The sharp lower bound for the volume of threefolds of general type with $${\chi({\mathcal O}_X)=1}$$

Abstract

Let V be a smooth projective threefold of general type. Denote by K 3 , a rational number, the self-intersection of the canonical sheaf of any minimal model of V . One defines K 3 as a canonical volume of V . The paper is devoted to proving the sharp lower bound K 3 ≥ 1 420 which can be reached by an example: X 46 ⊆ P(4, 5, 6, 7, 23).

Explicit birational geometry of 3-folds of general type, II

“…With a different approach, L. Zhu [29] also proved K 3 ≥ 1 420 . The proof of the last theorem gives the following: Corollary 3.12.…”

On quasismooth weighted complete intersections

“…Moreover, if α > 0, then X is of general type with χ ≤ 1. By [4] or [18], we have K 3 (X) ≥ 1 420 . By (2.6), we get s a 0 a 1 a 2 a 3 ≥ K 3 (X) ≥ 1 420…”