## On the log canonical ring of projective plt pairs with the Kodaira dimension two. (Sur L’anneau log canonique des paires plt projectives avec la dimension de Kodaira deux.)(English. French summary)Zbl 1465.14020

A pair $$(X,\Delta)$$ consists of a normal variety and a $$\mathbb Q$$-divisor $$\Delta$$ on $$X$$ such that $$K_X+\Delta$$ is $$\mathbb Q$$-Cartier. Given a projective birational morphism $$f:Y \to X$$ from a normal variety $$Y$$, one can write $$K_Y=f^*(K_X+\Delta) + \sum_i a_iE_i$$ with $$f_*(\sum_i a_iE_i)=-\Delta$$, where $$E_i$$ runs over prime divisors on $$Y$$. The discrepancy $$a_i=a_i(E_i,X,\Delta)$$ of $$E_i$$ with respect to $$(X,\Delta)$$ can be defined for any prime divisor $$E_i$$ over $$X$$ by taking as $$f$$ a suitable resolution of singularities of $$X$$. Assume that $$\Delta$$ is effective. Then $$(X,\Delta)$$ is called lc (log-canonical), or klt (Kawamata log terminal), according to whether $$a_i \geq -1$$, or $$a_i>-1$$ for every prime divisor $$E_i$$ over $$X$$, respectively. If $$a_i>-1$$ for every exceptional divisor $$E_i$$ over $$X$$, then $$(X,\Delta)$$ is called plt (purely log-terminal). In particular, plt implies lc. A relevant conjecture in higher dimensional algebraic geometry is the finite generation of the log canonical ring, namely the $$\mathbb C$$-algebra $$R(X,\Delta) = \bigoplus_{m \geq 0}H^0(X,\mathcal O_X(\lfloor(m(K_X+\Delta)\rfloor))$$, for any projective lc pair $$(X,\Delta)$$. It is closely related to the abundance conjecture, as shown by the first author and Y. Gongyo [Adv. Stud. Pure Math. 74, 159–169 (2017; Zbl 1388.14058)]. For projective klt pairs the conjecture is true, as proven by C. Birkar et al. [J. Am. Math. Soc. 23, No. 2, 405–468 (2010; Zbl 1210.14019)]. As to pairs $$(X,\Delta)$$ which are lc but not klt, the conjecture has been proven by the first author when $$\dim(X)=4$$ [Kyoto J. Math. 50, No. 4, 671–684 (2010; Zbl 1210.14020)] and by K. Hashizume when $$\dim(X)=5$$ and $$\kappa(X,K_X+\Delta)<5$$ [Ann. Inst. Fourier 68, 2069–2107 (2018; Zbl 1423.14111)]. The main result the authors prove is that $$R(X,\Delta)$$ is a finitely generated $$\mathbb C$$-algebra if $$(X,\Delta)$$ is a projective plt pair with $$\kappa(X,K_X+\Delta)=2$$.

### MSC:

 14E30 Minimal model program (Mori theory, extremal rays) 14N30 Adjunction problems

### Keywords:

log canonical ring; plt; canonical bundle formula

### Citations:

Zbl 1388.14058; Zbl 1210.14019; Zbl 1210.14020; Zbl 1423.14111
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### References:

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