I would like to know how much of the recent results on the MMP (due to Hacon, McKernan, Birkar, Cascini, Siu,…) which are usually only stated for varieties over the complex numbers, extend to varieties over arbitrary fields of characteristic zero or to the equivariant case.
I assume that the basic finite generation results hold for any such field, by base extending to an algebraic closure, so I would guess that most results should extend without too much difficulty. The particular questions that I am really interested in are:
1) Given a smooth projective rationally connected variety X over a field k of charaterisitic zero, can we perform a finite sequence of divisorial contractions and flips to obtain a Mori fibre space?
and
2) The equivariant version of 1) for the action of a finite group on X.
Best Answer
Both of these cases follow more or less automatically from the Minimal Model Program over an algebraically closed field $\bar k$. This is well, known, see for example the original Mori's paper "Threefolds whose canonical bundles are not numerically effective" or Kollár's paper on 3-folds over $\mathbb R$. Iskovskikh and Manin did both versions for surfaces in the 1970s, and their arguments still apply.
The point is that $K_X$ is invariant under the action of any group $G\subset Aut(X)$ and $Gal(\bar k/k)$. So if $\bar C$ is a curve on $\bar X= X\otimes_k \bar k$ with $K_X . \bar C<0$ then $C= \sum_{g\in G} g.\bar C$ (resp. the sum of the conjugates) also intersects $K_X$ negatively.
So you can work with the $G$-invariant (resp. $Gal$-invariant) part of the Mori cone $NE(X)\subset N_1(X)$. If $R$ is an extremal ray of $NE(X)^G$ then the supporting divisor $D$ can be chosen to be $G$-invariant, and it contracts a face of $NE(X)$ (instead of just a ray). It is either a divisorial contraction over $k$, or a flipping contraction. In the latter case there is a flip defined over $k$, since it is an appropriate relative canonical model, and every canonical model is automatically $G$-equivariant, resp. $Gal$-equivariant, by its uniqueness.
So you just do the MMP over $k$. Unless $K_X$ is pseudoeffective but not effective, MMP terminates by [BCHM], and you get either a minimal model (with $K_{X_{\rm min}}$ nef) of a Mori-Fano fibration.
Finally, if you started with a variety $X$ such that $\bar X$ is covered by rational curves then $K_X$ is not pseudoeffective.