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Almost analytic forms with respect to a quadratic endomorphism and their cohomology

The goal of this paper is to consider the notion of almost analytic form in a unifying setting for both almost complex and almost paracomplex geometries. We use a global formalism, which yields, in addition to generalizations of the main results of the previously known almost complex case, a relationship with the Frolicher–Nijenhuis theory. | Turk J Math (2015) 39: 322 – 334 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1404-29 Research Article Almost analytic forms with respect to a quadratic endomorphism and their cohomology Mircea CRASMAREANU, Cristian IDA∗ Faculty of Mathematics Department of Mathematics and Informatics University, “Al. I. Cuza” University Transilvania of Bra¸sov, Ia¸si Bra¸sov Romania, Romania Received: 09.04.2014 • Accepted/Published Online: 14.01.2015 • Printed: 29.05.2015 Abstract: The goal of this paper is to consider the notion of almost analytic form in a unifying setting for both almost complex and almost paracomplex geometries. We use a global formalism, which yields, in addition to generalizations of the main results of the previously known almost complex case, a relationship with the Fr¨ olicher–Nijenhuis theory. A cohomology of almost analytic forms is also introduced and studied as well as deformations of almost analytic forms with pairs of almost analytic functions. Key words: Quadratic endomorphism, almost F -analytic form, F -symmetric form, almost (para)complex structure, cohomology 1. Introduction The notion of almost analytic form was introduced a long time ago in the almost complex geometry and hence it was treated in local coordinates, especially by Japanese geometers [15, 16, 17, 18]. A global approach appeared in [14], unfortunately only in Romanian. Some of these global techniques were used in [9] and [13]; for example, in the former paper a differential is introduced in the algebra of pairs of almost analytic forms and a corresponding Poincar´e type lemma is proved. The present work aims to consider almost analytic forms in a unifying setting, which adds the almost paracomplex geometry. This type of even dimensional geometry is now in the mainstream of research as the surveys [1] and [4] and their several citations prove. In this way, we reveal the common parts of these geometries with respect