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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
##### Keywords:
complex manifold; Levi problem; holomorphic convexity
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##### References:
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