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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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