Derived Categories of Cubic 4-Folds

Derived Categories of Cubic 4-Folds

Tue, 08/02/2011
15:45 		Nicolas Addington (Imperial College London) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Derived Categories of Cubic 4-Folds
If $ X $ is a Fano variety with canonical bundle $ O(-k) $, its derived category has a semi-orthogonal decomposition (I will say what that means)
\[ D(X) = \langle O(-k+1), ..., O(-1), O, A \rangle, \]
where the subcategory $ A $ is the "interesting piece" of $ D(X) $. In the previous talk we saw that $ A $ can have very rich geometry. In this talk we will see a less well-understood example of this: when $ X $ is a smooth cubic in $ P^5 $, $ A $ looks like the derived category of a K3 surface. We will discuss Kuznetsov's conjecture that $ X $ is rational if and only if $ A $ is geometric, relate it to Hassett's earlier work on the Hodge theory of $ X $, and mention an autoequivalence of $ D(Hilb^2(K3)) $ that I came across while studying the problem.