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The suggested intuition behind mixed Hodge structures - developed in particular to generalize Hodge decomposition of cohomology groups from complex smooth complete varieties to more general algebraic varieties - is that one should think the cohomology groups $H^k(X)$ to be endowed with increasing filtrations whose successive quotients originate from cohomologies of appropriate smooth complete varieties, hence admit (pure) Hodge structures, but of different weights.

Here is an motivating and so rather 'informal' example which I took from here: Example of curves where I'm missing this decisive feature that successive quotients of the associated filtration should come from the cohomologies of smooth complete varieties.

The example works as follows:

To motivate the definition, consider the case of a reducible complex algebraic curve $X$ consisting of two nonsingular components, $X_1 $ and $X_2$, which transversally intersect at the points $Q_1$ and $Q_2$. Further, assume that the components are not compact, but can be compactified by adding the points $P_1 , ... , P_n$. The first cohomology group of the curve $X$ (with compact support) is dual to the first homology group, which is easier to visualize. There are three types of one-cycles in this group. First, there are elements $ \alpha_{i}, (i=1,..., n)$ representing small loops around the punctures $P_{i}$. Then there are elements $ \beta_{j} $ that are coming from the first homology of the compactification of each of the components. The one-cycle in $ X_{k}\subset X$ ( $ k=1,2$ ) corresponding to a cycle in the compactification of this component, is not canonical: these elements are determined modulo the span of $\alpha_{1} ,... , \alpha_{n}$. Finally, modulo the first two types, the group is generated by a combinatorial cycle $\gamma $ which goes from $ Q_{1}$ to $ Q_{2}$ along a path in one component $X_{1}$ and comes back along a path in the other component $X_{2}$. This suggests that $ H_{1}(X)$ admits an increasing filtration

$$ 0\subset W_{0}\subset W_{1}\subset W_{2}=H_{1}(X) $$

whose successive quotients $W_n/W_{n−1}$ originate from the cohomology of smooth complete varieties, hence admit (pure) Hodge structures, albeit of different weights.

Problem: The last point I not understand. From cohomology of which concrete smooth complete varieties originate the cycles $ \alpha _{i}$ generating $W_0$ in this example?
The same question about the combinatorical cycle $\gamma $ generating $W_2/W_{1}$.

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    $\begingroup$ Sorry, I don't have time to answer, but see here mathoverflow.net/questions/47852/… for other examples, which might help $\endgroup$
    – Donu Arapura
    Commented Aug 29, 2022 at 21:46
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    $\begingroup$ Suppose I have a non-compact, non-smooth curve. I can: resolve singularities, compactify (in some order, which doesn't actually matter). You get a smooth curve, and this is the "guts" of mixed Hodge structure. Now which points did you delete / identify to get the singular / noncompact beast? This gives you the other bits. This is all explained in a very clear way in Durfee's "A Naive Guide to Mixed Hodge Theory" core.ac.uk/download/pdf/39233249.pdf $\endgroup$
    – Geordie Williamson
    Commented Aug 29, 2022 at 22:55
  • $\begingroup$ Thank you two for references and the roadmap. Ok so roughly the cohomology of the variety I'm interested in made up of cycles of closed proper subvarieties of the resolution of the original variety? These are the 'pieces' that provide roughly the decomposition parts, right? $\endgroup$
    – user267839
    Commented Aug 30, 2022 at 0:16
  • $\begingroup$ But the main objective of this question is to understand from which smooth complete varieties the cycles $\alpha_i$ come from in the concrete example quoted above. So we are looking for a smooth complete variety from where the cycles $\alpha_i$ originate. My guess would be the blow up in the $P_i$ & compactification of the original curve, but I'm not sure. That's a beginner's example I doubt that there are some tricky resolving techniques involved without mentioning it explicitely. $\endgroup$
    – user267839
    Commented Aug 30, 2022 at 0:26
  • $\begingroup$ @GeordieWilliamson: I took a look in Durfee's notes and yes the example 4 on p 54 with variety / curve with two components - which are there projective & nonsingular - goes along with the intuition that the cycles on the cohomology of interest come from cycles of coho of complete nonsingular varieties. That fits in the picture. The problem I have with the example above is that it's components in contrast to Durfee's example are not compact. But if we compactify them the $\alpha_i $ become trivial there, so these cycles not even come from compactifications of the components. But on the other $\endgroup$
    – user267839
    Commented Aug 30, 2022 at 11:32

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We think of the cycle $\alpha_i$ as coming from a point, specifically, the point we need to add to compactify the puncture $p_i$.

Here "come from" refers to the excision exact sequence in compactly supported cohomology

When we have a variety $X$ obtained as the open subset of a variety $\overline{X}$ whose closed complement is $Z$, there is a long exact sequence

$$ H^{i-1}_c(Z) \to H^i_c( X) \to H^i_c(\overline{X} ) \to H^i_c(Z) $$

which in particular maps cycles in $H^0(Z)$ to cycles in $H^1(X)$. For $Z$ a finite union of points needed to compactify $X$, this produces the cycles $\alpha_i$.

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