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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].

##### MSC:
 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)