On canonical bundle formulas and subadjunctions. (English) Zbl 1260.14010

The authors generalize a subadjunction formula of Y. Kawamata [Am. J. Math. 120, No. 5, 893–899 (1998; Zbl 0919.14003)] proving the following result.
Let \(\mathbb K= \mathbb Q\) or \(\mathbb R\), and let \(X\) be a normal complex projective variety endowed with an effective \(\mathbb K\)-divisor \(D\) such that \((X,D)\) is log canonical. Let \(W\) be a minimal log canonical center with respect to \((X,D)\). Then there exists an effective \(\mathbb K\)-divisor \(D_W\) on \(W\) such that \((K_X+D)|_W\) is \(\mathbb K\)-linearly equivalent to \(K_W+D_W\) and the pair \((W,D_W)\) is Kawamata log terminal. In particular, \(W\) has only rational singularities.
The proof rests on a lemma providing a canonical bundle formula for generically finite proper surjective morphisms, a result which, as the authors say, is missing in the literature. The paper also contains applications to log-Fano varieties and to a new proof of the non-vanishing theorem for log canonical pairs, recently established by the first author [J. Algebr. Geom. 20, No. 4, 771–783 (2011; Zbl 1258.14018)]. Moreover, a local version of the above subadjunction formula is proven.


14C20 Divisors, linear systems, invertible sheaves
14N30 Adjunction problems
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
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