Let $X$ be a projective variety. Let $S \subset X$ be the nonsingular complete intersection of $k$ nonsingular divisors of $X$ of codimension $2k>2$. Denote $\tilde{X}$ the blow up of $X$ along $S$, and let $E$ be the exceptional divisor.
The blow-up/blow-down correspondence developed by Pandrepande and Maulik in "A topological view of Gromov-Witten theory" allows us to determine the Gromov-theory of the blowup from the restriction map $H^\bullet(X;\mathbb{Q}) \to H^\bullet(S;\mathbb{Q})$, and the GW theories of $X$ and $S$.
On the other hand, we know from Bondal-Orlov that there is a semi-orthogonal decomposition of $D^bCoh(\tilde{X})$ in terms of the derived category $D^bCoh(S)$ and $D^bCoh(X)$. This should give a long exact sequence relating the Hochschild cohomology of these factors via the results in Kuznetsov's Semiorthogonal decompositions in algebraic geometry.
My question:
In light of Kontsevich's conjecture that the closed-open map from Hochschild cohomology to quantum cohomology is always an isomorphism. How are the two facts above related? (It's very possible they are not, and I'm just missing something very simple...)
More generally (because there are also higher genus invariants here...) How does this fit into the whole story of non-commutative hodge structures and Gantara-Perutz-Sheridan "Hodge MS"?
