On the Hilbert scheme of points of an almost complex fourfold

Claire Voisin

Annales de l'institut Fourier (2000)

  • Volume: 50, Issue: 2, page 689-722
  • ISSN: 0373-0956

Abstract

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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 ) .

How to cite

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Voisin, Claire. "On the Hilbert scheme of points of an almost complex fourfold." Annales de l'institut Fourier 50.2 (2000): 689-722. <http://eudml.org/doc/75433>.

@article{Voisin2000,
abstract = {If $S$ is a complex surface, one has for each $k$ the Hilbert scheme $\{\rm Hilb\}^k(S)$, which is a desingularization of the symmetric product $S^\{(k)\}$. Here we construct more generally a differentiable variety $\{\rm Hilb\}^k(X)$ endowed with a stable almost complex structure, for every almost complex fourfold $X$. $\{\rm Hilb\}^k(X)$ is a desingularization of the symmetric product $X^\{(k)\}$.},
author = {Voisin, Claire},
journal = {Annales de l'institut Fourier},
keywords = {Hilbert scheme; almost complex structure; pseudoholomorphic curves; desingularization; symmetric product; Hilbert-Chow morphism},
language = {eng},
number = {2},
pages = {689-722},
publisher = {Association des Annales de l'Institut Fourier},
title = {On the Hilbert scheme of points of an almost complex fourfold},
url = {http://eudml.org/doc/75433},
volume = {50},
year = {2000},
}

TY - JOUR
AU - Voisin, Claire
TI - On the Hilbert scheme of points of an almost complex fourfold
JO - Annales de l'institut Fourier
PY - 2000
PB - Association des Annales de l'Institut Fourier
VL - 50
IS - 2
SP - 689
EP - 722
AB - If $S$ is a complex surface, one has for each $k$ the Hilbert scheme ${\rm Hilb}^k(S)$, which is a desingularization of the symmetric product $S^{(k)}$. Here we construct more generally a differentiable variety ${\rm Hilb}^k(X)$ endowed with a stable almost complex structure, for every almost complex fourfold $X$. ${\rm Hilb}^k(X)$ is a desingularization of the symmetric product $X^{(k)}$.
LA - eng
KW - Hilbert scheme; almost complex structure; pseudoholomorphic curves; desingularization; symmetric product; Hilbert-Chow morphism
UR - http://eudml.org/doc/75433
ER -

References

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  1. [1] J. CHEAH, On the Cohomology of the Hilbert scheme of points, J. Alg. Geom., 5 (1996), 479-511. Zbl0889.14001MR97b:14005
  2. [2] G. ELLINGSRUD, L. GÖTTSCHE, M. LEHN, On the cobordism class of Hilbert schemes of points on a surface, preprint. Zbl0976.14002
  3. [3] G. ELLINGSRUD, S-A. STRØMME, On the cohomology of the Hilbert schemes of points in the plane, Inventiones Math., 87 (1987), 343-352. Zbl0625.14002
  4. [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. Zbl0765.14010
  5. [5] J. FOGARTY, Algebraic families on an algebraic surface, Am. J. Math., 10 (1968), 511-521. Zbl0176.18401MR38 #5778
  6. [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. [7] L. GÖTTSCHE, The Betti numbers of the Hilbert scheme of points on a smooth projective surface, Math. Ann., 286 (1990), 193-207. Zbl0679.14007MR91h:14007
  8. [8] L. GÖTTSCHE, W. SOERGEL, Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces, Math. Ann., 296 (1993), 235-245. Zbl0789.14002MR94i:14026
  9. [9] A. IARROBINO, Hilbert scheme of points: Overview of last ten years, Proceedings symposia in Pure Math., Vol. 46 (1987), 297-320. Zbl0646.14002MR89b:14007
  10. [10] D. MCDUFF, The local behaviour of holomorphic curves in almost complex 4-manifolds, Jour. Diff. Geo., 34 (1991), 143-164. Zbl0736.53038MR93e:53050
  11. [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. [12] H. NAKAJIMA, Heisenberg algebra and Hilbert schemes of points on projective surfaces, Ann. Math., 145 (1997), 379-388. Zbl0915.14001MR98h:14006
  13. [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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