Topological sigma model and Donaldson-type invariants in Gromov theory
type: Item
instance of: scholarly article
title: Topological sigma model and Donaldson-type invariants in Gromov theory
zbMATH Open document ID: 0864.53032
DOI: 10.1215/S0012-7094-96-08316-7
author: Yongbin Ruan
published in: Duke Mathematical Journal
publication date: 1997-06-17T00:00:00Z
review text: The main purpose of the paper is to establish Donaldson-type invariants (coming from Donaldson's gauge theory) in Gromov's theory of the moduli space of pseudo-holomorphic curves. The author introduces two Donaldson-type invariants: \(\Phi\) counting the number of holomorphic curves \(\Phi_{(A,J,\omega)} (\alpha_1, \dots, \alpha_n)= \sum_im(C_i)\), and the invariant \(\widetilde\Phi\) counting the number of perturbed holomorphic maps \(f:S^2\to (V,\omega)\) (which fulfill the perturbed equation \(\overline \partial_Jf=g)\) with marked points. He gives a rigorous proof of the existence of such invariants, finds the most important properties of them and shows their most interesting applications to the geometry of Gross 3-folds and quantum cohomology rings. He proves that Gross 3-folds are diffeomorphic, have the same homotopy class of associated, almost complex structures up to diffeomorphism, but they are not symplectic deformation equivalent. Also he proves that there are no diffeomorphisms between Gross 3-folds which induce an isomorphism on the quantum cohomology ring.
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 37J99
zbMATH DE Number: 912133
zbMATH Keywords: symplectic invariants
zbMATH Keywords: Donaldson invariants
zbMATH Keywords: quantum cohomology ring
reviewed by: Stanisław Janeczko
MaRDI profile type: Publication
cites work: Q3708977
cites work: Q5331105
cites work: Elliptic Three-folds I: Ogg-Shafarevich Theory
cites work: Q3998602
cites work: Symplectic fixed points and holomorphic spheres
cites work: Pseudo holomorphic curves in symplectic manifolds
cites work: On finite groups of exponent \(p^ m q^ n\)
cites work: The Deformation Space of Calabi-Yau n-folds with canonical singularities can be obstructed
cites work: Examples of symplectic structures
cites work: Elliptic methods in symplectic geometry
cites work: The Structure of Rational and Ruled Symplectic 4-Manifolds
cites work: Symplectic manifolds with contact type boundaries
cites work: Q4307618
cites work: Q3871006
cites work: Mirror symmetry and rational curves on quintic threefolds: a guide for mathematicians
cites work: Q4039830
cites work: Q3688496
cites work: Symplectic topology on algebraic 3-folds
cites work: Symplectic topology and extremal rays
cites work: Q4353233
cites work: A mathematical theory of quantum cohomology
cites work: Q4840428
cites work: Topological sigma models
cites work: The Kähler cone on Calabi-Yau threefolds
cites work: Gromov's compactness of pseudo-holomorphic curves and symplectic geometry
cites work: Gromov's Compactness Theorem for Pseudo Holomorphic Curves
description: scientific article; zbMATH DE number 912133