Kronheimer, P. B. The genus-minimizing property of algebraic curves. (English) Zbl 0810.57014 Bull. Am. Math. Soc., New Ser. 29, No. 1, 63-69 (1993). 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)]. Reviewer: R.Stern (Salt Lake City) Cited in 2 ReviewsCited in 5 Documents MSC: 57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) 57R42 Immersions in differential topology 57R95 Realizing cycles by submanifolds Keywords:homology class in a complex surface; Thom conjecture; carried by a smooth algebraic curve; minimal genus PDFBibTeX XMLCite \textit{P. B. Kronheimer}, Bull. Am. Math. Soc., New Ser. 29, No. 1, 63--69 (1993; Zbl 0810.57014) Full Text: DOI arXiv References: [1] Simon K. Donaldson, Complex curves and surgery, Inst. Hautes Études Sci. Publ. Math. 68 (1988), 91 – 97 (1989). · Zbl 0696.57007 [2] S. K. Donaldson, Polynomial invariants for smooth four-manifolds, Topology 29 (1990), no. 3, 257 – 315. · Zbl 0715.57007 · doi:10.1016/0040-9383(90)90001-Z [3] Michael H. Freedman and Frank Quinn, Topology of 4-manifolds, Princeton Mathematical Series, vol. 39, Princeton University Press, Princeton, NJ, 1990. · Zbl 0705.57001 [4] P. B. Kronheimer, papers in preparation. [5] P. B. Kronheimer and T. S. Mrowka, Gauge theory for embedded surfaces. I, Topology 32 (1993), no. 4, 773 – 826. · Zbl 0799.57007 · doi:10.1016/0040-9383(93)90051-V [6] John W. Morgan, Comparison of the Donaldson polynomial invariants with their algebro-geometric analogues, Topology 32 (1993), no. 3, 449 – 488. · Zbl 0801.57014 · doi:10.1016/0040-9383(93)90001-C [7] John W. Morgan and Tomasz S. Mrowka, A note on Donaldson’s polynomial invariants, Internat. Math. Res. Notices 10 (1992), 223 – 230. · Zbl 0787.57011 · doi:10.1155/S1073792892000254 [8] Kieran G. O’Grady, Algebro-geometric analogues of Donaldson’s polynomials, Invent. Math. 107 (1992), no. 2, 351 – 395. · Zbl 0769.14008 · doi:10.1007/BF01231894 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.