Algebraic threefolds of general type with small volume

It is known that the optimal Noether inequality vol(X)≥43p_g(X)-103 holds for every 3-fold X of general type with p_g(X)≥11. In this paper, we give a complete classification of 3-folds X of general type with p_g(X)≥11 satisfying the above equality by giving the explicit structure of a relative canonical model of X. This model coincides with the canonical model of X when p_g(X)≥23. We also establish the second and third optimal Noether inequalities for 3-folds X of general type with p_g(X)≥11. A novel phenomenon shows that there is a one-to-one correspondence between the three Noether inequalities and three possible residues of p_g(X) modulo 3. These results answer two open questions by Chen et al. (Duke Math J 169(9):1603–1164, 2020), and in dimension three an open question by Chen and Lai (Int J Math 31(1):2050005, 2020).