Seshadri constants, Diophantine approximation, and Roth's theorem for arbitrary varieties (Q2346751)

Let \(k\) be a number field, let \(X\) be an irreducible projective variety defined over \(k\), and let \(L\) be an ample line bundle on \(X\). Fix a place \(v\) of \(k\), and choose a distance function \(d_v(x,x')\) on \(X(k_v)\), normalized in the same way as the contribution of \(v\) to the multiplicative height \(H_L\) relative to \(L\). The main object of study of the paper under review is the \textit{approximation constant} \(\alpha_{x,X}(L)=\alpha_x(L)\) for algebraic points \(x\in X(\bar k)\). This is defined to be the infimum of all \(\gamma\in\mathbb R\) for which there exists a sequence \(\{x_i\}\subseteq X(k)\) of distinct points that satisfy the conditions (i) \(d_v(x,x_i)\to 0\) as \(i\to\infty\) and (ii) \(d_v(x,x_i)^{\gamma} H_L(x_i)\) is bounded from above. (If there are no such \(\gamma\) then let \(\alpha_x(L)=\infty\).) This constant measures the cost in complexity required to get closer to \(x\), where complexity is measured by the height \(H_L\). For example, Roth's theorem says that if \(X=\mathbb P^1_k\) and \(L=\mathcal O(1)\), then \(\alpha_x(L)\geq 1/2\). It is closely related to a similar constant defined in an earlier paper by the first author [J. Algebr. Geom. 16, No. 2, 257--303 (2007; Zbl 1140.14016)]. The main purpose of this paper is to compare the approximation constant to the Seshadri constant \(\epsilon_x(L)\), which was defined by \textit{J.-P. Demailly} [Lect. Notes Math. 1507, 87--104 (1992; Zbl 0784.32024)]. It measures local positivity of \(L\) at \(x\). The main theorem of the paper is the following. Let \(X\), \(x\), and \(L\) be as above, and let \(n=\dim X\). Then either \(\alpha_x(L)\geq\frac{n}{n+1}\epsilon_x(L)\), or there is a proper closed subvariety \(Z\) of \(X\), irreducible over \(\bar k\) and containing \(x\), such that \(\alpha_{x,X}(L)=\alpha_{x,Z}(L\big|_Z)\) (i.e., ``\(\alpha_x(L)\) is attained on a proper subvariety \(Z\) of \(X\)''). This theorem is proved using an application of Schmidt's subspace theorem, as extended by \textit{G. Faltings} and \textit{G. Wüstholz} [Invent. Math. 116, No. 1--3, 109--138 (1994; Zbl 0805.14011)]. As a corollary, it also follows (by induction on dimension) that \(\alpha_x(L)\geq \frac{1}{2}\epsilon_x(L)\) for all varieties \(X\) over \(\text{Spec} k\) (possibly reducible), all \(x\in X(\bar k)\), and all ample \(L\) on \(X\).