On the positivity of the dimension of the global sections of adjoint bundle for quasi-polarized manifold with numerically trivial canonical bundle

Abstract

Let (𝑋, 𝐿) denote a quasi-polarized manifold of dimension 𝑛 ≥ 5 defined over the field of complex numbers such that the canonical line bundle 𝐾_𝑋 of 𝑋 is numerically equivalent to zero. In this paper, we consider the dimension of the global sections of 𝐾_𝑋 + 𝑚𝐿 in this case, and we prove that ℎ₀(𝐾_𝑋 + 𝑚𝐿) > 0 for every positive integer 𝑚 with 𝑚 ≥ 𝑛 −3. In particular, a Beltrametti–Sommese conjecture is true for quasi-polarized manifolds with numerically trivial canonical divisors.