A note on the dimension of global sections of adjoint bundles for polarized 4-folds

Abstract

Let $(X,L)$ be a polarized manifold defined over the field of complex numbers. In this paper, we consider the case where $\dim X=4$ and we prove that the second Hilbert coefficient $A_{2}(X,L)$ of $(X,L)$, which was defined in our previous paper, is non-negative. Furthermore we consider a question proposed by H. Tsuji for $\dim X=4$.