On property of complements of an algebraic curve with at least 4 irreducible components in \(P^{2}\) (Q2518778)

Let \(X\) be a connected complex manifold and \(M\subset X\) a relatively compact domain in \(X\). Let \(S\) be an analytic subset of \(X\). We say that \(M\) is tautly imbedded into \(X\) modulo \(S\), if, for arbitrary integer \(k \geq 1\) and for any sequence \((f_j)_j \subset \text{ Hol\,} (\Delta^k,X)\), at least one of the following statements holds: \smallskip (i) \((f_j)_j\) admits a subsequence that converges in \(\text{ Hol\,} (\Delta^k,X)\), \smallskip (ii) For any compact set \(K \subset \Delta^k\) and each compact subset \(L\) of \(X\setminus S\) there exists an integer \(N\) such that \(f_j(K)\cap L = \emptyset\), for \(j \geq N\). \smallskip Here, by \(\text{ Hol\,} (\Delta^k,X)\) we denote the family of holomorphic mappings \(f:\Delta^k \longrightarrow X\), where \(\Delta^k\) is the unit polydisc in \(\mathbb{C}^k\). \smallskip The domain \(M\) is said to be hyperbolically embedded into \(X\) modulo \(S\), if for any pair \(p,q\) of distinct points in \(\bar M\), not both lying in \(S\), one can find open neighborhoods \(V_p\) and \(V_q\), respectively, such that \(d_M( V_p\cap M , V_q \cap M)>0\), where \(d_M\) is the Kobayashi pseudodistance on \(M\). Here, for points \(x,y \in \bar M\), one defines \(d_M(x,y) = \lim \inf_{x'\to x,\,y' \to y} d_M (x',y')\). \smallskip On \(X\) the Kobayashi measure \(\mu_X\) is defined by \[ \mu_X (A):= \inf \sum_{i} \int_{A_i}dV \] where \(dV\) is the Poincaré volume form of \(\Delta^n\) and the inf is taken over all families \((A_i)_i\) of Borel subsets of \(\Delta^n\) and holomorphic mappings \(f_i \in \text{ Hol\,} (\Delta^n,X)\), such that \(A \subset \bigcup _i f_i(A_i) \). \smallskip We call \(X\) measure hyperbolic, if \(\mu_X(A)>0\) for any non-empty open set \(A \subset X\). Then the result of the present article is as follows: \smallskip Theorem: Let \(A(\ell)\) be a holomorphic curve in \(\mathbb{P}^2\) with \(\ell \geq 4\) irreducible components. Let \(S\) be the degeneracy set of the Kobayashi pseudodistance \(d_M\). Assume that \(S\) is a curve or the empty set. (The author has shown in an earlier paper, that this is always the case provided that \(S\neq \mathbb{P}^2\)). Then the following are equivalent: \smallskip (i) \(M\) is hyperbolically embedded modulo \(S\) in \(\mathbb{P}^2\), \smallskip (ii) \(M\) is tautly embedded modulo \(S\) in \(\mathbb{P}^2\), \smallskip (iii) The logarithmic Kodaira dimension of \(M\) is 2, \smallskip (iv) \(M\) is measure hyperbolic .