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Application of \(L^2\) methods to the Levi problem on complex manifolds. (English) Zbl 1439.32014
The paper provides a short proof of a sufficient condition for a weakly 1-complete manifold to be holomorphically convex, using the Ohsawa-Takegoshi \(L^2\) extension theorem for holomorphic sections in line bundles on Stein or weakly 1-complete Kähler manifolds [the author and K. Takegoshi, Math. Z. 195, 197–204 (1987; Zbl 0625.32011); the author, Publ. Res. Inst. Math. Sci. 24, No. 2, 265–275 (1988; Zbl 0653.32012); J.-P. Demailly, Ann. Sci. Éc. Norm. Supér. (4) 15, 457–511 (1982; Zbl 0507.32021)]. Here, a relevant equivalent criterion for a complex space \(X\) being holomorphically convex is that, for every discrete subset \(D\subset X\), there exists a holomorphic function \(h\) on \(X\) such that \(\sup_{D}|h| = \infty\). The space \(X\) is weakly 1-complete when there exists a \(\mathcal C^\infty\) plurisubharmonic exhaustion function on \(X\). The main result is as follows.
Theorem. Let \(X\) be a projectively embeddable weakly 1-complete manifold with semi-negative canonical bundle \(K_X\). Suppose \(X\) admits a non-constant holomorphic function \(f\) with no critical points. Then \(X\) is holomorphically convex.
This result is related to the previous result of the author [Publ. Res. Inst. Math. Sci. 17, 153–164 (1981; Zbl 0465.32011)] (a 2-dimensional weakly 1-complete manifold \(X\) is holomorphically convex if \(K_X\) is negative) as well as to the results of K. Takegoshi [Publ. Res. Inst. Math. Sci. 18, 1175–1183 (1982; Zbl 0536.32002)] and S. Takayama [J. Reine Angew. Math. 504, 139–157 (1998; Zbl 0911.32022)] (a 2-dimensional weakly 1-complete manifold is holomorphically convex if it admits a non-constant holomorphic function), and the result of [Zbl 0465.32011] for general dimensions.
The proof is done extending holomorphic functions on fibres of \(f\) to the whole of \(X\), a direct application of the Ohsawa-Takegoshi \(L^2\) extension theorem.
Using the extension theorem, the author also reproves a special case of the result of K. Knorr and M. Schneider [Math. Ann. 193, 238–254 (1971; Zbl 0222.32008)] (1-convex maps are locally holomorphically convex) without using the coherence of direct image sheaves.
With a careful analysis of the fibres of \(f\), the above theorem is strengthened in dimension 3 to allow singular fibres. The precise statement is as follows.
Theorem. A 3-dimensional weakly 1-complete manifold \(X\) with semi-negative \(K_X\) is holomorphically convex if it admits a non-constant holomorphic function with holomorphically convex fibres.
MSC:
32A36 Bergman spaces of functions in several complex variables
32T27 Geometric and analytic invariants on weakly pseudoconvex boundaries
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