New derived symmetries of some hyperkähler varieties

Abstract

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.

56 pages; corrected an error in Lemma 4.3, which did not affect the main results

References in corpus (3)

• Hochschild homology and semiorthogonal decompositions
• Integral Transforms and Deformations of K3 Surfaces
• Grassmannian twists on the derived category via spherical functors

Cited by in corpus (39)

• Hodge theory and derived categories of cubic fourfolds
• Categories of Massless D-Branes and del Pezzo Surfaces
• Moduli spaces of torsion sheaves on K3 surfaces and derived equivalences
• Derived autoequivalences of generalised Kummer varieties
• Equivalences of equivariant derived categories
• Integral Transforms and Deformations of K3 Surfaces
• Derived categories of (nested) Hilbert schemes
• Categorical Plücker Formula and Homological Projective Duality
• Mukai flops and P-twists
• Formality of $\mathbb{P}$-objects
• P-functor versions of the Nakajima operators
• A note on spherical functors
• Perverse sheaves of categories and some applications
• New derived autoequivalences of Hilbert schemes and generalised Kummer varieties
• ${\mathbb{P}}^n$-functors
• Automorphism groups of cubic fourfolds and K3 categories
• Fourier-Mukai transform of vector bundles on surfaces to Hilbert scheme
• Spherical functors on the Kummer surface
• Stable vector bundles on generalized Kummer varieties
• Rational singularities of nested Hilbert schemes
• Universal functors on symmetric quotient stacks of Abelian varieties
• Rational Hodge isometries of hyper-Kahler varieties of K3[n]-type are algebraic
• Serre functors and dimensions of residual categories
• Stability of some vector bundles on Hilbert schemes of points on K3 surfaces
• Perverse schobers and wall crossing
• Bar category of modules and homotopy adjunction for tensor functors
• Varieties with $\mathbb{P}$-units
• Spherical functors and the flop-flop autoequivalence
• Perverse schobers on Riemann surfaces: constructions and examples
• Remarks on the derived McKay correspondence for Hilbert schemes of points and tautological bundles
• Motives of moduli spaces on K3 surfaces and of special cubic fourfolds
• Relative stable categories and birationality
• Frobenius and spherical codomains and neighbourhoods
• On the derived category of the Hilbert scheme of points on an Enriques surface
• On some examples of spherical functors related to flops
• Exoflops in Two Dimensions
• Stability and certain ${\mathbb{P}}^n$-functors
• Asymmetry of $\mathbb{P}$-Functors
• Relations among $\mathbb{P}$-Twists