Characterization of products of projective spaces via nef complexity

TL;DR

This paper introduces the nef complexity invariant for projective varieties, characterizes when it is zero, classifies certain Fano threefolds, and proves Mukai's conjecture in specific cases, advancing understanding of the structure of Fano varieties.

Contribution

It defines nef complexity, proves its non-negativity, characterizes cases of zero complexity, classifies Fano threefolds with low nef complexity, and verifies Mukai's conjecture under new conditions.

Findings

Nef complexity is non-negative and zero only for products of projective spaces.

Classified smooth Fano threefolds with nef complexity at most one.

Proved Mukai's conjecture for Fano varieties with fiber type extremal contractions.

Abstract

We define the nef complexity of a projective variety $X$. This invariant compares $\dim X+\rho(X)$ with the sum of the coefficients of nef partitions of $-K_X$. We prove that the nef complexity is non-negative and it is zero precisely for products of projective spaces. We classify smooth Fano threefolds with nef complexity at most one. In a similar vein, we prove Mukai's conjecture for smooth Fano varieties for which every extremal contraction is of fiber type and study smooth images of products of projective spaces. Along the way, we answer positively a question of J. Starr regarding the nef cone of smooth Fano varieties.