Since this is the topic on diagonal, I like to ask a question: Let $X$ be a compact Kahler manifold of complex dimension $n$, and let $\Delta _X\subset X\times X$ be the diagonal of $X$. We denote by ${\Delta _X}$ its cohomology class in $H^{2n}(X\times X)$ (here I consider the cohomology with complex coefficients). In fact, we know that ${\Delta _X}$ lives in $H^{n,n}(X\times X)$. Let $\pi _1,\pi _2:X\times X\rightarrow X$ be the projections. By Kunneth's theorem and Hodge decomposition, we know that $H^{n,n}(X\times X)=\sum _{p+r=q+s=n}\pi _1^*(H^{p,q}(X))\otimes \pi _2^*(H^{r,s}(X))$, thus ${\Delta _X}$ lives in this direct sum.
My question is this: Sometime ago, I discussed with one mathematician about the possibility that $\Delta _X$ lives in $\sum _{p+r=n}\pi _1^*(H^{p,p}(X))\otimes \pi _2^*(H^{r,r}(X))$, but he said that this is not true for a general compact Kahler manifold, but we had little time to discuss that he could produce a specific example for the claim. Do any of you know of one such example?