A study on the dimension of global sections of adjoint bundles for polarized manifolds

Let X be a smooth complex projective variety of dimension n and let L be an ample line bundle on X. In this paper, in order to investigate the dimension of H0(KX+tL) more systematically, we introduce the invariant Ai(X,L) for every integer i with 0⩽i⩽n. Furthermore, we study this invariant for the case where L is ample and spanned by global sections. As applications we get a lower bound (resp. an upper bound) for the dimension of H0(KX+tL) if L is ample and spanned by global sections (resp. very ample).