## 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.

### MSC:

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

### Citations:

Zbl 0919.14003; Zbl 1258.14018
Full Text:

### References:

 [1] F. Ambro, Shokurov’s boundary property, J. Differential Geom. 67 (2004), 229-255. · Zbl 1097.14029 [2] —, The moduli $$b$$ -divisor of an lc-trivial fibration, Compositio Math. 141 (2005), 385-403. · Zbl 1094.14025 [3] C. Birkar, On existence of log minimal models II, J. Reine Angew. Math. 658 (2011), 99-113. · Zbl 1226.14021 [4] O. Fujino, Applications of Kawamata’s positivity theorem, Proc. Japan Acad. Ser. A Math. Sci. 75 (1999), 75-79. · Zbl 0967.14012 [5] —, A canonical bundle formula for certain algebraic fiber spaces and its applications, Nagoya Math. J. 172 (2003), 129-171. · Zbl 1072.14040 [6] —, On Kawamata’s theorem, Classification of algebraic varieties (Schiermonnikoog, 2009) EMS Ser. Congr. Rep., pp. 305-315, Eur. Math. Soc., Zürich, 2011. [7] —, Non-vanishing theorem for log canonical pairs, J. Algebraic Geom. 20 (2011), 771-783. · Zbl 1258.14018 [8] —, Fundamental theorems for the log minimal model program, Publ. Res. Inst. Math. Sci. 47 (2011), 727-789. · Zbl 1234.14013 [9] O. Fujino and Y. Gongyo, On images of weak Fano manifolds, Math. Z. 270 (2012), 531-544. · Zbl 1234.14033 [10] Y. Kawamata, Subadjunction of log canonical divisors for a subvariety of codimension 2, Birational algebraic geometry (Baltimore, 1996), Contemp. Math., 207, pp. 79-88, Amer. Math. Soc., Providence, RI, 1997. · Zbl 0901.14004 [11] —, On Fujita’s freeness conjecture for 3-folds and 4-folds, Math. Ann. 308 (1997), 491-505. · Zbl 0909.14001 [12] —, Subadjunction of log canonical divisors, II Amer. J. Math. 120 (1998), 893-899. · Zbl 0919.14003 [13] J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Math., 134, Cambridge Univ. Press, Cambridge, 1998. [14] N. Nakayama, Zariski-decomposition and abundance, MSJ Mem., 14, Math. Soc. Japan, Tokyo, 2004. · Zbl 1061.14018 [15] K. Schwede and K. E. Smith, Globally $$F$$ -regular and log Fano varieties, Adv. Math. 224 (2010), 863-894. · Zbl 1193.13004
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