Cohomology of the Hilbert scheme of points on a surface with values in the double tensor power of a tautological bundle

Some remarks on tautological sheaves on Hilbert schemes of points on a surface

“…[n] 2 [Dan00,Dan02], symmetric powers S k L [n] for k ≤ 2 and n ≤ 3 [Dan04], and then with sections of general tensor powers [Dan07]. In our previous work [Sca05,Sca06,Sca09b,Sca09a], among other things, we gave formulas for the cohomology of general double symmetric powers S 2 L [n] and for general exterior powers Λ k L [n] . Recently, Krug [Kru14] studied general tensor products of tautological bundles giving, among others, results for triple tensor products E…”

“…Recall that the Bridgeland-King-Reid transform [BKR01,Hai01,Sca05,Sca07] Φ : D b (X [n] ) -D b Sn (X n ) is the equivalence of categories between the derived category of the Hilbert scheme of n points over X and the S n -equivariant derived category of the product variety X n built (in this case) as the Fourier-Mukai transform through Haiman isospectral Hilbert scheme B n ⊂ X [n] ×X n . The image Φ(L [n] ) of a tautological bundle is resolved in [Sca05,Sca06,Sca09a] by a complex…”