On the geography of threefolds of general type

Abstract

Let $X$ be a complex nonsingular projective 3-fold of general type. We show that there are positive constants $c$, $c^{\prime }$ and $m_1$ such that $χ(ω_X)\geq -c\Vol (X)$ and $P_m(X)\geq c'm^3\Vol (X)$ for all $m\ge m_1$.