On the dimension of global sections of adjoint bundles for polarized 3-folds and 4-folds

Let (X,L)(X,L) be a polarized manifold of dimension nn defined over the field of complex numbers. In this paper, we treat the case where n=3n=3 and 4. First we study the case of n=3n=3 and we give an explicit lower bound for h0(KX+L)h0(KX+L) if κ(X)≥0κ(X)≥0. Moreover, we show the following: if κ(KX+L)≥0κ(KX+L)≥0, then h0(KX+L)>0h0(KX+L)>0 unless κ(X)=−∞κ(X)=−∞ and h1(OX)=0h1(OX)=0. This gives us a partial answer of Effective Non-vanishing Conjecture for polarized 3-folds. Next for n=4n=4 we investigate the dimension of H0(KX+mL)H0(KX+mL) for m≥2m≥2. If n=4n=4 and κ(X)≥0κ(X)≥0, then a lower bound for h0(KX+mL)h0(KX+mL) is obtained. We also consider a conjecture of Beltrametti–Sommese for 4-folds and we can prove that this conjecture is true unless κ(X)=−∞κ(X)=−∞ and h1(OX)=0h1(OX)=0. Furthermore we prove the following: if (X,L)(X,L) is a polarized 4-fold with κ(X)≥0κ(X)≥0 and h1(OX)>0h1(OX)>0, then h0(KX+L)>0h0(KX+L)>0.