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Let $X$ be a projective Fano 3-fold or 4-fold and let $D^b(X)$ be the bounded derived category of coherent sheaves on $X$. For what $X$ is it known a semi orthogonal decomposition into indecomposable ...

It is known that in general Globally Torelli does not imply the local Torelli theorem, see Is the Torelli map an immersion? Globally Torelli means that the period map $\mathcal{P}$ is injective and ...