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On Brody and entire curves. (English) Zbl 1159.32015
Let \(X\) be a compact complex manifold. Using chains of holomorphic discs, one can define on \(X\) a pseudo-distance \(d_X\) named after Kobayashi. By definition \(X\) is Kobayshi hyperbolic if \(d_X\) is a distance (i.e., \(x\neq y \Rightarrow d_X(x,y)>0\)).
The pseudo-distance \(d_X\) can be reconstructed from infinitesimal data: one can define a seminorm on \(TX\), the Kobayashi-Royden pseudo-metric, such that \(d_X\) is recovered from \(|\cdot |\) in the usual way by taking the infimum of the length of curves.
A fundamental result of R. Brody [Trans. Am. Math. Soc. 235, 213–219 (1978; Zbl 0416.32013)] asserts that \(X\) is Kobayashi hyperbolic iff there is no non-constant entire curve \(f: \mathbb{C} \rightarrow X\) with bounded derivative (a so called Brody curve) iff the Kobayashi-Royden pseudo-metric is nondegenerate (i.e., it is a norm: \(v\in T_xX \setminus \{0\} \Rightarrow |v|_x >0\)). In particular, if \(|\cdot |_x\) is degenerate for some \(x\in X\), then there is a Brody curve somewhere in \(X\).
The beautiful paper under review addresses the following question: if \(|\cdot |_x\) is degenerate for a point \(x\in X\), can we find a Brody curve passing through \(x\), or at least such that \(x\in \overline{f(\mathbb{C}})\)?
A related problem is the following. If \(f:\mathbb{C} \rightarrow X\) is a non-constant entire curve, and we apply the construction in the proof of Brody lemma to the restrictions of \(f\) to larger and larger discs, we get a non-constant Brody curve \(\varphi: \mathbb{C} \rightarrow X\) with \(\varphi(\mathbb{C}) \subset \overline{f(\mathbb{C}})\). It is natural to ask if varying \(\varphi\), one can in fact cover all of \( \overline{f(\mathbb{C})}\) or not.
Both statements are answered in the negative in the paper under review, by means of two extremely interesting examples.
In the first example (discussed in §3), the author constructs an abelian surface \(T\) with two open subsets \(\Omega _2 \subset \Omega_1 \subset T\) having the following properties: a) \(\Omega_2 \) is not dense in \(\Omega_1\) and \(\Omega_1 \) is not dense in \(T\); b) there is an entire curve in \(\Omega_1\) whose image is dense in \(\Omega_1\) (one can impose that this curve passes through any fixed point of \(\Omega_1\) with prescribed tangent); c) \(\Omega_2\) is covered by non-constant Brody curves contained in \(\Omega_2\); d) any non-constant Brody curve contained in \(\Omega_1\) is in fact contained in \(\Omega_2\) (together with its closure). This shows that the answer to the second problem mentioned above is negative: if we start with an entire curve sweeping all \(\Omega_1\) and apply Brody’s construction, in the process of reparametrizing and passing to a subsequence the curve gets confined to \(\Omega_2\). The abelian surface \(T\) is a product of two elliptic curves and the construction of \(\Omega_1\) and \(\Omega_2\) boils down to finding entire functions with some special properties. This is accomplished through very ingeneous explicit arguments and by application of the Arakelyan approximation theorem [see e.g., J. P. Rosay and W. Rudin, Am. Math. Mon. 96, No. 5, 432–434 (1989; Zbl 0678.30025)].
The second example is given by a \(3\)-fold \(X\) which is the blow-up of an abelian \(3\)-fold \(A\) along a smooth curve \(C\subset A\). Since affine maps \(\mathbb{C} \rightarrow A\) lift to \(X\), the Kobayashi-Royden pseudo-metric vanishes identically on any such \(X\). On the other hand, if \(A\) and \(C\) are carefully chosen, any non-constant Brody curve in \(X\) is contained in the exceptional divisor \(E\). In particular for any \(x\in X \setminus E\), we have \(|\cdot |_x\equiv 0\), but there is no non-constant Brody curve through \(x\). This provides a negative answer to both problems mentioned above. The proof is this time very geometrical, and proceeds roughly as follows. In §§5.2–5.3 the author proves the following: given any \(A,C \) and \(X\) as above, and a non-constant affine map \(f: \mathbb{C} \rightarrow A\), denote by \(B\) the (real) closure of \(f(\mathbb{C})\) in \(A\); if \(B\) and \(C\) intersect transversally at some point, then \(f\) cannot lift to a non-constant Brody curve in \(X\). Next, some properties of real subtori in abelian \(3\)-folds are established (§§5.4–5.5), which are of independent interest. As a result of this analysis the author proves the existence of \(A_0\) and \(C_0\) with the following property: given a 1-parameter subgroup \(P\) of \(A_0\) and a point \(p\in A_0\), let \(B\) denote the (real) closure of the translate of \(P\) through \(p\); then \(B\) intersects \(C_0\) transversally at some point (Prop. 7). The proof is then completed as follows (§5.1). Let \(\pi: X_0\rightarrow A_0\) be the blow-up map. If \(\widehat{f}: \mathbb{C} \rightarrow X_0\) is a Brody curve, then \(f=\pi\circ \widehat{f}\) is Brody as well, therefore affine. Let \(B\) be the (real) closure of \(f(\mathbb{C})\). By Prop. 7, \(B\) and \(C_0\) intersect transversally at some point. Therefore, \(f\) must be constant and \(\widehat{f}(\mathbb{C}) \subset E\).
§6 contains some remarks related to the analogy with diophantine geometry. If the second example can be constructed over a number field, then the proper analogue of infinite sequences of rational points might be arbitrary entire curves, not necessarily with bounded derivative.
As the author notices (Question 1 p.26) the following remains open: if \(X\) is a compact complex manifold and for some \(x\in X \) the Kobayashi-Royden pseudometric \(|\cdot|_x\) is degenerate, is there a non-constant entire curve \(f: \mathbb{C} \rightarrow X\) with \(x\in f(\mathbb{C})\)?
Questions of a similar nature are addressed also in a recent paper by J. Duval [Invent. Math. 173, No. 2, 305–314 (2008; Zbl 1155.32017)].

32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
32A22 Nevanlinna theory; growth estimates; other inequalities of several complex variables
14K12 Subvarieties of abelian varieties
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