Minimal threefolds with small slope and the Noether inequality for canonically polarized threefolds

Abstract

1) We give a 3-dimensional analogue of M. Noether&#39;s inequality for canonically polarized threefolds: $K^3\ge 2(2p_g-5)/3$. This inequality is sharp by known examples of M. Kobayashi. 2) Given a minimal 3-fold $X$ of general type with canonical singularities, if $K^3< (3p_g-5)/2$, we show that $X$ must be birationally fibred by curves of genus 2.