In the book Principles of Algebraic Geometry by Griffiths and Harris, the authors state that the Hodge
decomposition reflects the analytic structure but that the Lefschetz decomposition is essentially topological. Are there any examples or proofs of these statements？
More precisely, for a complex manifold $M$, they say that the rank of the group $(H^{p,q}(M)\oplus H^{q,p}(M))\bigcap H^{p+q}(M)$ may jump when we change the complex structure. Is there any example? If changing complex structure may lead to the jumping of the rank of the group mentioned above, won't the rank of the group $P^{p,q}(M)$ change as we change the complex structure? Here $P^{p,q}(M)$ means the group of primitive forms of type$(p,q)$. Thank you for your answer!
Best Answer
Yes, there are such examples. See the MO discussion and references here.
But there is more to the dependence of the Hodge decomposition on the complex structure than just change of Hodge numbers. Even for Riemann surfaces, when you vary the complex structure (on a fixed smooth surface), the Hodge decomposition of $H^1$ (in general) changes, while Hodge numbers remain constant. This is what G&H are really talking about.
Edit. Regarding the Lefschetz decomposition: Suppose that $X$ is Kahler. Then you get the Kahler class $[\omega]\in H^{1,1}(M)$. In the most interesting case, when $M$ is projective, the class $[\omega]$ is integral, hence, if you have a continuous family $M_t$ of projective manifolds, the class is constant. Kahler class uniquely determines the Lefschetz decomposition (which is given by kernels of the cup-products with powers of the Kahler class). Thus, this is what mean by saying that the decomposition is "topological" (constant under a continuous variation).
However, there are examples (which were unknown when they wrote their book) when the Kahler class (not just the form!) actually changes under some "large" change of the complex structure. The resulting manifolds are homeomorphic but not diffeomorphic, so the word "topological" should be taken with a grain of salt.