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Effective very ampleness. (English) Zbl 0853.32034

Let \(L\) be an ample line bundle over a compact complex manifold of dimension \(n\). Fujita conjectured the freeness of \((n + 1) L+K_X\) and the very ampleness of \((n+2)L+K_X\), [see T. Fujita, Proc. 1985 Sendai Conf. Alg. Geometry, Adv. Stud. Math. 10, 167-178 (1987; Zbl 0659.14002)]. The conjecture was validated in dimensions 1, 2 and the freeness part in dimension 3, too.
The note under review is a continuation of several recent articles by the author and his collaborators. The main result is the very ampleness of \(mL + 2K_X\) with the order of \(m\) no more than \((3e)^n\); as a consequence the very ampleness of \(mL + K\) is proved for \(m\) greater or equal than \(2(n+2+n {3n+1\choose n})\). The proofs are algebraic geometric in nature, but they also contain Nadel’s multiplier ideal sheaf and positive currents ideas.

MSC:

32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
14F17 Vanishing theorems in algebraic geometry
32C30 Integration on analytic sets and spaces, currents
32J15 Compact complex surfaces

Citations:

Zbl 0659.14002
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