Siu, Yum-Tong An effective Matsusaka big theorem. (English) Zbl 0803.32017 Ann. Inst. Fourier 43, No. 5, 1387-1405 (1993). We prove the following effective form of Matsusaka’s Big Theorem. For an ample line bundle \(L\) over a compact complex manifold \(X\) of complex dimension \(n\) with canonical line bundle \(K_ X\), the line bundle \(mL\) is very ample for \(m\) no less than \[ (2^{3n-1} 5n)^{4^{n-1}} (3(3n- 2)^ nL^ n + K_ X \cdot L^{n - 1})^{4^{n - 1}3n} \over (6(3n - 2)^ n - 2n - 2)^{4^{n - 1} n - {2 \over 3}} (L^ n)^{4^{n-1} 3(n - 1)}. \] Reviewer: Y.T.Siu (Cambridge / Mass.) Cited in 3 ReviewsCited in 24 Documents MSC: 32J25 Transcendental methods of algebraic geometry (complex-analytic aspects) 14E25 Embeddings in algebraic geometry 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) Keywords:positive line bundle; holomorphic line bundle; compact complex manifold; estimates; closed positive current; Lelong number; strong Morse inequality; Matsusaka’s big theorem; ample line bundle PDFBibTeX XMLCite \textit{Y.-T. Siu}, Ann. Inst. Fourier 43, No. 5, 1387--1405 (1993; Zbl 0803.32017) Full Text: DOI Numdam EuDML Link References: [1] [D1] , Champs magnétiques et inégalités de Morse pour la \(d``\) cohomologie, Compte-Rendus Acad. Sci, Série I, 301 (1985), 119-122 and Ann. Inst. Fourier, 35-4 (1985), 189-229. · Zbl 0565.58017 [2] [D2] , A numerical criterion for very ample line bundles, J. Diff. Geom., 37 (1993), 323-374. · Zbl 0783.32013 [3] [18] , et , Approche de la résurgence, Hermann, 1993 · Zbl 0803.14004 [4] [F] , On polarized manifolds whose adjoint bundles are not semipositive, Proceedings of the 1985 Sendai Conference on Algebraic Geometry, Advanced Studies in Pure Mathematics, 10 (1987), 167-178. · Zbl 0659.14002 [5] [K] , Effective base point freeness, Math. Ann., to appear. · Zbl 0818.14002 [6] [KM] and , Riemann-Roch type inequalities, Amer. J. Math., 105 (1983), 229-252. · Zbl 0538.14006 [7] [L] , Plurisubharmonic functions and positive differential forms, Gordon and Breach, New York, 1969. · Zbl 0195.11604 [8] [LM] and , Matsusaka’s Big Theorem (Algebraic Geometry, Arcata 1974), Proceedings of Symposia in Pure Math., 29 (1975), 513-530. · Zbl 0321.14004 [9] [M1] , On canonically polarized varieties II, Amer. J. Math., 92 (1970), 283-292. · Zbl 0195.22802 [10] [M2] , Polarized varieties with a given Hilbert polynomial, Amer. J. Math., 94 (1972), 1027-1077. · Zbl 0256.14004 [11] [N] , Multiplier ideal sheaves and existence of Kähler-Einstein metrics of positive scalar curvature, Proc. Nat. Acad. Sci. U.S.A., 86 (1989), 7299-7300 and Ann. of Math., 132 (1990), 549-596. · Zbl 0711.53056 [12] [S] , Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math., 27 (1974), 53-156. · Zbl 0289.32003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.