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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. $$ \mathrm{Gr}^W_{k+1} H^k(U,\mathbf Q)?$$

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up vote 5 down vote accepted

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.

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