Definition of Mixed Hodge Structure

Question

One definition of Mixed Hodge Structure could be found at Wikipedia

https://en.wikipedia.org/wiki/Hodge_structure#Mixed_Hodge_structures

The definition assumes an abelian group $H_{\mathbb{Z}}$, but the Hodge filtration $F^p$ and weight filtration $W_j$ are associated with $H_{\mathbb{C}}=H_{\mathbb{Z}} \otimes \mathbb{C}$ and $H_{\mathbb{Q}}=H_{\mathbb{Z}} \otimes \mathbb{Q}$. In the process of tensorring with $\mathbb{Q}$ or $\mathbb{C}$, the torsion part of $H_{\mathbb{Z}}$ is killed. So my questions are

1. Does the torsion of $H_{\mathbb{Z}}$ matter? What is its importance?

2. Why use a lattice instead of the $\mathbb{Q}$ vector space $H_{\mathbb{Q}}$ in the defition?

Answer 1

The cohomology of an algebraic variety is not a mere abelian group. It has way more structure. In particular, it has a mixed Hodge structure on it. This structure is indeed the data of some filtrations on $H_{\mathbb{Q}}$ and $H_{\mathbb{C}}$. But the cohomology of an algebraic variety in not only a $\mathbb{Q}$-vector space with a weight filtration and Hodge filtration after tensoring by $\mathbb{C}$. It also has an integral structure $H_{\mathbb{Z}}$.

Of course, we can forget about the Hodge structures. There are nice theorems which do not need them. Or we can forget about integral coefficients for other purposes. We may also want a structure which forgets none of them. This is why there is a definition of integral mixed Hodge stucture.

Here is an important example where it plays a role : consider a proper flat map $f:X\rightarrow D$ where $D$ is a small disk centered at 0, whose fibers are projective varieties, smooth except above 0 where it is a normal crossing divisor. Then the groups $H^*(X_t)$ carry a Hodge structure depending on $t$. If we "fix" the Hodge filtration, then the weight filtration is also fixed, but the lattices vary. (We can also "fix" the lattices, then the weight filtration is also fixed, and this is the Hodge filtration that varies).

However, it is true that the Hodge structure does not give anything on the torsion in cohomology. But it does not mean torsion is not interesting. It is just unrelated to the Hodge structure.