# Log canonical counterexample to Kawamata-Viehweg vanishing

I found in the literature that, in characteristic 0, Kodaira vanishing holds for log-canonical pairs. On the other hand, the usual statement for Kawamata-Viehweg vanishing talks about a klt pair $(X,\Delta)$.

Question: Are there (good) examples of failure of Kawamata-Viehweg vanishing for log-canonical pairs (even better, for $\Delta=0$)?

In Prop. 3.13 of http://arxiv.org/pdf/1212.5105.pdf there is an example of a generically finite map $\lambda : T\to A$ where $A$ is an abelian variety and $T$ is a Gorenstein variety with log canonical singularites, and $R^1\lambda _* \omega _T\ne 0$ is supported at a closed point of $A$ (everything is defined over a field of characteristic 0). But then if $H$ is sufficiently ample on $A$, $\lambda ^*H$ is nef and big on $T$ and $H^j(R^i\lambda _* \omega _T)=0$ for $j>0$ and $i\geq 0$. Thus $H^1(T,\omega _T\otimes \lambda ^*H)\cong H^0(R^1\lambda _* \omega _T \otimes H)\ne 0$.

There are surely more straightforward examples (eg. there is no need to go to the trouble of guaranteeing that there is a generically finite map to an abelian variety).

• I'm a little confused by this answer. It seems to contradict Theorem 1.7 in the article linked in the question. If $T$ has a single isolated lc singularity, then this point is the only log canonical center of $T$ so any big and nef line bundle $L$ is big when restricted to the log canonical centers of $(T,0)$. The above theorem then guarantees that $H^j(T, \omega_T \otimes L) = 0$ for $j > 0$. What am I missing? – Dori Bejleri Sep 24 '16 at 16:27
• @Dori Thank!. You are right, it is not an isolated singularity, but the singularity does live over a single point on the abelian variety. Also thanks for adding the reference to Thm 1.10 of Fujino's paper. – Hacon Sep 25 '16 at 3:03
• Ah okay that makes more sense. Thanks! – Dori Bejleri Sep 25 '16 at 3:16

Here is a straightforward example with $\Delta \neq 0$. I think it can be adapted to give an example without a boundary.

Let $X_0 = C \times C'$ be a product of elliptic curves viewed as an elliptic fibration $X_0 \to C$ with a fixed section $S_0$ and fiber $E_0$. Let $\mu : X \to X_0$ be the blowup of the intersection point $p = S_0 \cap E_0$. I'll denote the exceptional divisor by $A$ and the two strict tranforms by $S$ and $E$.

Now consider the log canonical pair $(X, \Delta = S + F + E)$ where $F$ is the strict transform of a fiber disjoint from $p$. We have that $K_X \sim_\mathbb{Q} A$ and $$L = K_X + \Delta = K_X + S + F + E$$ is big, nef and even semiample. $L$ is zero precisely on $E$ so the basepoint free linear series $|mL|$ for $m \gg 0$ induces a log canonical contraction $f : X \to Y$ that contracts $E$ to an elliptic singularity and $Y$ is the log canonical model of $(X, \Delta)$.

Now we want to show that $\mathcal{O}_X(mL) = \omega_X(\Delta) \otimes \mathcal{O}_X((m-1)L)$ has nonzero higher cohomology. First note that $f_*\mathcal{O}_X(mL) = \mathcal{O}_Y(m(K_X + f_*\Delta))$ is sufficiently ample on $Y$ so $H^j(f_*\mathcal{O}_X(mL)) = 0$ for $j > 0$. Therefore $$H^1(X, \mathcal{O}_X(mL)) = H^0(Y, R^1f_*\mathcal{O}_X(mL))$$ from Leray spectral sequence. Now $R^1f_*\mathcal{O}_X(mL)$ is a skyscraper sheaf at $f(E)$ and we can compute this is nonzero by the theorem on formal functions using: $(1)$ $E$ is an elliptic curve, $(2)$ $mL|_E = \mathcal{O}_E$.

Here the restriction of $mL$ to $E$ contributes to the cohomology since $E$ is a log canonical center of the pair. If I understand correctly, this is the only way that KV vanishing can fail for log canonical pairs:

$\textbf{Theorem:}$ Let $(X, \Delta)$ be a projective log canonical pair. Let $L$ be a big and nef $\mathbb{Q}$-Cartier divisor such that $L$ remains big when restricted to each log canonical center of $(X,\Delta)$. Then $$H^j(X, \mathcal{O}_X(K_X + \Delta + L)) = 0$$ for $j > 0$.

This is a special cause of Theorem 1.10 in this paper of Fujino. See also Theorem 1.7 in the paper linked in the question.