On the Hilbert scheme of points of an almost complex fourfold
Voisin, Claire
Annales de l'Institut Fourier, Tome 50 (2000), p. 689-722 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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) .

@article{AIF_2000__50_2_689_0,
     author = {Voisin, Claire},
     title = {On the Hilbert scheme of points of an almost complex fourfold},
     journal = {Annales de l'Institut Fourier},
     volume = {50},
     year = {2000},
     pages = {689-722},
     doi = {10.5802/aif.1769},
     mrnumber = {2001k:32048},
     zbl = {0954.14002},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2000__50_2_689_0}
}

Voisin, Claire. On the Hilbert scheme of points of an almost complex fourfold. Annales de l'Institut Fourier, Tome 50 (2000) pp. 689-722. doi : 10.5802/aif.1769. http://gdmltest.u-ga.fr/item/AIF_2000__50_2_689_0/

[1] J. Cheah, On the Cohomology of the Hilbert scheme of points, J. Alg. Geom., 5 (1996), 479-511. | MR 97b:14005 | Zbl 0889.14001

[2] G. Ellingsrud, L. Göttsche, M. Lehn, On the cobordism class of Hilbert schemes of points on a surface, preprint. | Zbl 0976.14002

[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 0625.14002

[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 0765.14010

[5] J. Fogarty, Algebraic families on an algebraic surface, Am. J. Math., 10 (1968), 511-521. | MR 38 #5778 | Zbl 0176.18401

[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 91h:14007 | Zbl 0679.14007

[8] L. Göttsche, W. Soergel, Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces, Math. Ann., 296 (1993), 235-245. | MR 94i:14026 | Zbl 0789.14002

[9] A. Iarrobino, Hilbert scheme of points: Overview of last ten years, Proceedings symposia in Pure Math., Vol. 46 (1987), 297-320. | MR 89b:14007 | Zbl 0646.14002

[10] D. Mcduff, The local behaviour of holomorphic curves in almost complex 4-manifolds, Jour. Diff. Geo., 34 (1991), 143-164. | MR 93e:53050 | Zbl 0736.53038

[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 98h:14006 | Zbl 0915.14001

[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.