Kawamata--Miyaoka type inequality for varieties with nef anticanonical divisor

Abstract

Let $X$ be an $\epsilon$-lc projective variety of dimension $n\geq 2$ such that $-K_X$ is nef and $0<\epsilon \leq 1$ is a real number. Then for any nef divisors $D_1,\dots,D_{n-2}$, \[ c_1(X)^{2}\cdot D_1\cdots D_{n-2}\leq \frac{2(1+\epsilon)}{\epsilon}\,\hat c_2(X)\cdot D_1\cdots D_{n-2}. \] In particular, there exists a Kawamata--Miyaoka type inequality \[ c_1(X)^n\leq \frac{2(1+\epsilon)}{\epsilon}\,\hat c_2(X)\cdot c_1(X)^{n-2}. \]