We start from a short summary of our principal result from [KK]: An example of a complex algebraic surface which is not deformation equivalent to its complex conjugate and which, moreover, has no homeomorphisms reversing the canonical class. Then, we generalize this result to higher dimensions and construct several series of higher dimensional compact complex manifolds having no homeomorphisms reversing the canonical class. | Turk J Math 26 (2002) , 1 – 25 ¨ ITAK ˙ c TUB Deformation inequivalent complex conjugated complex structures and applications Viatcheslav Kharlamov and Viktor Kulikov∗ Abstract We start from a short summary of our principal result from [KK]: an example of a complex algebraic surface which is not deformation equivalent to its complex conjugate and which, moreover, has no homeomorphisms reversing the canonical class. Then, we generalize this result to higher dimensions and construct several series of higher dimensional compact complex manifolds having no homeomorphisms reversing the canonical class. After that, we resume and broaden the applications given in [KK] and [KK2], in particular, as a new application, we propose examples of (deformation) non equivalent symplectic structures with opposite canonical classes. 1. Introduction Many of achievements in real algebraic geometry appeared as applications of complex geometry. By contrary, the results of the present paper, which has grown from a solution of some questions from the real algebraic geometry, can be considered as applications in the opposite direction. They may be of an interest for symplectic geometry too. To state the principal questions we need to fix some definitions. We choose the language of complex analytic geometry since in this setting the results sometimes look stronger than if we restricted ourselves to algebraic or K¨ahler varieties, though the K¨ ahler hypothesis could simplify several proofs and could allow one to extend the set of examples. Thus, we define a real structure on a complex manifold X as an antiholomorphic involution c : X → X. By a deformation equivalence we mean the equivalence generated by local deformations of complex manifolds. Since we are interested only in compact manifolds, this, commonly used, local deformation equivalence relation can be defined in the following way: given a proper holomorphic submersion f : W → B1 , B1 = {|z| 2 there exists a compact complex .
