Let $X$ be a smooth variety and let $Y\subset X$ be a closed smooth subvariety. We have the local cohomology $H^i_Y(X)$, which becomes a MHS via the mixed cone construction, as explained in Section 5.5 from "Mixed hodge structures" by Peters and Steenbrink.

One also has the Thom isomorphism $H_Y^i(X)\cong H^{i-c}(Y)$, where $c$ is the codimension of $Y$ in $X$. $H^{i-c}(Y)$ is itself also equipped with a (canonical) MHS.

Now I wonder what the relation between these two MHS's is. I am particularly puzzled by this because the MHS on $H^{i-c}(Y)$ is canonical (i.e. independent of the embedding $Y\subset X$), while the cone construction is not canonical at all.

I have not been able to find a reference for this in the literature, and working out the details of the cone construction and combining it with the Thom isomorphism map proved quite intricate and I have not been successful with this analysis.

  • 1
    $\begingroup$ In Akira Fujiki's Duality of Mixed Hodge Structures of Algebraic Varieties, it is proved that there is a natural perfect pairing $H^i_Y(Y,\mathbb Q)\times H^{2n-i}(Y,\mathbb Q)\to \mathbb Q$ which induces duality of mixed $\mathbb Q$-Hodge structures. Together with the standard pairing $H^{i-2c}(Y,\mathbb Q)\times H^{2n-i}(Y,\mathbb Q)\to \mathbb Q$, the isomorphism $H^i_Y(X)\cong H^{i-2c}(Y)$ preserves mixed Hodge structures. This should be compatible with Thom isomorphism. $\endgroup$
    – AG learner
    Nov 13 '20 at 4:37

The Thom isomorphism is an isomorphism of mixed Hodge structures up to a Tate twist: $$ H^k_Y(X,\mathbf Q) \cong H^{k-2c}(Y,\mathbf Q) \otimes \mathbf Q(-c).$$ I don't know what's the canonical reference for this fact - how you prove it will in any case depend on how you choose to define the MHS on the left hand side - but it certainly follows from Saito's theory of mixed Hodge modules.

Saito proves for $a_X \colon X \to \mathrm{Spec}(\mathbf C)$, $X$ a smooth complex variety of dimension $d$, that $a_X^! \mathbf Q \simeq \mathbf Q_X(-d)[-2d]$, an isomorphism in the derived category of mixed Hodge modules. It follows from this that if $i \colon Y \hookrightarrow X$ the inclusion of a smooth subvariety of codimension $c$ that $i^! \mathbf Q_X \simeq \mathbf Q_Y(-c)[-2c]$ (use that $i^!a_X^! = a_Y^!$). But taking derived global sections of this isomorphism gives the result: on the right hand side we get precisely $H^{k-2c}(Y,\mathbf Q) \otimes \mathbf Q(-c)$ and on the left hand side we get $H^k_Y(X,\mathbf Q)$, defined as in Peters-Steenbrink in terms of a cone, more precisely a mapping cone in the derived category of mixed Hodge modules on $X$, since $i_!i^! \mathbf Q_X$ is the fiber (i.e. cone up to a shift) of $\mathbf Q_X \to j_\ast j^\ast \mathbf Q_X$ where $j \colon (X \setminus Y) \hookrightarrow X$.


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