The Derived Category of the Intersection of Four Quadrics

The derived category of a general complete intersection of four quadrics in P^{2n-1} has a semi-orthogonal decomposition < O(-2n+9), …, O(-1), O, D >, where D is the derived category of twisted sheaves on a certain non-algebraic complex 3-fold coming from a moduli problem. In particular, when n=4 we obtain a (twisted) derived equivalence of Calabi-Yau 3-folds predicted by Gross. This differs from Kuznetsov’s result in that our construction is geometric and avoids non-commutative varieties.