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On Kodaira energy and adjoint reduction of polarized manifolds. (English) Zbl 0766.14027

The author classifies polarized complex \(n\)-folds \((M,L)\) (respectively their Sommese second reduction \((M'',A))\) when \(n\geq 5\) (respectively \(n=4)\) and \(-\kappa\varepsilon(M,L)\geq n=3\). Here, \(\kappa\varepsilon(M,L)\) denotes the Kodaira energy, defined by: \(- \kappa\varepsilon(M,L)=\text{Inf}\{t\in\mathbb{Q}\mid \kappa(M,K_ M+tL)\geq 0\}\).
The case \(n\geq 6\) was settled by Beltrami and Sommese (the author precizes their result), but the cases \(n=4\) and 5 need a special analysis, since many new possibilities appear. This classification allows the author to check the spectrum and the fibration conjectures in that range. For example, when \(n=5\) and \(-\kappa\varepsilon(M,L)\geq 2\), then \(-\kappa\varepsilon(M,L)\) is one of the following: \(6,5,4,3,{5\over 2},{7\over 3},2\).

MSC:

14J10 Families, moduli, classification: algebraic theory
32C15 Complex spaces
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14J15 Moduli, classification: analytic theory; relations with modular forms
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