## 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: $[K:\mathbb{Q}]\cdot h(X)\geq \left(8+ {4\over g} \right)^{-1} \left(\sum_P \delta_P \log NP+ \sum_{\sigma: K\to\mathbb{C}} \psi \bigl([X_\sigma] \bigr) \right).$

### MSC:

 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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