A variety that cannot be dominated by one that lifts (Q2037857)

Let \(X\) be a smooth proper variety defined over an algebraically closed field \(k\) of characteristic \(p>0\). We say that \(X\) is liftable to characteristic \(0\) if there exists a discrete valuation ring \(R\) of characteristic \(0\) with the residue field \(k\) and a flat proper \(R\)-scheme with the special fiber isomorphic to \(X\). The main aim of the paper is to show an example of a smooth projective variety \(X\) in characteristic \(p\) such that if a smooth proper variety \(Y\) admits a dominant rational map to \(X\) then \(Y\) cannot be lifted to characteristic \(0\). \(X\) can be explicitly described as a general ample divisor in the self-product of \(3\) copies of a supersingular curve of genus \(\ge 2\) defined over an algebraic closure of a finite field. The method of proof depends on a new strong version of Siu-Beauville's theorem on characterization of morphisms from a smooth proper variety in characteristic \(0\) to curves of genus \(\ge 2\) in terms of maps of fundamental groups. This is used to prove that if \(X\) is a smooth proper variety in characteristic \(p\) with a lifting to characteristic \(0\) then any map from \(X\) to a smooth projective curve \(C\) of genus \(g\ge 2\) can be lifted to a morphism in characteristic \(0\), possibly after extending \(R\) and changing \(C\) by its Frobenius twist. Finally, the author uses this result and existence of many morphisms to curves of higher genus to arrive at a contradiction in his example. The author's method leaves open the question if there exists smooth proper \(X\) in characteristic \(p\) such that no smooth proper \(Y\) dominating \(X\) can be formally lifted to characteristic zero.