Effective non-vanishing of global sections of multiple adjoint bundles for polarized 3-folds

Original · EN

Let X be a smooth complex projective variety of dimension three and let L be an ample line bundle on X. In this paper, we provide a lower bound of the dimension of the global sections of m(Kₓ+L) under the assumption that κ(Kₓ+L) is non-negative. In particular, we get the following: (1) if κ(Kₓ+L) is greater than or equal to zero and less than or equal to two, then h⁰(Kₓ+L) is positive. (2) If κ(Kₓ+L) is equal to three, then h⁰(2(Kₓ+L)) is greater than or equal to three. Moreover we get a classification of (X,L) such that κ(Kₓ+L) is equal to three and h⁰(2(Kₓ+L)) is equal to three or four.