Preliminaries on (Almost-)Complex Manifolds

  • Daniele Angella
Part of the Lecture Notes in Mathematics book series (LNM, volume 2095)


In this preliminary chapter, we summarize some basic notions and some classical results in (almost-)complex and symplectic geometry. In particular, we start by setting some definitions and notation concerning (almost-)complex structures, Sect. 1.1, symplectic structures, Sect. 1.2, and generalized complex structures, Sect. 1.3; then we recall the main results in the Hodge theory for Kähler manifolds, Sect. 1.4, and in the Kodaira, Spencer, Nirenberg, and Kuranishi theory of deformations of complex structures, Sect. 1.5; furthermore, we summarize some basic definitions and some useful facts about currents and de Rham homology, Sect. 1.6, and about nilmanifolds and solvmanifolds, Sect. 1.7, in order to set the notation for the following chapters. (As a matter of notation, unless otherwise stated, by “manifold” we mean “connected differentiable manifold”, and by “compact manifold” we mean “closed manifold”.)


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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Daniele Angella
    • 1
  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly

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