On the Hilbert scheme of points of an almost complex fourfold

Voisin, Claire

Annales de l'Institut Fourier, Tome 50 (2000) no. 2, pp. 689-722.

Si $S$ est une surface complexe, on peut définir pour chaque entier $k$ le schéma de Hilbert ${\mathrm{Hilb}}^k\left(S\right)$, qui est une désingularisation du produit symétrique $S^{\left(k\right)}$. On construit ici plus généralement une variété différentiable ${\mathrm{Hilb}}^k\left(X\right)$ munie d’une structure presque complexe stable, pour toute variété différentiable $X$ de dimension $4$ munie d’une structure presque complexe. ${\mathrm{Hilb}}^k\left(X\right)$ est une désingularisation du produit symétrique $X^{\left(k\right)}$.

If $S$ is a complex surface, one has for each $k$ the Hilbert scheme ${\mathrm{Hilb}}^k\left(S\right)$, which is a desingularization of the symmetric product $S^{\left(k\right)}$. Here we construct more generally a differentiable variety ${\mathrm{Hilb}}^k\left(X\right)$ endowed with a stable almost complex structure, for every almost complex fourfold $X$. ${\mathrm{Hilb}}^k\left(X\right)$ is a desingularization of the symmetric product $X^{\left(k\right)}$.

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