There is a conjecture (often attributed to Mumford) I believe which states that if, on a smooth proper variety $X$ (over an algebraically closed field of characteristic zero), there are no pluricanonical forms (that is $H^0(X, mK_X) = 0$ for all positive $m$) then $X$ is uniruled.
Apparently, this conjecture follows from a "well known" conjecture arising from the minimal model program. I believe, but am not entirely sure, that this is the Abundance Conjecture, which says (in one formulation) that the Kodaira dimension of $X$ agrees with the so called numerical Kodaira dimension of $X$. There are by now many well written introductions to the MMP, here is one.
At the same time, Professor Siu has recently posted a sketch of the proof the Abundance Conejcture. Unfortunately, I am not sufficiently equipped to read the proof which uses L2 estimates of d-bar equations. Here are my questions.
Is it true that over the complex number Siu's result implies Mumford's conjecture? He doesn't mention this in the preprint, but is there a reference?
Is anyone well versed enough in both the analytic techniques and algebraic geometry to explain what Siu did to someone more algebraically minded?
Do people have an opinion (vague or otherwise) as to whether techniques coming from analysis are just stronger than techniques coming from algebra? An if so, why is that? (An obvious example, example: Hodge decomposition, but also Siu's proof of invariance of plurigenera, and now the abundance conjecture).