathcal O}_E (1)}\). To any algebraic cycle \(Z\) of (absolute) dimension \(d\) on \(\mathbb{P}(E)\), we can attach its height with respect to \(\overline {{\mathcal O}_E(1)}\), namely the real number \(h_{\overline {{\mathcal O}_E(1)}} (Z)= \widehat {\deg} \bigl(\widehat c_1 (\overline {{\mathcal O}_E(1)})^d |Z)\). When \(d\geq 1\), we may also introduce the degree of its generic fiber \(\deg_K(Z)= \deg_K (c_1 ({\mathcal O}_E(1))^{d-1} \cdot [Z_K])\). On the other hand, we shall denote by \(\text{rk} E\) the rank of \(E\), and by \(\widehat {\deg} \overline E\) the Arakelov degree of \(\overline E\), that is the real number \(\widehat {\deg} \overline E= \widehat {\deg} \widehat c_1 (\overline E)\). – Our main result is the following:
Theorem I. Let \(Z\) be an effective cycle of dimension \(d\geq 1\) on \(\mathbb{P}(E)\). If the restriction \(Z_K\) of \(Z\) to the generic fiber \(\mathbb{P}(E_K)\) is not zero and has a semi-stable Chow point, then the following inequality holds: \[ {1\over d} {h_{\overline {{\mathcal O}_E (1)}} (Z) \over \deg_K(Z)} \geq- {\widehat {\deg} \overline E\over \text{rk} E} -[K:\mathbb{Q}] C(\text{rk} E,d, \deg_KZ), \] where \(C(\text{rk} E,d, \deg_KZ)\) denotes a constant which depends only on \(\text{rk} E,d\), and \(\deg_KZ\).
For various applications, it is convenient to have a more “intrinsic” version of theorem I, as follows:
Theorem II. Let \(\pi: X\to S=\text{Spec} {\mathcal O}_K\) be any projective arithmetic variety and let \(L\) be an invertible sheaf on \(X\), and \(F\) a subsheaf of \(\pi_*(L)\), then under some hypotheses the following inequality holds: \[ {1\over d} {h_{\overline L} (X)\over \deg_{L_K} (X)}\geq {\widehat {\deg} \overline F\over \text{rk} F}- [K:\mathbb{Q}] C(\text{rk} F,d, \deg_{L_K}X). \] We now formulate the function field version of theorem I.
Let \(C\) be a smooth connected projective curve over an algebraically closed field \(k\), let \(E\) be a non-zero vector bundle over \(C\). Let \(K\) denote the function field of \(C\). To any cycle \(Z\) of dimension \(d\) on \(\mathbb{P}(E)\) are attached its geometric height, namely the intersection number computed in \(\text{CH}^* (\mathbb{P} (E))\), \(h_{{\mathcal O}_E(1)} (Z)= \deg_k(c_1 ({\mathcal O}_E(1))^d \cdot [Z])\), and, if \(d\geq 1\), its generic degree, namely the intersection number computed in \(\text{CH}^* (\mathbb{P}(E_K))\), \(\deg_K (Z) =\deg_K (c_1({\mathcal O}_E (1))^{d-1} \cdot [Z_K])\).
Theorem III. Let \(Z\) be an effective cycle of dimension \(d\geq 1\) on \(\mathbb{P}(E)\). If the restriction \(Z_K\) of \(Z\) to the generic fiber \(\mathbb{P}(E_K)\) is not zero and has a semistable Chow point, then the following inequality holds: \[ {1\over d} {h_{{\mathcal O}_E(1)} (Z)\over \deg_K(Z)} \geq- {\deg E\over \text{rk} E}. \] Let us state a consequence of theorem II concerning arithmetic surfaces.
Let \(\pi: X\to S\) be a semistable regular arithmetic surface of genus \(g\geq 2\) over \(S\). For any closed point \(P\in S\), let \(\mathbb{F}_P={\mathcal O}_K/P\) be the residue field of \(P\) and \(NP=\# \mathbb{F}_P\) the norm of \(P\), and let \(X_P\) be the vertical fiber \(\pi^{-1}(P)\), and \(\delta_P\) the number of singular points in \(X_P(\overline {\mathbb{F}}_P)\) (it is zero for almost every \(P)\). For any complex embedding \(\sigma: K\to \mathbb{C}\), let \(X_\sigma\) be the complex curve \(X\times_{K,\sigma} \mathbb{C}\).
Theorem IV. For any \(g\geq 2\), there exists a continuous exhaustion function \(\psi: M_g(\mathbb{C}) \to\mathbb{R}\) on the moduli space of complex smooth projective curves of genus \(g\) such that, for any semistable regular arithmetic surface \(\pi: X\to S\) of genus \(g\) as above, the following inequality holds: \[ [K:\mathbb{Q}]\cdot h(X)\geq \left(8+ {4\over g} \right)^{-1} \left(\sum_P \delta_P \log NP+ \sum_{\sigma: K\to\mathbb{C}} \psi \bigl([X_\sigma] \bigr) \right). \]


14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14C25 Algebraic cycles
14D20 Algebraic moduli problems, moduli of vector bundles
14G25 Global ground fields in algebraic geometry
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