Inequality for Gorenstein minimal 3-folds of general type (Q1749221)

All varieties in this paper are defined over the complex number field $\mathbb{C}$. \par Let $X$ be a Gorenstein minimal threefold of general type, i.e., $X$ is a three-dimensional normal projective variety with Kodaira dimension 3 and has at worst ${\mathbb{Q}}$-factorial terminal singularities such that the canonical divisor $K_X$ is a nef Cartier divisor. \par A Weil divisor is a $\mathbb{Q}$-Cartier divisor if some non-zero multiple of it is a Cartier divisor and a variety has $\mathbb{Q}$-factorial singularities if each Weil divisor on it is a $\mathbb{Q}$-Cartier divisor. \par $X$ has at worst terminal singularities if there is a resolution of singularities $f: Y\rightarrow X$, such that $K_Y= f^*K_X+ \sum a_iE_i $ with $a_i> 0$, where each $E_i\subset Y$ is a reduced exceptional divisor. (Here $K_X$ is a nef Cartier divisor.) \par For all canonically polarized smooth varieties of dimension $n$, in [Proc. Natl. Acad. Sci. USA 74, 1798--1799 (1977; Zbl 0355.32028)], \textit{S.-T. Yau} proved the optimal inequality \[ (-1)^n c_1^{n-2}\cdot c_2\geq (-1)^n \frac{n}{2(n+1)} c_1^n. \] Miyaoka proved that the inequality $c^2_1\leq 3c_2$ holds for all smooth projective surfaces of general type in 1977 and in 1985, he proved that $3c_2-c_1^2$ is a limit of those of effective rational 1- cycles for all smooth minimal projective varieties of general type. Later, Kobayashi, M. Chen, Catanese, De-Qi Zhang and J. A. Chen generalized Noether's inequalities for surfaces and obtained various versions of Noether inequalities for threefolds. \par \textit{M. Chen} [Math. Res. Lett. 11, No. 5--6, 833--852 (2004; Zbl 1079.14047)] conjectured that for a Gorenstein minimal threefold $X$ of general type, there should exist positive rational numbers $a$ and $b$ such that \[ K_X^3\geq a \chi (\omega_X)-b. \] In this paper, the author proves two main theorems. \par Theorem 1.2. Let $X$ be an irregular Gorenstein minimal threefold of general type. Then \[ K_X^3\geq \frac{4}{3} \chi (\omega_X). \] A direct application of Theorem 1.2 is to prove the following optimal inequality: \par Theorem 1.3. Let $X$ be a Gorenstein minimal threefold of general type. Then \[ K_X^3\geq \frac{4}{3} \chi (\omega_X)-2. \] The main idea to prove Theorem 1.2 is to first show \[ K_X^3\geq \frac{4}{3} \chi (\omega_X)-6. \] Since $X$ is irregular, for every integer $m\geq 2$, there is a cyclic unramified covering $\pi: X'\rightarrow X$ of degree $m$ such that \[ K_X^3\geq \frac{4}{3} \chi (\omega_X)-\frac{6}{m} \] holds for every $m\geq 2$. Let $m\rightarrow \infty,$ Theorem 1.2 is proved.