2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/154309Let $X$ be a complex nonsingular projective 3-fold of general type. We show that there are positive constants $c$, $c'$ and $m_1$ such that $χ(ω_X)\geq -c\Vol (X)$ and $P_m(X)\geq c'm^3\Vol (X)$ for all $m\geq m_1$.Algebraic Geometry14J30; 14J45text