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Vestislav Apostolov : Old and new trends in Bihermitian geometry

A bihermitian structure is a Riemannian metric compatible with two distinct orthogonal complex structures. In the mathematical literature this notion appeared in 90's in the study of the curvature of conformal 4-manifolds. However, bihermitian metrics were already studied in the physics literature in the 80's, as a building bloc of what Gates, Hull and Rocek call `the target space for a (2,2) super-symmetric sigma model'. There has been a great deal of interest in bihermitian geometry more recently, motivated by its link with the notion of generalized Kaehler geometry, introduced by Gualtieri and Hitchin. In this talk I will explain some main features of 4-dimensional bihermitian manifolds, as developed in the 90's, and report on recent classification results that I obtained with M. Gualtieri and G. Dloussky.

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