The celebrated Kodaira Vanishing Theorem asserts that
$$H^i(X, \mathcal{O}(K_X +A))=0$$
for $i>0$, where $X$ is a nonsingular projective variety over $\mathbb{C}$ and $A$ is an ample divisor on $X$.
According to the Iitaka philosophy, one obtains its logarithmic version by adding a boundary divisor
$$H^i(X, \mathcal{O}(K_X+D+A))=0$$
for $i>0$, where $D=\sum D_i$ is an SNC divisor on $X$. Now the Kawamata-Viehweg Vanishing Theorem allows the boundary divisor to have a fractional part
$$H^i(X,\mathcal{O}(K_X +B+F+A))=H^i(X,\mathcal{O}(K_X +B+\lceil A \rceil )=0$$
for $i>0$, where $D=B\cup F$ is an SNC divisor on $X$, $F=\sum f_j F_j \ (0 \dim X$,
where $X$ is a nonsingular projective variety over $\mathbb{C}$ and $A$ is an ample divisor on $X$. Again according to the Iitaka philosophy, one obtains its logarithmic version by replacing the usual differential forms with the lopgarithmic differentail forms
$$H^i(X,\Omega^j(\log(D))(A)) = 0$$
for $i + j > \dim X$, where $D=\sum D_i$ is an SNC divisor on $X$. This is nothing but the Esnault-Viehweg Vanishing Theorem.
Then what is the alleged Kawamata-Viehweg type formulation of the (logarithmic) Akizuki-Nakano Vanishing Theorem?
The answer is the theme of this talk, and it was somewhat different from the author naively expected at the beginning.
This is a joint work with D. Arapura, D. Patel and J. Wlodarczyk.