Semiorthogonal decomposition and Hochschild homology conjecture for Fano varieties

Let $X$ be a complex Fano variety such that the condition

holds. Assume the $\bm{\mu}_m$-invariant \subset

\bm{\kappa}_X(\QS(X)) \cap ({\mathbb{A}}^1 \setminus {0}) = {z_1,\dots,z_N} \subset {\mathbb{A}}^1 \setminus {0}

$$is ordered so that$$

holds. Semiorthogonal decomposition and Hochschild homology conjecture. There is an $\operatorname{Aut}(X)$-invariant semiorthogonal decomposition

$$\Db(X) = \langle \cR, \cB, \cB \otimes \cL, \dots, \cB \otimes \cL^{m-1} \rangle$$

and

$$\operatorname{HH}_\bullet(\cR) = \QH_{\mathrm{can}}(X) \otimes_{\QH_{\mathrm{can}}^{\mathrm{even}}(X)} \QH_{\mathrm{can}}^{\mathrm{even}}(X)_{\bm{\kappa}_X^{-1}(0)},$$ $$\operatorname{HH}_\bullet(\cB) = \bigoplus_{i=1}^{N/m} \QH_{\mathrm{can}}(X) \otimes_{\QH_{\mathrm{can}}^{\mathrm{even}}(X)} \QH_{\mathrm{can}}^{\mathrm{even}}(X)_{\bm{\kappa}_X^{-1}(z_i)}.$$

Here $\QH_{\mathrm{can}}(X)$ is identified with $H^\bullet(X,\mathbb{C})=\operatorname{HH}_\bullet(\Db(X))$ with its Hochschild homology grading. This conjecture predicts a geometric semiorthogonal decomposition whose components reflect the decomposition of canonical quantum cohomology over the critical values of the quantum cohomology spectrum; its general validity remains open.