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

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 $X^{\left(k\right)}$.

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 $X^{\left(k\right)}$.

[1] J. Cheah, On the Cohomology of the Hilbert scheme of points, J. Alg. Geom., 5 (1996), 479-511. | Zbl | MR

[2] G. Ellingsrud, L. Göttsche, M. Lehn, On the cobordism class of Hilbert schemes of points on a surface, preprint. | Zbl

[3] G. Ellingsrud, S-A. Strømme, On the cohomology of the Hilbert schemes of points in the plane, Inventiones Math., 87 (1987), 343-352. | Zbl

[4] B. Fantechi, L. Göttsche, The cohomology ring of the Hilbert scheme of three points on a smooth projective variety, J. reine angew. Math., 439 (1993), 147-158. | Zbl

[5] J. Fogarty, Algebraic families on an algebraic surface, Am. J. Math., 10 (1968), 511-521. | Zbl | MR

[6] P. Gauduchon, The canonical almost complex structure on the manifold of 1-jets of pseudoholomorphic mappings between two almost complex manifolds, in Holomorphic curves in symplectic geometry, M. Audin, J. Lafontaine Eds, Progress in Math. 117, Birkhäuser 1994, 69-74.

[7] L. Göttsche, The Betti numbers of the Hilbert scheme of points on a smooth projective surface, Math. Ann., 286 (1990), 193-207. | Zbl | MR

[8] L. Göttsche, W. Soergel, Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces, Math. Ann., 296 (1993), 235-245. | Zbl | MR

[9] A. Iarrobino, Hilbert scheme of points: Overview of last ten years, Proceedings symposia in Pure Math., Vol. 46 (1987), 297-320. | Zbl | MR

[10] D. Mcduff, The local behaviour of holomorphic curves in almost complex 4-manifolds, Jour. Diff. Geo., 34 (1991), 143-164. | Zbl | MR

[11] D. Mcduff, Singularities and positivity of intersection of J-holomorphic curves, in Holomorphic curves in symplectic geometry, M. Audin, J. Lafontaine Eds, Progress in Math. 117, Birkhäuser 1994, 191-215.

[12] H. Nakajima, Heisenberg algebra and Hilbert schemes of points on projective surfaces, Ann. Math., 145 (1997), 379-388. | Zbl | MR

[13] J.C. Sikorav, Some properties of holomorphic curves in almost complex manifolds, in Holomorphic curves in symplectic geometry, M. Audin, J. Lafontaine Eds, Progress in Math. 117, Birkhäuser 1994, 165-189.