A note on the ampleness of numerically positive log canonical and anti-log canonical divisors

الأصل · EN

In this short note, we consider the conjecture that the log canonical divisor (resp. the anti-log canonical divisor) Kₓ + Δ (resp. -(Kₓ + Δ)) on a pair (X, Δ) consisting of a complex projective manifold X and a reduced simply normal crossing divisor Δ on X is ample if it is numerically positive. More precisely, we prove the conjecture for Kₓ + Δ with Δ≠ 0 in dimension 4 and for -(Kₓ + Δ) with Δ≠ 0 in dimension 3 or 4.