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Very ampleness criterion of double adjoints of ample line bundles. (English) Zbl 0853.32035
Bloom, Thomas (ed.) et al., Modern methods in complex analysis. The Princeton conference in honor of Robert C. Gunning and Joseph J. Kohn, Princeton University, Princeton, NJ, USA, Mar. 16-20, 1992. Princeton, NJ: Princeton University Press. Ann. Math. Stud. 137, 291-318 (1995).
Let $$X$$ be a compact complex manifold of complex dimension $$n$$, and let $$L$$ be an ample line bundle on $$X$$. Fujita has conjectured in 1987 that $$(n+1)L+K_X$$ is free and that $$(n+2)L+K_X$$ is very ample. Evidence for this conjecture comes from its validity in dimensions 1 and 2 and the proof by Ein-Lazarsfeld of the freeness part in dimension 3. A great deal of effort was put in proving the freeness and very ampleness of bundles of the form $$aL + bK_X$$, with $$a$$ and $$b$$ much larger than in the conjecture [see J.-P. Demailly, J. Differ. Geom. 37, No. 2, 323-374 (1993; Zbl 0783.32013) and L. Ein, R. Lazarsfeld and M. Nakayame, preprint (1994)].
The paper under review contains a numerical criterion for $$L + 2K_X$$ to be very ample. The proofs use Shokurov’s technique for a non-vanishing theorem [V. V. Shokurov, Math. USSR, Izv. 26, 591-604 (1986); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 49, No. 3, 635-651 (1985; Zbl 0605.14006)] and Nadel’s multiplier ideal sheaves [cf. N. Nadel, Proc. Natl. Acad. Sci. USA 86, No. 19, 7299-7300 (1989; Zbl 0711.53056) and J. Koller, Ann. Math. 296, No. 4, 595-605 (1993; Zbl 0818.14002)].
The author has improved the results of this paper in several more recent articles, to mention only the latest in Invent. Math. 124, No. 1-3, 563-571 (1996; see the paper above).
For the entire collection see [Zbl 0852.00026].

##### MSC:
 32L20 Vanishing theorems 14F17 Vanishing theorems in algebraic geometry 32C30 Integration on analytic sets and spaces, currents 32J15 Compact complex surfaces