Local invariants of singular surfaces in an almost complex four-manifold
Let \(f: S^2\to (M^4, J)\) be a generic mapping, that is, \(f\) is a finite cover of an almost everywhere immersion (except for a finite number of points) which has only finite complex points. To such a mapping \(f\) the authors assign local invariants \(i(x)\) and \(m(x)\) at every point \(x\in f (S^2)\). The invariant \(i(x)\) is defined as the local intersection number of \(V\) and its canonical perturbation \(V'\) around \(x\) (an idea of McDuff in her paper on local behaviour of pseudo-holomorphic curves in symplectic 4-manifolds), and the invariant \(m(x)\) is a generalization of the Maslov index by Givental (for Lagrangian embedding in symplectic 4-manifolds). The authors obtain formulae relating these local invariants at singular points of \(f\) and topological invariant of \(V\) (through Euler class, first Chern class and self-intersection number).
pseudo-holomorphic curves in symplectic 4-manifolds
Lagrangian embedding in symplectic 4-manifolds
scientific article; zbMATH DE number 727084
scientific article