Remarks on the derived McKay correspondence for Hilbert schemes of points and tautological bundles

Abstract

We study the images of tautological bundles on Hilbert schemes of points on surfaces and their wedge powers under the derived McKay correspondence. The main observation of the paper is that using a derived equivalence differing slightly from the standard one considerably simplifies both the results and their proofs. As an application, we obtain shorter proofs for known results as well as new formulae for homological invariants of tautological sheaves. In particular, we compute the extension groups between wedge powers of tautological bundles associated to line bundles on the surface.

Universal functors on symmetric quotient stacks of Abelian varieties

“…The proof of F F is similar to the argument in [30,Theorem 3.6]. Indeed, the key observation of [20, Section 4] and [30,Section 3.1] is the relationship between the universal families:…”

“…is an equivalence too 4 ; see [30,Proposition 2.9]. Now let Z ⊂ A × A [n] be the universal subscheme and consider the Fourier-Mukai functor FM O Z : D(A) → D(A [n] ).…”

Equivariant characteristic classes of external and symmetric products of varieties

“…In fact, our prior work on symmetric products from [13] has already been used for the study of (pushforwards under the Hilbert-Chow morphism of) characteristic classes of Hilbert schemes of points on smooth quasi-projective varieties, see [12]. But for smooth surfaces, results of [12] may be improved via the McKay correspondence [23,36,37], by using the stronger equivariant results of the present paper. In addition, the present work can also be used for obtaining generating series formulae for the singular Todd classes td * (F [n] ) of tautological sheaves F [n] (associated to a given F ∈ Coh(Z )) on the Hilbert scheme Z [n] of n points on a smooth quasi-projective algebraic surface Z.…”

Stability of some vector bundles on Hilbert schemes of points on K3 surfaces

“…Since the kernel of each composition factor in (12), viewed as an object in D b (X × X ), remains the same under the permutation of the two copies of X , it follows that…”

“…Proof First of all, by (12) and [8,Lemma 8.12], a standard computation of the cohomological Fourier-Mukai transform shows that…”