Anticanonical Volumes of Fano 4-Folds

Abstract

In this paper, we study the explicit geometry of threefolds, in particular, Fano varieties. We find an explicitly computable positive integer N , such that all but a bounded family of Fano threefolds have N -complements. This result has many applications on finding explicit bounds of algebraic invariants for threefolds. We provide explicit lower bounds for the first gap of the R-complementary thresholds for threefolds, the first gap of the global lc thresholds, the smallest minimal log discrepancy of exceptional threefolds, and the volume of log threefolds with reduced boundary and ample log canonical divisor. We also provide an explicit upper bound of the anti-canonical volume of exceptional threefolds. While the bounds in this paper may not and are not expected to be optimal, they are the first explicit bounds of these invariants in dimension three. Contents 1. Introduction 2. Preliminaries 2.1. Sets 2.2. Pairs and singularities 2.3. Complements 2.4. Stein degree 2.5. Generalized pairs 3. Explicit bounds of invariants and their general behavior 3.1. Some bounds of algebraic invariants 3.2. Global lc threshold and mld of exceptional pairs 3.3. Canonical bundle formulas 3.4. A lemma on Cartier index 4. Curves 4.1. Bound of the lc threshold 4.2. Bound of the global lc threshold and exceptional mlds 4.3. Bound of complements and indices 5. Explicit bounds on surfaces 5.1. Bound of the lc threshold 5.2. Bound of complements and indices: the standard coefficient case 5.3. Lower bound of volumes 6. Proof of Theorem 1.1 6.1. Reduction of the theorem 6.2. The plt case 6.3. The non-plt case 6.4. The general case 7. Proof of the main theorems 7.1.