the fundamental group $\pi_1(X)$ is necessary, because in (1) the $M: = \lceil L \rceil \ge L$ is big and $\pi_1(X) = (1)$.

\(3) The example also shows the necessity to assume the nefness of the Cartier integral divisor $D$ (with $(X, B)$ klt and $D - (K_X
+ B)$ nef and big) in Kawamata’s conjecture \[Ka\] for the non-vanishing of $H^0(X, D)$. Indeed, in the example, we have $\lceil L
\rceil = L + B$ with $B$ a simple normal crossing effective divisor so that $[B] = 0$, whence $(X, B)$ is klt. To be precise, let $D : = \lceil L \rceil$. Then $D - (K_X + B) = \lceil L \rceil
- B = L$ is nef and big, $D = K_X + \lceil L \rceil$, and $D$ is not nef for $D . D_i = -1$ with the notation in the example.

We end the Introduction with:

Consider a fibred space $f:V\longrightarrow C$ where $V$ is a nonsingular projective variety and $C$ a complete curve. Assume $L$ is a nef and big normal crossing ${\bf Q}$-divisor such that $K_V+L$ is nef. The well-known positivity says that $f_*(\omega_{V/C}\otimes {\mathcal O}_V(\lceil L\rceil))$ is positive whenever it is not 