The minimal model program aims to find a minimal representative in the birational class of a given variety with reasonable singularities. Assuming this has been done, it seems natural to ask what these minimal models look like.
Is it feasible to ask for a classification of minimal (complex) varieties, say for threefolds? If so, has this been done? Or are there known wild examples which show that seeking a classification is too ambitious?
Best Answer
It depends what you mean by classification. The key results for surfaces IMO are: 1) Any surface $S$ of general type has a canonical model given by $S_{can}:={\rm Proj} R(K_S)$ and a unique minimal model given by the minimal resolution of $S_{can}$. 2) The canonical volume ${\rm vol}(S)=K_{S_{can}}^2$ is an integer and 3) $5K_{S_{can}}$ is very ample. In particular for fixed $v$, canonical modules of surfaces of general type of volume ${\rm vol}(S)=v$ can be parametrized by a variety of finite type. On the other hand, for any fixed invariants ${\rm vol}(S)$, $h^1(\mathcal O _S)$ etc it can be impossibly hard to determine the moduli space of canonical models of such surfaces of general type.
All of this actually generalizes to all dimensions (with a few caveats / changes). 1) holds by Birkar-Cascini-Hacon-McKernan: $R(K_X)=\oplus H^0(mK_X)$ is finitely generated and we take $X_{can}={\rm Proj}(R(K_X))$; there are also minimal models which have terminal singularities. One canonical model can have multiple minimal models (but only finitely many and they are connected by flops). 2-3) By a result of of Hacon-McKernan, Takayama, Tsuji, for any fixed dimension, the canonical volumes ${\rm vol}(X)=K_{X_{can}}^{\rm dim(X)}$ belong to a discrete set and for fixed dimension $d$ and volume $v$ there is an integer $m$ such that $mK_{X_{can}}$ is very ample.
Explicit results are few and far apart (the work of J.Chen and M.Chen comes to mind).