4454 articles – 13148 Notices  [english version]
 HAL : hal-00137742, version 1
 arXiv : math.AG/0301131
 (2004)
 Gauge theoretical Gromov-Witten invariants and virtual fundamental classes
 (2004)
 This article is an expanded version of talks given by the authors in Oberwolfach, Bochum, and at the Fano Conference in Torino. Some new results (e. g. the material concerning flag varieties, Quot spaces over $\P^1$, and the generalized quiver representations) were included. The main goal is the construction of gauge theoretical Gromov-Witten type invariants of arbitrary genus associated with certain symplectic factorization problems with additional symmetry, and the computation of these invariants in terms of complex geometric objects. The main tool for describing moduli spaces associated with symplectic factorization problems is the "universal Kobayashi-Hitchin correspondence", which gives canonical isomorphisms ${\cal M}^*\to{\cal M}^{\rm st}$ between gauge theoretic moduli spaces of irreducible solutions of certain PDE's and complex geometric moduli spaces of stable framed holomorphic objects. We state a conjecture for the general situation: When the gauge theoretic problem is of Fredholm type, and the data for ${\cal M}^{\rm st}$ are algebraic, then ${\cal M}^{\rm st}$ admits a canonical perfect obstruction theory in the sense of Behrend-Fantechi, and the Kobayashi-Hitchin isomorphism ${\cal M}^*\to{\cal M}^{\rm st}$ identifies the gauge theoretic and the algebraic virtual fundamental classes. The conjecture was checked for the symplectic factorization problems which yield the toric varieties.
 1 : Laboratoire d'Analyse, Topologie, Probabilités (LATP) CNRS : UMR6632 – Université de Provence - Aix-Marseille I – Université Paul Cézanne - Aix-Marseille III
 Domaine : Mathématiques/Géométrie algébriqueMathématiques/Géométrie différentielleMathématiques/Géométrie symplectique
 Lien vers le texte intégral : http://fr.arXiv.org/abs/math.AG/0301131
 hal-00137742, version 1 http://hal.archives-ouvertes.fr/hal-00137742 oai:hal.archives-ouvertes.fr:hal-00137742 Contributeur : Import Arxiv <> Soumis le : Mercredi 21 Mars 2007, 16:45:14 Dernière modification le : Mercredi 21 Mars 2007, 16:45:14