Quasi elementary contractions of Fano manifolds

TL;DR

This paper introduces and studies 'quasi elementary' contractions of Fano manifolds, establishing bounds on Picard numbers and characterizing the structure of the manifolds in various dimensions.

Contribution

It defines quasi elementary contractions, proves smoothness and Fano properties of the base Y under certain conditions, and characterizes the structure of X when specific Picard number bounds are met.

Findings

Y is smooth and Fano when dim(Y) ≤ 3 and ρ(Y) ≥ 4

X is a product if ρ(Y) ≥ 6

Sharp bounds on ρ(X) for 4-dimensional X with quasi elementary contractions

Abstract

Let X be a smooth complex Fano variety. We define and study 'quasi elementary' contractions of fiber type f: X -> Y. These have the property that rho(X) is at most rho(Y)+rho(F), where rho is the Picard number and F is a general fiber of f. In particular any elementary extremal contraction of fiber type is quasi elementary. We show that if Y has dimension at most 3 and Picard number at least 4, then Y is smooth and Fano; if moreover rho(Y) is at least 6, then X is a product. This yields sharp bounds on rho(X) when dim(X)=4 and X has a quasi elementary contraction, and other applications in higher dimensions.