Miyaoka type inequality for terminal threefolds with nef anti-canonical divisors
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916567613177856 |
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| author | Iwai, Masataka Jiang, Chen Liu, Haidong |
| author_facet | Iwai, Masataka Jiang, Chen Liu, Haidong |
| contents | In this paper, we study the Miyaoka type inequality on Chern classes of terminal projective $3$-folds with nef anti-canonical divisors. Let $X$ be a terminal projective $3$-fold such that $-K_X$ is nef. We show that if $c_1(X)\cdot c_2(X)\neq 0$, then $c_1(X)\cdot c_2(X)\geq \frac{1}{252}$; if further $X$ is not rationally connected, then $c_1(X)\cdot c_2(X)\geq \frac{4}{5}$ and this inequality is sharp. In order to prove this, we give a partial classification of such varieties along with many examples. We also study the nonvanishing of $c_1(X)^{\dim X-2}\cdot c_2(X)$ for terminal weak Fano varieties and prove a Miyaoka--Kawamata type inequality for terminal weak Fano $3$-folds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_00268 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Miyaoka type inequality for terminal threefolds with nef anti-canonical divisors Iwai, Masataka Jiang, Chen Liu, Haidong Algebraic Geometry In this paper, we study the Miyaoka type inequality on Chern classes of terminal projective $3$-folds with nef anti-canonical divisors. Let $X$ be a terminal projective $3$-fold such that $-K_X$ is nef. We show that if $c_1(X)\cdot c_2(X)\neq 0$, then $c_1(X)\cdot c_2(X)\geq \frac{1}{252}$; if further $X$ is not rationally connected, then $c_1(X)\cdot c_2(X)\geq \frac{4}{5}$ and this inequality is sharp. In order to prove this, we give a partial classification of such varieties along with many examples. We also study the nonvanishing of $c_1(X)^{\dim X-2}\cdot c_2(X)$ for terminal weak Fano varieties and prove a Miyaoka--Kawamata type inequality for terminal weak Fano $3$-folds. |
| title | Miyaoka type inequality for terminal threefolds with nef anti-canonical divisors |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2303.00268 |