Positivity of the Cotangent Bundle of Complex Hyperbolic Manifolds with Cusps
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866912114511183872 |
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| author | Memariansorkhabi, Soheil |
| author_facet | Memariansorkhabi, Soheil |
| contents | Let $\overline{X}$ be the toroidal compactification of a cusped complex hyperbolic manifold $X=\mathbb{B}^n/Γ$ with the boundary divisor $D=\overline{X}\setminus X$. The main goal of this paper is to find the positivity properties of $Ω^{1}_{\overline{X}}$ and $Ω^{1}_{\overline{X}}\big(\log(D)\big)$ depending intrinsically on $X$. We prove that $Ω^{1}_{\overline{X}}\big(\log(D)\big) \langle -r D \rangle$ is ample for all sufficiently small rational numbers $r >0$, and $Ω^{1}_{\overline{X}}\big(\log(D)\big)$ is ample modulo $D.$ Further, we conclude that if the cusps of $X$ have uniform depth greater than $4π$, then $Ω^{1}_{\overline{X}}$ is semi-ample and is ample modulo $D$, all subvarieties of $X$ are of general type, and every smooth subvariety $V\subset \overline{X}$ intersecting $\overline{X}$ has ample $K_{V}$. Finally, we show that the minimum volume of subvarieties of $\overline{X}$ intersecting both $X$ and $D$ tends to infinity in towers of normal covering of $X.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_10816 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Positivity of the Cotangent Bundle of Complex Hyperbolic Manifolds with Cusps Memariansorkhabi, Soheil Algebraic Geometry Complex Variables Differential Geometry 32Q45, 32M15, 14F10, 32M15, 14G35 Let $\overline{X}$ be the toroidal compactification of a cusped complex hyperbolic manifold $X=\mathbb{B}^n/Γ$ with the boundary divisor $D=\overline{X}\setminus X$. The main goal of this paper is to find the positivity properties of $Ω^{1}_{\overline{X}}$ and $Ω^{1}_{\overline{X}}\big(\log(D)\big)$ depending intrinsically on $X$. We prove that $Ω^{1}_{\overline{X}}\big(\log(D)\big) \langle -r D \rangle$ is ample for all sufficiently small rational numbers $r >0$, and $Ω^{1}_{\overline{X}}\big(\log(D)\big)$ is ample modulo $D.$ Further, we conclude that if the cusps of $X$ have uniform depth greater than $4π$, then $Ω^{1}_{\overline{X}}$ is semi-ample and is ample modulo $D$, all subvarieties of $X$ are of general type, and every smooth subvariety $V\subset \overline{X}$ intersecting $\overline{X}$ has ample $K_{V}$. Finally, we show that the minimum volume of subvarieties of $\overline{X}$ intersecting both $X$ and $D$ tends to infinity in towers of normal covering of $X.$ |
| title | Positivity of the Cotangent Bundle of Complex Hyperbolic Manifolds with Cusps |
| topic | Algebraic Geometry Complex Variables Differential Geometry 32Q45, 32M15, 14F10, 32M15, 14G35 |
| url | https://arxiv.org/abs/2212.10816 |