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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.
32A36 Bergman spaces of functions in several complex variables
32T27 Geometric and analytic invariants on weakly pseudoconvex boundaries
Full Text: DOI
[1] C. Bănică and O. Stănăşilă, Algebraic Methods in the Global Theory of Complex Spaces, Wiley, London, 1976. · Zbl 0334.32001
[2] Z. Błocki, Suita conjecture and the Ohsawa-Takegoshi extension theorem, Invent. Math. 193 (2013), 149-158. · Zbl 1282.32014
[3] J.-P. Demailly, Estimations L2pour l’opérateur ¯∂ d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète, Ann. Sci. École Norm. Sup. (4) 15 (1982), 457-511. · Zbl 0507.32021
[4] J.-P. Demailly, Analytic Methods in Algebraic Geometry, Surveys Modern Math. 1, Int. Press, Somerville, MA, and Higher Education Press, Beijing, 2012.
[5] A. Fujiki, On the blowing down of analytic spaces, Publ. RIMS Kyoto Univ. 10 (1974/75), 473-507. · Zbl 0316.32009
[6] A. Fujiki and S. Nakano, Supplement to “On the inverse of monoidal transformation”, Publ. RIMS Kyoto Univ. 7 (1971/72), 637-644. · Zbl 0234.32019
[7] H. Grauert, On Levi’s problem and the imbedding of real-analytic manifolds, Ann. of Math. (2) 68 (1958), 460-472. · Zbl 0108.07804
[8] H. Grauert, Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962), 331-368.
[9] H. Grauert, Bemerkenswerte pseudokonvexe Mannigfaltigkeiten, Math. Z. 81 (1963), 377-391.
[10] Q.-A. Guan and X.-Y. Zhou, A solution of an L2extension problem with an optimal estimate and applications, Ann. of Math. (2) 181 (2015), 1139-1208.
[11] H. Hironaka, Flattening theorem in complex-analytic geometry, Amer. J. Math. 97 (1975), 503-547. · Zbl 0307.32011
[12] H. Hironaka and H. Rossi, On the equivalence of imbeddings of exceptional complex spaces, Math. Ann. 156 (1964), 313-333. · Zbl 0136.20801
[13] K. Knorr, Noch ein Theorem der analytischen Garbentheorie, Habilitationsschrift, Regensburg, 1970.
[14] K. Knorr and M. Schneider, Relativexzeptionelle analytische Mengen, Math. Ann. 193 (1971), 238-254.
[15] K. Kodaira, A theorem of completeness of characteristic systems for analytic families of compact submanifolds of complex manifolds, Ann. of Math. (2) 75 (1962), 146-162. · Zbl 0112.38404
[16] S. Nakano, On the inverse of monoidal transformation, Publ. RIMS Kyoto Univ. 6 (1970/71), 483-502; Supplement, ibid., 7 (1971/72), 637-644.
[17] T. Ohsawa, Finiteness theorems on weakly 1-complete manifolds, Publ. RIMS Kyoto Univ. 15 (1979), 853-870. · Zbl 0434.32014
[18] T. Ohsawa, Weakly 1-complete manifold and Levi problem, Publ. RIMS Kyoto Univ. 17 (1981), 153-164; Supplement, see · Zbl 0465.32011
[19] .
[20] T. Ohsawa, Vanishing theorems on complete Kähler manifolds, Publ. RIMS Kyoto Univ. 20 (1984), 21-38. · Zbl 0568.32018
[21] T. Ohsawa, On the extension of L2holomorphic functions. II, Publ. RIMS Kyoto Univ. 24 (1988), 265-275. · Zbl 0653.32012
[22] T. Ohsawa, On the extension of L2holomorphic functions VIII—a remark on a theorem of Guan and Zhou, Int. J. Math. 28 (2017), no. 9, 1740005, 12 pp. · Zbl 1380.32015
[23] T. Ohsawa and K. Takegoshi, On the extension of L2holomorphic functions, Math. Z. 195 (1987), 197-204. · Zbl 0625.32011
[24] K. Oka, Collected Papers, reprint of the 1984 edition, Springer Collected Works in Math., Springer, Heidelberg, 2014.
[25] Y.-T. Siu, A pseudoconvex-pseudoconcave generalization of Grauert’s direct image theorem, Ann. Scuola Norm. Sup. Pisa (3) 26 (1972), 649-664. · Zbl 0248.32014
[26] K. Takegoshi, On weakly 1-complete surfaces without nonconstant holomorphic functions, Publ. RIMS Kyoto Univ. 18 (1982), 1175-1183. · Zbl 0536.32002
[27] S. Takayama, The Levi problem and the structure theorem for non-negatively curved complete Kääler manifolds, J. Reine Angew. Math. 504 (1998), 139-157. · Zbl 0911.32022
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