Nicolas Addington

New derived symmetries of some hyperkähler varieties

We construct a new autoequivalence of the derived category of the Hilbert scheme of n points on a K3 surface, and of the variety of lines on a smooth cubic 4-fold. For Hilb^2 and the variety of lines, we use the theory of spherical functors; to deal with Hilb^n for n > 2 we develop a theory of P-functors. We conjecture that the same construction yields an autoequivalence for any moduli space of sheaves on a K3 surface. In an appendix we give a cohomology and base change criterion which is well-known to experts, but not well-documented.

On the symplectic eightfold associated to a Pfaffian cubic fourfold

We show that the irreducible holomorphic symplectic eightfold Z associated to a cubic fourfold Y not containing a plane is deformation-equivalent to the Hilbert scheme of four points on a K3 surface. We do this by constructing for a generic Pfaffian cubic Y a birational map Z ---> Hilb^4(X), where X is the K3 surface associated to Y by Beauville and Donagi. We interpret Z as a moduli space of complexes on X and observe that at some point of Z, hence on a Zariski open subset, the complex is just the ideal sheaf of four points.

Categories of massless D-branes and del Pezzo surfaces

A bstractIn analogy with the physical concept of a massless D-brane, we define a notion of “

Mukai flops and ℙ-twists

\mathbb{Q}\hbox{-}\mathrm{masslessness}

Moduli spaces of torsion sheaves on K3 surfaces and derived equivalences

” for objects in the derived category. This is defined in terms of monodromy around singularities in the stringy Kähler moduli space and is relatively easy to study using “spherical functors”. We consider several examples in which del Pezzo surfaces and other rational surfaces in Calabi-Yau threefolds are contracted. For precisely the del Pezzo surfaces that can be written as hypersurfaces in weighted

D-brane probes, branched double covers, and noncommutative resolutions

{{\mathbb{P}}^3}

The Derived Category of the Intersection of Four Quadrics

, the category of

THE PFAFFIAN-GRASSMANNIAN EQUIVALENCE REVISITED

\mathbb{Q}\hbox{-}\mathrm{massless}

Cubic fourfolds fibered in sextic del Pezzo surfaces

objects is a “fractional Calabi-Yau” category of graded matrix factorizations.

Some non-special cubic fourfolds

Associated to a Mukai flop X⇢X′ is on the one hand a sequence of equivalences Db(X)→Db(X′), due to Kawamata and Namikawa, and on the other hand a sequence of autoequivalences of Db(X), due to Huybrechts and Thomas. We work out a complete picture of the relationship between the two. We do the same for standard flops, relating Bondal and Orlov’s derived equivalences to spherical twists, extending a well-known story for the Atiyah flop to higher dimensions.