TITLE: Examples of Mixed Hodge Structures
QUESTION [32 upvotes]: Does anyone know a user-friendly, example-laden introduction to mixed Hodge structures?  I get from Wikipedia how to calculate for a punctured and pinched curve (http://en.wikipedia.org/wiki/Hodge_structure#Mixed_Hodge_structures), but I want more!  I want tables and numbers and everything explicit and spoonfed.  Thanks!

REPLY [17 votes]: I just noticed the question. The references mentioned so far are good. So I'll just do
the example that I normally do on a blackboard when anyone asks me.
Take a smooth complex projective variety $X$ with a smooth divisor $D$. Set $U=X-D$. There's
a long exact sequence
$$ \ldots H^i(X)\to H^i(U)\to H^{i-1}(D)\to H^{i+1}(X)\ldots$$
say with rational or complex coefficients. What are the maps?
The first is restriction, the second using $\mathbb{C}$ coefficients is a residue map,
and the third is  the Gysin map which is of type $(1,1)$ (or you want to want to get
fancy you need a Tate twist here). The mixed Hodge theory of $U$ can be read off from this.
For example, for the Hodge numbers
$$\dim H^{pq}(U)= \dim im[H^{p-1,q-1}(D)\to H^{pq}(X)]+\dim ker[H^{pq}(D)\to H^{p+1,q+1}(X)]$$
and so on. (By the way, $H^{pq}$ is taken to be the $(p,q)$ part of the $p+q$ 
weight graded  quotient.)
OK, let me make it more concrete. Let $X$ be a surface with irregularity $q=0$, perhaps $\mathbb{P}^2$, then $D \subset X$ is a curve of say genus $g$.
Then from above, the interesting Hodge numbers are
$$h^{20}(U)=h^{02}(U)=h^{20}(X)$$
$$h^{11}(U)=h^{11}(X)-1$$
$$h^{12}(U)=h^{21}(U)=g$$
 Maybe that's enough for now.
I can't resist squeezing in one more example. Suppose $D$ on  the above surface $X$
can be contracted to a point in a normal surface $Y$. So for example, $Y$ might be a cone over a plane curve, and $X$ the blow up of the vertex. Using duality  and the  standard exact sequence for compactly supported cohomology, we can  conclude 
$$H^i(Y)=H_c^i(U)= H^{4-i}(U)^*(-2),\quad i>0$$
As far as Hodge numbers are concerned, the  dual means $(p,q)\mapsto (-p,-q)$ and $(-2)$ means shift by $(2,2)$.