Mixed Hodge structure with examples

Question

The Hodge structure provides a decomposition of the cohomology of compact Kahler manifolds. The mixed Hodge structure provides a generalization so that the Hodge filtration on each graded component(corresponding to the weight filtration) gives a Hodge structure.

1) I want to see some examples of mixed Hodge structure on the cohomology of some spaces.

2) What do we mean by saying that the mixed Hodge structure on $H^i(X)$ is not pure?

Thanks

Answer 1

See the book here by James Carlson, Stefan Müller-Stach, and Chris Peters, especially the first chapter.

Carlson, James; Müller-Stach, Stefan; Peters, Chris. Period mappings and period domains. Cambridge Studies in Advanced Mathematics, 85. Cambridge University Press, Cambridge, 2003. xvi+430 pp.

There are several concrete examples of mixed Hodge structure in it.

A short answer is this. It is not so much the Hodge structure being mixed but the weights that play a role. For a smooth projective variety $X$, $H^k(X)$ has a pure weight $k$ Hodge structure. But if $X$ is no longer compact or smooth this need not be true.

Simple example: $\mathbb{C}^\times $ (complex plane deprived of origin), $H^1 = \mathbb{Z}$, but its weight cannot be one (if so, it would have to have even rank). In fact, the Deligne mixed Hodge structure is pure of weight $2$ in this case. For more complicated punctured curves such as once punctured torus, you get a mixed Hodge structure on $H^1$ with $H^2 = W_2 \supset W_1$. Then $W_1$ comes from the compact curve and $W_2/W_1$ from the punctures.

Similarly when you have a singular curve, $H^1$ has weights $0$ and $1$ with $W_0$ coming from the singularities and $W_1/W_0$ from the desingularization. But if the desingularization is a rational curve, this piece is missing, and the Hodge structure is pure of weight $0$.

In both of these two cases, if the weight filtration has $2$ steps or more with nontrivial gradings, the Hodge structure is not pure. But you can still have a pure Hodge structure on the cohomology of a singular or noncompact variety, as demonstrated above.