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Effective freeness and point separation for adjoint bundles. (English) Zbl 0847.32035
Let $$X$$ be a compact complex manifold of dimension $$n$$ and let $$K_X$$ be its canonical bundle. A conjecture of Fujita raised about ten years ago asks whether for an ample line bundle $$L$$ on $$X$$, the bundle $$(n + 1) L + K_X$$ is free and $$(n + 2) L + K_X$$ is very ample. A great deal of effort was put recently to prove these conjectures. So far the answer is known for combinations of the form $$pL + qK_X$$ with $$p$$ roughly of the order of $$n^n$$ and similarly for $$q$$ [see J.-P. Demailly, J. Differ. Geom. 37, No. 2, 323-374 (1993; Zbl 0783.32013)], L. Ein, R. Lazarsfeld and M. Nakamaye, Zero-estimates, intersection theory, and a theorem of Demailly, preprint (1995), and J. Kollar, Math. Ann. 296, No. 4, 595-605 (1993; Zbl 0818.14002)].
Two main applications of the results contained in the paper by authors’ state that $$mL + K_X$$ is free for $$m \geq {1 \over 2} (n^2 + n + 2)$$ and more generally, the global sections of $$mL + K_X$$ separate any $$r$$ distinct points if $$m \geq {1 \over 2} (n^2 + 2rn - n + 2)$$. The innovation in the proofs consists in combining the algebro-geometric (traditional) methods with some new transcendental methods. Quite specifically, starting from fractionar powers of $$L$$, the authors construct a singular metric whose (Nadel) multiplier ideal sheaf has a finite and controllable set of zeroes. Then a deformation argument, plus the recent $$L^2$$-estimates of Ohsawa and Takegoshi and an induction argument lead to the bounds of the necessary fractions of $$L$$.
[See for completeness and comparison: the second author, Effective very ampleness, Invent. Math. 124, No. 1-3, 563-571 (1996)].

##### MSC:
 32L20 Vanishing theorems 32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results 14C20 Divisors, linear systems, invertible sheaves
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##### References:
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