K3 Categories and Mukai Homomorphisms for Debarre-Voisin Varieties

Let M=X₆^σ be a smooth Debarre-Voisin hyperkähler fourfold and X=X₃^σ the associated smooth hyperplane section of Gr(3,10). Using the incidence correspondence between X and M, we construct a family E of objects in the Kuznetsov component Ku(X). We write v for the common Mukai vector of its fibers. We prove that the associated Mukai homomorphism \[ ΘE|v^⊥ v^⊥{~}H^2(M, Z) \] is an integral Hodge isometry and explicitly determine the inverse image of the Plücker polarization. For a very general point of the trivector divisor corresponding to discriminant $24$, we also construct an integral Hodge isometry between the full Mukai lattices of Ku(X) and the associated twisted K3 surface.