BIHERMITIAN STRUCTURES ON COMPLEX SURFACES

• Published in: Proceedings of the London Mathematical Society Vol. 79; no. 2; pp. 414 - 428
• Main Authors: APOSTOLOV, V., GAUDUCHON, P., GRANTCHAROV, G.
• Format: Journal Article
• Language: English
• Published: Cambridge University Press 01.09.1999; Oxford University Press
• Subjects: bihermitian structures; complex surfaces; Hermitian geometry; bihermitian structures; complex surfaces; Hermitian geometry
• ISSN: 0024-6115; 1460-244X
• DOI: 10.1112/S0024611599012058

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.