On the geography of Gorenstein minimal 3-folds of general type

Abstract

Let $X$ be a minimal projective Gorenstein 3-fold of general type. We give two applications of an inequality between $\chi ({\omega }_X)$ and $p_g(X)$: 1) Assume that the canonical map ${\Phi }_{|K_X|}$ is of fiber type. Let $F$ be a smooth model of a generic irreducible component in the general fiber of ${\Phi }_{|K_X|}$. Then the birational invariants of $F$ are bounded from above. 2) If $X$ is nonsingular, then $c_1^3\le 1/27c_1{c}_2+10/3$ where $c_1$, $c_2$ are Chern invariants of $X$.

To appear in Asian J. Math