Hyperbolicity of adjoint linear series on varieties with positive tangent bundle
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911342402732032 |
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| author | Ito, Atsushi Moraga, Joaquín Raychaudhury, Debaditya Yeong, Wern |
| author_facet | Ito, Atsushi Moraga, Joaquín Raychaudhury, Debaditya Yeong, Wern |
| contents | Let $X$ be a smooth projective variety of dimension $n\geq 3$, and let $L$ be an ample line bundle on $X$. In this article, we study the algebraic hyperbolicity of a very general section of the adjoint linear series $|K_X+mL|$ when the tangent bundle $T_X$ of $X$ has suitable positivity properties. As a consequence, we show that the linear system $|K_X+mL|$ is hyperbolic (or pseudo-hyperbolic) for $m\geq 3n+1$, for various classes of polarized pairs $(X,L)$, thus providing new evidence of a conjecture that was proposed by the second and fourth authors. Moreover, when $X$ is abelian, we show that the linear system $|mL|$ is hyperbolic for $m\geq n$, and the same holds when $m\geq n-1$, if $|L|$ has no base divisors. It turns out that these bounds for abelian varieties are sharp. We also prove analogous statements for Kummer varieties and certain classes of hyperelliptic varieties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_20827 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hyperbolicity of adjoint linear series on varieties with positive tangent bundle Ito, Atsushi Moraga, Joaquín Raychaudhury, Debaditya Yeong, Wern Algebraic Geometry Primary: 32J27, 14E30, Secondary: 14M25, 14J28 Let $X$ be a smooth projective variety of dimension $n\geq 3$, and let $L$ be an ample line bundle on $X$. In this article, we study the algebraic hyperbolicity of a very general section of the adjoint linear series $|K_X+mL|$ when the tangent bundle $T_X$ of $X$ has suitable positivity properties. As a consequence, we show that the linear system $|K_X+mL|$ is hyperbolic (or pseudo-hyperbolic) for $m\geq 3n+1$, for various classes of polarized pairs $(X,L)$, thus providing new evidence of a conjecture that was proposed by the second and fourth authors. Moreover, when $X$ is abelian, we show that the linear system $|mL|$ is hyperbolic for $m\geq n$, and the same holds when $m\geq n-1$, if $|L|$ has no base divisors. It turns out that these bounds for abelian varieties are sharp. We also prove analogous statements for Kummer varieties and certain classes of hyperelliptic varieties. |
| title | Hyperbolicity of adjoint linear series on varieties with positive tangent bundle |
| topic | Algebraic Geometry Primary: 32J27, 14E30, Secondary: 14M25, 14J28 |
| url | https://arxiv.org/abs/2511.20827 |