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Twisted complex geometry

Part of: Surfaces and higher-dimensional varieties General theory of differentiable manifolds Global differential geometry

Published online by Cambridge University Press:  09 April 2009

Shuguang Wang
Shuguang Wang
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211, USA, e-mail: [email protected]
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Abstract

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We introduce complex differential geometry twisted by a real line bundle. This provides a new approach to understand the various real objects that are associated with an anti-holomorphic involution. We also generalize a result of Greenleaf about real analytic sheaves from dimension 2 to higher dimensions.

MSC classification

Secondary: 58A05: Differentiable manifolds, foundations 53C07: Special connections and metrics on vector bundles (Hermite-Einstein-Yang-Mills) 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli 53C56: Other complex differential geometry
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 80 , Issue 2 , April 2006 , pp. 273 - 296
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Alling, N. L. and Greenleaf, N., Foundations of the theory of Klein surfaces, Lecture Notes in Math. 219 (Springer, Berlin, 1971).CrossRefGoogle Scholar
[2]Donaldson, S. K., ‘A new proof of a theorem of Narasimhan and Seshadri’, J. Differential Geom. 18 (1983), 269–278.CrossRefGoogle Scholar
[3]Greenleaf, N., ‘Analytic sheaves on Klein surfaces’, Pacific J. Math. 37 (1971), 671–675.CrossRefGoogle Scholar
[4]Gualtieri, M., Generalized complex geometry (Ph.D. Thesis, Oxford University, 2003).Google Scholar
[5]Katz, S. and Liu, C., ‘Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc’, Adv. Theor. Math. Phys. 5 (2001), 1–49.CrossRefGoogle Scholar
[6]Li, J. and Song, Y., ‘Open string instantons and relative stable morphisms’, Adv. Theor. Math. Phys. 5 (2001), 67–91.CrossRefGoogle Scholar
[7]Mackaay, M., ‘A note on the holonomy of connections in twisted bundles’, Can. Topol.Géom. Différ Catég. 44 (2003), 39–62.Google Scholar
[8]Natanzon, S. M., ‘Klein surfaces’, Russian Math. Surveys 45 (1990), 53–108.CrossRefGoogle Scholar
[9]Ooguri, H. and Vafa, C., ‘Knot invariants and topological strings’, Nuclear Phys. B 577 (2000), 69–108.CrossRefGoogle Scholar
[10]Sinha, S. and Vafa, C., ‘SO and Sp Chern-Simons at large N’, preprint, hep-th/0012136.Google Scholar
[11]Tian, G. and Wang, S., ‘Orientability and real Seiberg-Witten invariants’, in preparation.Google Scholar
[12]Wang, S., ‘A vanishing theorem for Seiberg-Witten invariants’, Math. Res. Lett. 2 (1995), 305–310.CrossRefGoogle Scholar
[13]Wang, S., ‘A Narasimhan-Seshadri-Donaldson correspondence over non-orientable surfaces’, Forum Math. 8 (1996), 461–474.CrossRefGoogle Scholar
[14]Welschinger, J.-Y., ‘Invariants of real rational symplectic 4-manifolds and lower bounds in real enumerative geometry’, C.R.Math. Acad. Sci. Paris 336 (2003), 341–344.CrossRefGoogle Scholar