How does the MHS on $H_Y^i(X)$ behave with respect to the Thom isomorphism?

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.

• 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. 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 $$j \colon (X \setminus Y) \hookrightarrow X$$.