Recently the paper http://de.arxiv.org/PS_cache/arxiv/pdf/1011/1011.1464v1.pdf by Hacon, McKernan and Xu appeared on the arXiv. There the authors prove that the number of birational automorphisms of a variety of general type can be bounded by using the volume of the variety itself. Moreover the authors claim to be able to use the same techniques, that appear in the paper, to prove Kollar's conjecture about DCC of the volumes (conjecture 1.4). I'd like to know if there are important applications of these statements, expecially consequences that follow directly from the boundedness of the automorphisms, regarding for example moduli of varieties. Thanks a lot.
I don't have any direct/immediate consequence of the main result of the paper. However the techniques that we develop here are very useful. In work in progress, we plan to use these techniques to prove the boundedness of (semi) log canonical pairs (generalizing Alexeev's results do dimension \geq 3).