We study general symmetric powers $S^k L^{ [n]}$ of a tautological bundle $L^{ [n]}$ 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 $V_L$ be the $\mathfrak{ S}_n$-vector bundle on $X^n$ defined as the exterior direct sum $L \boxplus \cdots \boxplus L$. We prove that the Bridgeland-King-Reid transform $\mathbf{ \Phi}(S^k L^{ [n]})$ of symmetric powers $S^k L^{ [n]}$ is quasi isomorphic to the last term of a finite decreasing filtration on the natural vector bundle $S^k V_L$, defined by kernels of operators $D^l_L$, which operate locally as higher order restrictions to pairwise diagonals. We use this description and the natural filtration on $(S^k V_L)^{ \mathfrak{ S}_n}$ induced by the decomposition in direct sum, to obtain, for $n =2$ or $k \leq 4$, a finite decreasing filtration $\mathcal{ W}^\bullet$ on the direct image $\mu_*(S^k L^{ [n]})$ 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 $S^k L^{ [n]}$, an effective vanishing theorem for the cohomology of symmetric powers $S^k L^{ [n]} \otimes \mathcal{ D}_A$ twisted by the determinant, in presence of adequate positivity hypothesis on $L$ and $A$, as well as universal formulas for their Euler-Poincar\'e characteristic.
Categories: math.AG