Canonical Volumes of Threefolds with Positive Geometric Genus

We prove that a complex projective threefold of general type with canonical singularities and positive geometric genus has canonical volume at least $1/30$, with strict inequality when its Euler characteristic is positive. We describe the equality cases as canonical hypersurfaces of degree $28$ in P(1,3,4,5,14). These canonical models and their smooth projective resolutions are simply connected. Their coarse moduli space is irreducible and rational of dimension $103$, with finite quotient singularities; all admissible Q-Gorenstein deformations are unobstructed and arise by varying the equation.