The Derived Category of the Intersection of Four Quadrics

TL;DR

This paper describes the semi-orthogonal decomposition of the derived category of a general complete intersection of four quadrics, revealing a twisted derived equivalence of Calabi-Yau 3-folds with a geometric approach.

Contribution

It provides a geometric construction of the derived category decomposition for intersections of four quadrics, avoiding non-commutative varieties and confirming a predicted Calabi-Yau equivalence.

Findings

Semi-orthogonal decomposition of derived categories established

Twisted derived equivalence of Calabi-Yau 3-folds demonstrated for n=4

Geometric construction differs from previous non-commutative approaches

Abstract

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.