Log pluricanonical representations and the abundance conjecture (Q2877493)

Let \((X, \Delta)\) be a projective log canonical pair with \(K_X + \Delta\) semi-ample and \(\mathbb Q\)-Cartier. The group \(\text{Bir}(X,\Delta)\) of boundary birational maps is defined as a subgroup of \(\text{Bir}(X)\), and contains maps \(f\) such that for any resolution \(X \stackrel{p}{\longrightarrow} Y \stackrel{q}{\longrightarrow} X \) of \(f\), we have \(p^*(K_X + \Delta) = q^*(K_X + \Delta)\). Any such map induces an action on the space of sections \(H^0(X, m(K_X + \Delta))\), where \(m \geq 1\) is such that \(m(K_X + \Delta)\) is Cartier. So we get a linear ``log pluricanonical'' representation \(\rho_m\) of \(\text{Bir}(X,\Delta)\) for each such \(m\). The main result of the paper is that under the above hypothesis \(\rho_m(\text{Bir}(X,\Delta))\) is a finite group. A corollary is that if \((X, \Delta)\) is a log canonical pair with \(K_X + \Delta\) big, then the group \(\text{Bir}(X)\) itself is finite. This is a generalization of the classical result that the group of birational selfmaps of a projective variety of general type is finite. As an application the authors prove many particular partial instances of the abundance conjecture, that is, conditions that garanty that a nef adjoint divisor \(K_X + \Delta\) is semi-ample. The key intermediate step, which relies on the finiteness result about the representation \(\rho_m\), is that the adjoint divisor \(K_X + \Delta\) is semi-ample if and only if the corresponding adjoint divisor on the normalization of \(X\) is semi-ample.