The local behaviour of holomorphic curves in almost complex 4-manifolds. (English) Zbl 0736.53038

A \(J\)-holomorphic curve in an almost complex manifold \((V,J)\) is a map \(f: \varepsilon\rightarrow V\) defined on the Riemannian surface \((\Sigma,J_ 0)\) such that \(df\cdot J_ 0=J\cdot df\). The following result stated by M. Gromov in [ Invent. Math. 82, 307-347 (1985; Zbl 0592.53025)]is proved here: Two closed distinct \(J\)-holomorphic curves \(C\) and \(C'\) in an almost complex 4-manifold \((V,J)\) have only a finite number of intersection points. Each such point \(x\) contributes a number \(k_ x\geq1\) to the algebraic intersection number \(C\cdot C'\). Moreover \(k_ x=1\) only if \(C\) and \(C'\) intersect transversally at \(x\). Another result proved here states that the virtual genus \(g(C)\) of the \(J\)-holomorphic image of the closed Riemann surface \(\Sigma\) is an integer which is not less than the genus of \(\Sigma\) and the equality holds if and only if \(C\) is embedded.
Reviewer: C.-L.Bejan (Iaşi)


53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)


Zbl 0592.53025
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