On a numerical criterion for Fano fourfolds

Abstract

In this paper, we prove a special case of Campana-Peternell's conjecture in dimension 4. Specifically, we show that a smooth projective fourfold X with c 2 1 (X) • c 2 (X) = 0 and strictly nef anti-canonical divisor −K X is a Fano fourfold.

The Nonvanishing problem for varieties with nef anticanonical bundle

“…if −K X intersects every curve on X positively, then it is expected that X is a Fano variety. This has been confirmed in dimension 3 in [LOW + 21] and in many cases in dimension 4 in [Liu21].…”

The Nonvanishing Problem for varieties with nef anticanonical bundle

“…if K X intersects every curve on X positively, then it is expected that X is a Fano variety. This was confirmed in dimension 3 in [63,73] and in many cases in dimension 4 in [62]. When X is a smooth, projective, rationally connected fourfold with K X strictly nef, then Ä.X; K X / 0 by [62].…”