Remark \[remk=1\] shows that it is hard to replace the coefficient $"4"$ in the theorem above by $"1"$.

In Sections 3 and 6 (Theorems \[irreg\] and Theorem \[ratell\]), we prove the following non-vanishing results without assuming the condition (3) in Conjecture \[mainconj\], and the proof presented for the first assertion is applicable to higher dimensional varieties. The Fourier-Mukai transforms are applied in the proof.

Let $X$ and $L$ be as in Conjecture \[mainconj\] satisfying the first two conditions only. Then $H^0(X, K_X + \lceil L \rceil) \ne 0$ if either

$(i)$ $X$ is a surface with irregularity $q(X) > 0$, or

$(ii)$ $X$ is a relatively minimal elliptic surface with $\kappa(X) = -\infty$ and $K_X + L$ nef and big.

$(1)$ In Example \[example\], we construct an example of pair $(X, L)$ satisfying the conditions (1) and (2) in Conjecture \[mainconj\] (indeed, both $L$ and $K_X + L$ are nef and big) but with $H^0(X, K_X + \lceil L \rceil) = 0$. So an extra condition such as the (3) in Conjecture \[mainconj\] is necessary.

\(2) The same example shows that in Kollar’s result \[Ko\] on non-vanishing of $H^0(X, K_X + M)$ for big divisor $M$, the “bigness” assumption on 