Topological sigma model and Donaldson-type invariants in Gromov theory
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.
scientific article; zbMATH DE number 912133
scientific article