Varieties with free tangent sheaves

TL;DR

This work studies $T$-trivial varieties, i.e., smooth proper schemes with free tangent sheaf, and analyzes when such varieties arise from or are dominated by abelian structures. The authors develop a framework using automorphisms of abelian varieties, para-abelian targets, and relative Albanese maps to constrain positive-characteristic examples (notably showing $p\le 3$ for certain quotients and Igusa-type constructions). They prove a fibered decomposition: after a finite étale cover, the Albanese fibration has fibers that are $T$-trivial with $b_1=0$, linking the geometry to potential abelian or para-abelian components. Furthermore, liftability results connect characteristic $p>0$ and characteristic zero via Chern-class vanishing and a dimension-dependent bound MN$(n)$, yielding finite étale covers by abelian varieties under suitable hypotheses, with concrete implications in low dimensions. The work blends Albanese theory, automorphism groups, and étale coverings to advance a positive-characteristic classification of tangent-free varieties and their relation to abelian geometry.

Abstract

We coin the term \emph{$T$-trivial varieties} to denote smooth proper schemes over ground fields $k$ whose tangent sheaf is free. Over the complex numbers, this are precisely the abelian varieties. However, Igusa observed that in characteristic $p\leq 3$ certain bielliptic surfaces are $T$-trivial. We show that $T$-trivial varieties $X$ separably dominated by abelian varieties $A$ can exist only for $p\leq 3$. Furthermore, we prove that every $T$-trivial variety, after passing to a finite étale covering, is fibered in $T$-trivial varieties with Betti number $b_1=0$. We also show that if some $n$-dimensional $T$-trivial $X$ lifts to characteristic zero and $p\geq 2n+2$ holds, it 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.

Theorems & Definitions (42)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem E
• Lemma 1.1
• Lemma 1.2
• Proposition 1.3
• Proposition 1.4
• Lemma 1.5