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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.12585 |
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| _version_ | 1866913811801309184 |
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| author | Rössler, Damian Schröer, Stefan |
| author_facet | Rössler, Damian Schröer, Stefan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_12585 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Varieties with free tangent sheaves Rössler, Damian Schröer, Stefan Algebraic Geometry 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. |
| title | Varieties with free tangent sheaves |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2503.12585 |