$\mathbb{P}$-functor versions of the Nakajima operators

Abstract

For a smooth quasi-projective surface X we construct a series of P n−1 -functors H ℓ,n : D b (X × X [ℓ] ) → D b (X [n+ℓ] ) for n > ℓ and n > 1 using the derived McKay correspondence. They can be considered as analogues of the Nakajima operators. The functors also restrict to P n−1 -functors on the generalised Kummer varieties. We also study the induced autoequivalences and obtain, for example, a universal braid relation in the groups of derived autoequivalences of Hilbert squares of K3 surfaces and Kummer fourfolds.given by the Fourier-Mukai transform along the ideal sheaf of the universal family Ξ ⊂ X ×X [n] for X a K3 surface and n ≥ 2. Similarly, for X = A an abelian surface andΞ ⊂ A×K n A the universal family of the generalised Kummer variety K n A ⊂ A [n+1] the functorF n = FM IΞ : D b (A) → D b (K n A) is a P n−1 -functor for n ≥ 2 and a P 1 -functor for n = 1; see [Mea12] and [KM14]. We will refer to these examples of P-functors as the universal ideal functors.An important tool for the investigation of the derived categories of the Hilbert schemes of points on surfaces is the Bridgeland-King-Reid-Haiman equivalence (also known as derived McKay correspondence; see [BKR01] and [Hai01])Sn (X n ) . This provides an identification of D b (X [n] ) with the derived category of equivariant coherent sheaves on the product X n , i.e. of coherent sheaves equipped with a linearisation by the symmetric group S n .

Universal functors on symmetric quotient stacks of Abelian varieties

“…When n = 3, our universal functor F : D(A × A) → D(A [3] ) is spherical and we can directly compare it with a similar spherical functor H : D(A × A) → D(A [3] ); constructed as part of a series of P-functors by the first author in [32], whose image is supported on the exceptional divisor. Theorem (2.9) The autoequivalences of D(A [3] ) associated to the two spherical functors: [3] ), satisfy the braid relation:…”

Automorphisms of Hilbert schemes of points on surfaces

“…But by a result of Krug [14, Theorem 1.1(ii)] 1 there always exist non-standard auto-equivalences on the Hilbert scheme. For Hilbert squares and Hilbert cubes of surfaces with ample or anti-ample canonical bundle a 1 The numbering refers to the (non yet publicly available) published version of the paper [14]. The corresponding results in the arXiv version are Theorem 1C and Conjecture 5.…”

“…The corresponding results in the arXiv version are Theorem 1C and Conjecture 5. 14. proof of [14,Conjecture 7.5] combined with Theorems 1 and 2 would give a full description of the derived auto-equivalence group.…”

Semiorthogonal decompositions of the categories of equivariant coherent sheaves for some reflection groups

“…Our main motivating example for Conjecture A is the case when X = C n , where C is a smooth curve over C, and G = S n , the symmetric group acting on the cartesian power C n by permutations of factors. We prove Conjecture A in this case by constructing an explicit semiorthogonal decomposition of D b Sn (C n ), numbered by partitions of n. Note that a less refined decomposition of D b Sn (C n ) was constructed by Krug in [45,Cor. B,Sec.…”