The Miyaoka-Yau inequality for singular varieties with big canonical or anticanonical divisors
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| Format: | Preprint |
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2025
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| author | Iwai, Masataka Jinnouchi, Satoshi Zhang, Shiyu |
| author_facet | Iwai, Masataka Jinnouchi, Satoshi Zhang, Shiyu |
| contents | We establish the Miyaoka-Yau inequality for $n$-dimensional projective klt varieties with big canonical divisor $K_X$: \[ (2(n+1)\widehat{c}_2(X) - n \widehat{c}_1(X)^2) \cdot \langle c_1(K_X)^{n-2} \rangle \ge 0. \] We also prove the Miyaoka-Yau inequality for K-semistable projective klt varieties with big anticanonical divisor $-K_X$. As part of our approach, we define the non-pluripolar product $\langle α_1 \cdots α_p \rangle$ on singular varieties, and establish the Bogomolov-Gieseker type inequality for $\langle α^{n-1} \rangle$-semistable Higgs sheaves with respect to a big class $α$. In addition, we investigate second Chern class inequalities in the cases where $K_X$ or $-K_X$ is nef. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_08522 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Miyaoka-Yau inequality for singular varieties with big canonical or anticanonical divisors Iwai, Masataka Jinnouchi, Satoshi Zhang, Shiyu Algebraic Geometry Complex Variables Differential Geometry Primary 32J25, Secondary 32Q15, 14C30, 14E30 We establish the Miyaoka-Yau inequality for $n$-dimensional projective klt varieties with big canonical divisor $K_X$: \[ (2(n+1)\widehat{c}_2(X) - n \widehat{c}_1(X)^2) \cdot \langle c_1(K_X)^{n-2} \rangle \ge 0. \] We also prove the Miyaoka-Yau inequality for K-semistable projective klt varieties with big anticanonical divisor $-K_X$. As part of our approach, we define the non-pluripolar product $\langle α_1 \cdots α_p \rangle$ on singular varieties, and establish the Bogomolov-Gieseker type inequality for $\langle α^{n-1} \rangle$-semistable Higgs sheaves with respect to a big class $α$. In addition, we investigate second Chern class inequalities in the cases where $K_X$ or $-K_X$ is nef. |
| title | The Miyaoka-Yau inequality for singular varieties with big canonical or anticanonical divisors |
| topic | Algebraic Geometry Complex Variables Differential Geometry Primary 32J25, Secondary 32Q15, 14C30, 14E30 |
| url | https://arxiv.org/abs/2507.08522 |