where $\lceil L \rceil$ is the round up of $L$.

This kind of non-vanishing problem has been considered by Ambro \[Am\], A. Chen-Hacon \[CH\], Kawamata \[Ka\], Kollar \[Ko\], Takayama \[Ta\], and others. When $L$ is an integral Cartier divisor, Kawamata \[Ka\] has proved the above Conjecture \[mainconj\] if either $\dim X =
2$, or $\dim X = 3$ and $X$ is minimal (i.e., the canonical divisor $K_X$ is nef).

Conjecture \[mainconj\] is slightly different from that of Kawamata’s in [@Ka]. It is somewhat general in the sense that the divisor $L$ in question is not assumed to have integral coefficients. It is precisely this non-Cartierness of $L$ that causes a lot of trouble when estimating $h^0(X, K_X + \lceil L \rceil)$. To elaborate, the Kawamata-Viehweg vanishing ([@KV], [@Vi1]) implies that $h^0(X, K_X + \lceil L \rceil) =
\chi(K_X + \lceil L \rceil)$ when the fractional part of $L$ is of normal crossings. However, the Riemann-Roch formula for $\chi$ may not be effective because $\lceil L \rceil$ may not be nef and hence $\lceil L \rceil . (K_X + \lceil L \rceil)$ may not be non-negative when $X$ is a nonsingular surface. The worse thing is that as remarked in a 