Higher symmetric powers of tautological bundles on Hilbert schemes of points on a surface

We study general symmetric powers SL of a tautological bundle L on the Hilbert scheme X [n] of n points over a smooth quasi-projective surface X, associated to a line bundle L on X. Let VL be the Sn-vector bundle on X n defined as the exterior direct sum L · · · L. We prove that the Bridgeland-King-Reid transform Φ(SL) of symmetric powers SL is quasi isomorphic to the last term of a finite decreasing filtration on the natural vector bundle SVL, defined by kernels of operators D L, which operate locally as higher order restrictions to pairwise diagonals. We use this description and the natural filtration on (SVL) Sn induced by the decomposition in direct sum, to obtain, for n = 2 or k ≤ 4, a finite decreasing filtration W• on the direct image μ∗(SL) for the Hilbert-Chow morphism whose graded sheaves we control completely. As a consequence of this structural result, we obtain a chain of cohomological consequences, like a spectral sequence abutting to the cohomology of symmetric powers SL, an effective vanishing theorem for the cohomology of symmetric powers SL⊗DA twisted by the determinant, in presence of adequate positivity hypothesis on L and A, as well as universal formulas for their Euler-Poincaré characteristic.