The Verra involution on special cubic fourfolds

Source abstract

We present a detailed study of the Hodge theory, geometry and derived categories of Verra fourfolds, and their relationship with twisted K3 surfaces and cubic fourfolds. We classify Brauer classes of Verra type on K3 surfaces of degree 2, and use this to derive an explicit counting formula for Fourier--Mukai partners of a large class of Verra fourfolds. We develop the basic theory of cohomology of Verra fourfolds and apply it to prove a derived Torelli theorem for a large class of Verra fourfolds. As an application, we prove that L-equivalent Verra fourfolds in this class are Fourier--Mukai partners, and conjecture that this implication holds for all Verra fourfolds. We then give a construction of cubic fourfolds birational to Verra fourfolds. The construction yields examples of cubic and Verra fourfolds whose associated twisted K3 surfaces have non-trivial Brauer classes. Therefore, these fourfolds are expected to be irrational.