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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 ...

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 ...