## 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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