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Let $X\subset P^5$ be a smooth cubic fourfold. It is well known that its variety of lines $F(X)$ is a smooth fourfold Fano variety. Hence its derived category should have a semi-orthogonal decomposition. Is a SOD of $D^b(F(X))$ known?

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    $\begingroup$ What do you mean by "Fano variety"? The classical name for a Hilbert scheme parameterizing linear subschemes of a given projective variety is a "Fano scheme". However, the dualizing sheaf of $F(X)$ is trivial. Thus $F(X)$ is not "Fano" in the sense of having anti-ample dualizing sheaf. $\endgroup$
    – Jason Starr
    Commented Jul 25, 2018 at 17:03
  • $\begingroup$ Of course you are right! I was mistaken by a comment on another question, my apologies. $\endgroup$
    – IMeasy
    Commented Jul 25, 2018 at 17:34
  • $\begingroup$ $F(X)$ is a famous example of hyper-K\"ahler manifold deformation equivalent to $\mathrm{Hilb}^2(K3)$ (see Beauville-Donagi). The derived category of a (connected) variety with trivial canonical bundle has no non-trivial SOD. This is a standard trick due to Bridgeland. $\endgroup$
    – Libli
    Commented Jul 26, 2018 at 8:59
  • $\begingroup$ Of course you are both right. It's embarrassing how I messed it out in my mind! Thanks anyway for caring to show me I was nonsense. $\endgroup$
    – IMeasy
    Commented Jul 27, 2018 at 16:11

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