Noncommutative K3 conjecture for the Debarre–Voisin hyperplane section

Let YGr(3,V10)Y\subset\operatorname{Gr}(3,V_{10}) be a very general hyperplane section, and let A\mathsf{A} be the K3 category in

Db(Y)=A,E1,,E108,\operatorname{D}^b(Y)=\langle\mathsf{A},E_1,\ldots,E_{108}\rangle,

where the EiE_i are exceptional objects. Noncommutative K3 conjecture. There is no smooth projective K3 surface WW and no Brauer class α\alpha on WW such that

ADb(W,α).\mathsf{A}\simeq\operatorname{D}^b(W,\alpha).

The claim expresses the expectation that the K3 category of a very general Debarre–Voisin hyperplane section is a genuine deformation of a K3 category rather than the derived category of a twisted K3 surface; the source states that this is not known in general.

Sources & referencesView supporting material

Primary source

Marcello Bernardara, Enrico Fatighenti and Laurent Manivel, “Nested varieties of K3 type”, arXiv:1912.03144 (2019).

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