# zbMATH — the first resource for mathematics

Compactification of the moduli spaces of vortices and coupled vortices. (English) Zbl 1022.53025
In the study of the coupled vortex equations after O. García-Prada [see Int. J. Math. 5, 1-52 (1994; Zbl 0799.32022) and Commun. Math. Phys. 156, 527-546 (1993; Zbl 0790.53031)], compactification problems are discussed. It is shown that the moduli space of ideal couple vortices $$IV_\tau$$ (new notion) is compact, that the moduli space $$V_\tau$$ on Hermitian vector bundles admits a compactification embedded in $$IV_\tau$$, that the space of vortices on a Hermitian bundle over a compact Kähler manifold admits a compactification in the space of ideal coupled vortices. These theorems give a compactification of the moduli space of corresponding holomorphic objects, stable pairs and triples. Hence [see the theorems on removable singularities for admissible Hermitian-Yang-Mills (HYM) connections and ideal coupled vortices characterize the non-removable singularities precisely, see also A. Vanzurová, Acta Univ. Palacki Olomuc., Fac. Rerum. Nat. 100, Math. 30, 257-271 (1991; Zbl 0762.53014)]. Possible directions for further studies are: boundary points of the compactified moduli spaces, stability and semistability for reflexive sheaf pairs and triples via Hitchin-Kobayashi correspondence, topology of the compactified moduli space.
Reviewer: M.Rahula (Tartu)

##### MSC:
 53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) 53C55 Global differential geometry of Hermitian and Kählerian manifolds 57R20 Characteristic classes and numbers in differential topology
Full Text:
##### References:
 [1] S. Bando, Removable singularities for holomorphic vector bundles, To^hoku Math. J. (2) 43 (1991), 61-67. · Zbl 0736.32011 [2] S. Bando, Y.T. Siu, Stable sheaves and Einstein-Hermitian metrics, Geometry and analysis on complex manifolds, World Sci. Publishing (1994), 39-50. · Zbl 0880.32004 [3] F. Bethuel, H. Brezis, F. Helein, Ginzburg-Landau vortices, Progr. Nonlin. Di . Equ. Appl. 13, Birkh user, Boston 1994. [4] S., Comm. Math. Phys. 135 pp 1– (1990) [5] S., Geom. 33 pp 169– (1991) [6] S. B. Bradlow, G. D. Daskalopoulos, O. Garci?a-Prada, R. Wentworth, Stable augmented bundles over Riemann surfaces, Vector bundles in algebraic geometry, Cambridge Univ. Press (1995), 15-67. · Zbl 0827.14010 [7] Bradlow S. B., Internat. J. Math. 4 pp 903– (1993) [8] Bradlow S. B., Internat. J. Math. 2 pp 477– (1991) [9] S. K. Donaldson, P. B. Kronheimer, The geometry of four-manifolds, The Clarendon Press, Oxford University Press, 1990. · Zbl 0820.57002 [10] Internat. J. Math. 5 pp 1– (1994) [11] Comm. Math. Phys. 156 pp 527– (1993) [12] R. Harvey, B. Shi man, A characterization of holomorphic chains, Ann. Math. (2) 99 (1974), 553-587. · Zbl 0287.32008 [13] A. Ja e, C. Taubes, Vortices and monopoles, Structure of static gauge theories, Progr. Phys. 2, Birkh user, 1980. [14] J., Acta Math. 127 pp 185– (1971) [15] S. Kobayashi, Di erential geometry of complex vector bundles, Princeton University Press, 1987. [16] J. Math. Soc. Japan pp 40– (1988) [17] Manuscr. Math. 43 pp 131– (1983) [18] R. Schoen, S. T. Yau, Lectures on harmonic maps, International Press, 1997. · Zbl 0886.53004 [19] Y.T. Siu, A Hartogs type extension theorem for coherent analytic sheaves, Ann. Math. (2) 93 (1971), 166-188. · Zbl 0208.10404 [20] Tian G., Ann. Math. 151 pp 193– (2000) [21] Uhlenbeck K. K., Comm. Math. Phys. 83 pp 11– (1982) [22] Uhlenbeck K. K., Comm. Math. Phys. 83 pp 31– (1982)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.