The Miyaoka-Yau inequality for singular varieties with big canonical or anticanonical divisors

Fuente: arXiv
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Autori principali: Iwai, Masataka, Jinnouchi, Satoshi, Zhang, Shiyu
Natura: Preprint
Pubblicazione: 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
id 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