Varieties with free tangent sheaves

Source abstract

Abstract In this text, we will consider schemes, which are smooth and proper over a field, and whose tangent sheaf is free. We will call such schemes T-trivial varieties . Over the complex numbers, the T -trivial varieties are precisely the abelian varieties. Igusa observed however that in characteristic p ≤ 3 $p\leq 3$ p less than or equals 3 there are T -trivial bielliptic surfaces, which are not isomorphic to abelian varieties. In this context, we show that T -trivial varieties X separably dominated by abelian varieties A can exist only for p ≤ 3 $p\leq 3$ p less than or equals 3 . Furthermore, we prove that every T -trivial variety, after passing to a finite étale covering, can be fibered in T -trivial varieties with Betti number b 1 = 0 $b_1=0$ b 1 equals 0 . We also show that if some n -dimensional T -trivial variety X lifts to characteristic zero and p ≥ 2 n + 1 $p\geq 2n+1$ p greater than or equals 2 n plus 1 , then X admits a finite étale covering by an abelian variety. Along the way, we establish several results about the automorphism group of abelian varieties, and the existence of relative Albanese maps.