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](https://en.wikipedia.org/wiki/Hodge_structure#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 $W\_2/W\_{1}$.



Example motivating mixed Hodge structures
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 $\alpha\_i$.


