Hostname: page-component-848d4c4894-2xdlg Total loading time: 0 Render date: 2024-07-03T20:46:02.192Z Has data issue: false hasContentIssue false

BIHERMITIAN STRUCTURES ON COMPLEX SURFACES

Published online by Cambridge University Press:  01 September 1999

V. APOSTOLOV
Affiliation:
Mathematical Institute, 24–28 St Giles', Oxford, OX1 3LB. apostolo@maths.ox.ac.uk
P. GAUDUCHON
Affiliation:
Centre de Mathématiques, UMR 7640 du CNRS, École Polytechnique, 91128 Palaiseau Cedex, France. pg@math.polytechnique.fr
G. GRANTCHAROV
Affiliation:
Department of Mathematics, University of California at Riverside, Riverside CA 52521 U.S.A.geogran@math.ucr.edu
Get access

Abstract

Bihermitian complex surfaces are oriented conformal four-manifolds admitting two independent compatible complex structures. Non-anti-self-dual bihermitian structures on ${\mathbb R}^4$ and the four-dimensional torus $T^4$ have recently been discovered by P. Kobak. We show that an oriented compact 4-manifold, admitting a non-anti-self-dual bihermitian structure, is a torus or K3 surface in the strongly bihermitian case (when the two complex structures are independent at each point) or, otherwise, must be obtained from the complex projective plane or a minimal ruled surface of genus less than 2 by blowing up points along some anti-canonical divisor (but the actual existence of bihermitian structures in the latter case is still an open question). The paper includes a general method for constructing non-anti-self-dual bihermitian structures on tori, K3 surfaces and $S^1\times S^3$. Further properties of compact bihermitian surfaces are also investigated.

1991 Mathematics Subject Classification: 53C12, 53C55, 32J15.

Type
Research Article
Copyright
1999 London Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)