Invariants of real symplectic four-manifolds out of reducible and cuspidal curves

Jean-Yves Welschinger

Bulletin de la Société Mathématique de France (2006)

  • Volume: 134, Issue: 2, page 287-325
  • ISSN: 0037-9484

Abstract

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We construct invariants under deformation of real symplectic four-manifolds. These invariants are obtained by counting three different kinds of real rational J -holomorphic curves which realize a given homology class and pass through a given real configuration of (the appropriate number of) points. These curves are cuspidal curves, reducible curves and curves with a prescribed tangent line at some real point of the configuration. They are counted with respect to some sign defined by the parity of their number of isolated real double points and in the case of reducible curves, with respect to some mutiplicity. In the case of the complex projective plane equipped with its standard symplectic form and real structure, these invariants coincide with the ones previously constructed in [Welschinger 2005]. This leads to a relation between the count of real rational J -holomorphic curves done in [Welschinger 2005] and the count of real rational reducible J -holomorphic curves presented here.

How to cite

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Welschinger, Jean-Yves. "Invariants of real symplectic four-manifolds out of reducible and cuspidal curves." Bulletin de la Société Mathématique de France 134.2 (2006): 287-325. <http://eudml.org/doc/272451>.

@article{Welschinger2006,
abstract = {We construct invariants under deformation of real symplectic four-manifolds. These invariants are obtained by counting three different kinds of real rational $J$-holomorphic curves which realize a given homology class and pass through a given real configuration of (the appropriate number of) points. These curves are cuspidal curves, reducible curves and curves with a prescribed tangent line at some real point of the configuration. They are counted with respect to some sign defined by the parity of their number of isolated real double points and in the case of reducible curves, with respect to some mutiplicity. In the case of the complex projective plane equipped with its standard symplectic form and real structure, these invariants coincide with the ones previously constructed in [Welschinger 2005]. This leads to a relation between the count of real rational $J$-holomorphic curves done in [Welschinger 2005] and the count of real rational reducible $J$-holomorphic curves presented here.},
author = {Welschinger, Jean-Yves},
journal = {Bulletin de la Société Mathématique de France},
keywords = {real symplectic manifold; rational curve; enumerative geometry},
language = {eng},
number = {2},
pages = {287-325},
publisher = {Société mathématique de France},
title = {Invariants of real symplectic four-manifolds out of reducible and cuspidal curves},
url = {http://eudml.org/doc/272451},
volume = {134},
year = {2006},
}

TY - JOUR
AU - Welschinger, Jean-Yves
TI - Invariants of real symplectic four-manifolds out of reducible and cuspidal curves
JO - Bulletin de la Société Mathématique de France
PY - 2006
PB - Société mathématique de France
VL - 134
IS - 2
SP - 287
EP - 325
AB - We construct invariants under deformation of real symplectic four-manifolds. These invariants are obtained by counting three different kinds of real rational $J$-holomorphic curves which realize a given homology class and pass through a given real configuration of (the appropriate number of) points. These curves are cuspidal curves, reducible curves and curves with a prescribed tangent line at some real point of the configuration. They are counted with respect to some sign defined by the parity of their number of isolated real double points and in the case of reducible curves, with respect to some mutiplicity. In the case of the complex projective plane equipped with its standard symplectic form and real structure, these invariants coincide with the ones previously constructed in [Welschinger 2005]. This leads to a relation between the count of real rational $J$-holomorphic curves done in [Welschinger 2005] and the count of real rational reducible $J$-holomorphic curves presented here.
LA - eng
KW - real symplectic manifold; rational curve; enumerative geometry
UR - http://eudml.org/doc/272451
ER -

References

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  9. [9] J.-Y. Welschinger – « Towards relative invariants of real symplectic four-manifolds », Geom. Funct. Anal., To appear, see math.SG/0502358. Zbl1107.53059MR2276536
  10. [10] —, « Enumerative invariants of strongly semipositive real symplectic manifolds », Preprint math.AG/0509121 (2005). 
  11. [11] —, « Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry », Invent. Math.162 (2005), p. 195–234. Zbl1082.14052MR2198329
  12. [12] —, « Spinor states of real rational curves in real algebraic convex 3 -manifolds and enumerative invariants », Duke Math. J.127 (2005), p. 89–121. Zbl1084.14056MR2126497
  13. [13] —, « Invariant count of holomorphic disks in the cotangent bundles of the two-sphere and real projective plane », Research announcement (2006). Zbl1118.53057

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