On the Hilbert scheme of points of an almost complex fourfold
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 Hilb k (S), qui est une désingularisation du produit symétrique S (k) . On construit ici plus généralement une variété différentiable Hilb k (X) munie d’une structure presque complexe stable, pour toute variété différentiable X de dimension 4 munie d’une structure presque complexe. Hilb k (X) est une désingularisation du produit symétrique X (k) .

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

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     title = {On the {Hilbert} scheme of points of an almost complex fourfold},
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Voisin, Claire. On the Hilbert scheme of points of an almost complex fourfold. Annales de l'Institut Fourier, Tome 50 (2000) no. 2, pp. 689-722. doi : 10.5802/aif.1769. http://www.numdam.org/articles/10.5802/aif.1769/

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

[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. | MR | Zbl

[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. | MR | Zbl

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

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

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

[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. | MR | Zbl

[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.

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