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The Fujita conjecture and the extension theorem of Ohsawa-Takegoshi. (English) Zbl 0941.32021
Noguchi, J. (ed.) et al., Geometric complex analysis. Proceedings of the conference held at the 3rd International Research Institute of the Mathematical Society of Japan, Hayama, March 19-29, 1995. Singapore: World Scientific. 577-592 (1996).
From the introduction: “Let \(L\) be an ample line bundle over a compact complex manifold \(X\) of complex dimension \(n\). We discuss here the most recent result of myself and U. Angehrn [Invent. Math. 122, No. 2, 291-308 (1995; Zbl 0847.32035)] on the conjecture of Fujita on freeness. Fujita’s conjecture states that \((n+1)L +K_X\) is free. The conjecture of Fujita has a second part on very ampleness which states that \((n+2) L+K_X\) is very ample. We will confine ourselves to the freeness part of the Fujita conjecture. My result with Angehrn is the following.
Main Theorem. Let \(\kappa\) be a positive number. If \((L^d\cdot W)^{1\over d}\geq{1\over 2} n(n+2r_1) +\kappa\) for any irreducible subvariety \(W\) of dimension \(1\leq d\leq n\) in \(X\), then the global holomorphic sections of \(L+K_X\) over \(X\) separate any set of \(f\) distinct points \(P_1, \dots, P_r\) of \(X\). In other words, the restriction map \(\Gamma(X,L) \to \oplus^r_{\nu=1} {\mathcal O}_X/ {\mathfrak m}_{P_\nu}\) is surjective, where \({\mathfrak m}_{P_\nu}\) is the maximum ideal at \(P_\nu\).
Corollary. \(mL+K_X\) is free for \(m\geq{1\over 2} (n^2+n+2)\)”.
For the entire collection see [Zbl 0903.00037].

32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
14C20 Divisors, linear systems, invertible sheaves
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)