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

Fujino, Osamu; Liu, Haidong

Ann. Inst. Fourier 70, No. 4, 1775-1789 (2020).

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 \(\kappa(X,K_X+\Delta)=2\).

Reviewer: Antonio Lanteri (Milano)

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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