As far as I understand not all experts in birational geometry would agree that Siu 
settles in his preprint Abundance conjecture, and this conjecture is considered for the moment as open. 
When Kawamata mentions in his recent paper the work of Siu 
  
On the abundance theorem in the case $\nu=0$ 

http://arxiv.org/PS_cache/arxiv/pdf/1002/1002.2682v3.pdf

his says that in his work Siu  using analytic methods reproved the abundance conjecture for a non-minimal algebraic variety whose numerical Kodaira dimension is equal to 0.
I would guess Kawamata would not write his paper if it were accepted that Siu gives complete proof of abundance conjecture.

So for the moment the  answer to you question should be "It is unknown" (in dimensions higher than three).
As for analysis been stronger than algebra, there are examples when this is not quite the case. A famous one is bend and break, that uses characteristics $p$ and is a corner stone of minimal model program 
http://www.math.ens.fr/~debarre/Grenoble.pdf
The existence of minimal models is still unknown even for 3-dimensional Kahler manifolds that are not algebraic...