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Log pluricanonical representations and the abundance conjecture. (English) Zbl 1314.14029
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.

##### MSC:
 1.4e+31 Minimal model program (Mori theory, extremal rays) 1.4e+08 Birational automorphisms, Cremona group and generalizations
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