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

Abstract

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 $L⊞\cdots ⊞L$. 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 $k\le 4$, a finite decreasing filtration ${\mathcal{W}}^{\bullet }$ on the direct image ${\mu }_{\ast }(S^k{L}^{[n]})$ 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-Poincaré characteristic.

pdflatex, 70 pages

Cited by in corpus (3)

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• K-Theoretic Descendent Series for Hilbert Schemes of Points on Surfaces