On the log canonical ring of projective plt pairs with the Kodaira dimension two (Q2027749)

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 \textit{Y. Gongyo} [Adv. Stud. Pure Math. 74, 159--169 (2017; Zbl 1388.14058)]. For projective klt pairs the conjecture is true, as proven by \textit{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 \textit{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 \(\kappa(X,K_X+\Delta)=2\).