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The genus-minimizing property of algebraic curves. (English) Zbl 0810.57014

An intriguing conjecture, often attributed to Thom, is that any homology class \(\zeta\) in a complex surface which is carried by a smooth algebraic curve realizes the minimal genus of all smooth embedded 2-manifolds representing \(\zeta\). This paper announces results which confirm this conjecture for a broad class of complex surfaces. Most recently, the author and T. Mrowka have verified this conjecture in general using the new equations of N. Seiberg and E. Witten [cf. “The genus of embedded surfaces in the projective plane”, Math. Res. Letters (to appear)].

MSC:

57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57R42 Immersions in differential topology
57R95 Realizing cycles by submanifolds
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References:

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