If $U$ is a smooth algebraic variety, then one can give a simple description of the lowest weight part of its cohomology: if $X$ is a smooth compactification and $j \colon U \to X$ the inclusion, then 
$$ W\_k H^k(U,\mathbf Q) = \mathrm{Im} \left( j^\ast \colon H^k(X,\mathbf Q) \to H^k(U,\mathbf Q)\right). $$ 


Is there a similar direct description of the second lowest weight part, i.e.
$$ \mathrm{Gr}^W\_{k+1} H^k(U,\mathbf Q)?$$



Second lowest weight piece of the cohomology of an algebraic variety
Here is a geometric description of $W\_{k+1}H^k(U)/W\_k H^k(U)$. Set $d=\dim U$. Suppose $X$ is a compactification of $U$ such that the complement $X\setminus U$ is the union of normal crossing divisors, or, more generally, union of smooth varieties $D\_i$ all of whose intersections are smooth. Take submanifolds $Y\_i\subset D\_i$ (in the $C^\infty$ sense) of dimension $2d-k-1$ such that $\sum [Y\_i]$ vanishes in $X$. Then $\bigcup Y\_i$ bounds a chain $c$ in $X$. It is useful to think of $c$ as a $C^\infty$ submanifold with boundary in $X\setminus U$. If we intersect this submanifold with $U$ we get a submanifold without boundary. The Poincar\'e dual of it will be in $W\_{k+1}$ and all elements of $W\_{k+1}$ arise in this way.


Notice that $c$ is not uniquely defined: there may be several chains that bound $\bigcup Y\_i$. But the difference between resulting classes $\in H^k(U)$ will be the Poincar\'e dual class of a $C^\infty$ submanifold of $X$, and so will be of weight $k$, as expected.


