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Holomorphic curves in complex spaces. (English) Zbl 1133.32002
The authors study the existence of topologically closed complex curves normalized by bordered Riemann surfaces in complex spaces. The main result is the following:
Theorem. Let \(X\) be an irreducible complex space of dimension \(>1\) and let \(\rho:X\to\mathbb{R}\) be a smooth exhaustion function that is \((n-1)\)-convex on the set \(\{ x\in X : \rho(x)>c\}\) for some \(c\in\mathbb{R}\). Given a bordered Riemann surface \(D\) and a \(\mathcal{C}^2\)-map \(f:\overline{D}\to X\) which is holomorphic in \(D\) and satisfies \(f(D)\not\subset X_{\text{sing}}\) and \(f(bD)\subset X_c\), there exists a sequence of proper holomorphic maps \(g_{\nu}:D\to X\) homotopic to \(f|_D\) and converging to \(f\) uniformly on compact sets in \(D\) as \(\nu\to\infty\). Given an integer \(k\in\mathbb{N}\) and finitely many points \(\{z_i\}\subset D\), each \(g_{\nu}\) can be chosen to have the same \(k\)-jet as \(f\) at each of the points \(z_j\).
The authors show by examples that the conditions in this theorem are essentially optimal. They also prove that a compact complex curve with \(\mathcal{C}^2\)-boundary in a complex space admits a basis of open Stein neighborhoods. They obtain several consequences of these results, including new approximation theorems.

MSC:
32C25 Analytic subsets and submanifolds
32F32 Analytical consequences of geometric convexity (vanishing theorems, etc.)
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
32H35 Proper holomorphic mappings, finiteness theorems
14H55 Riemann surfaces; Weierstrass points; gap sequences
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