Explicit birational geometry of threefolds of general type

Let $V$ be a complex nonsingular projective 3-fold of general type. We prove $P_{12}(V)>0$ and $P_{24}(V)>1$ (which answers an open problem of J. Kollar and S. Mori). We also prove that the canonical volume has an universal lower bound $\text{Vol}(V)\ge 1\mathrm{/}2660$ and that the pluri-canonical map ${\mathrm{\Phi }}_m$ is birational onto its image for all $m\ge 77$. As an application of our method, we prove Fletcher’s conjecture on weighted hyper-surface 3-folds with terminal quotient singularities. Another featured result is the optimal lower bound $\text{Vol}(V)\ge 1\mathrm{/}420$ among all those 3-folds $V$ with $\chi({\mathcal O}_V)\leq 1$.