Abstract
The aim of this chapter is to give a partial and informal introduction to classical deformation theory of complex manifolds and moves around the notion of a family of compact complex manifolds, defined as a proper holomorphic submersion \(f:M\rightarrow B\) of complex manifolds. The origin of the problem lies in the fact that, while the differential structure of the fibres \(M_t:=f^{-1}(t)\) remains unchanged when B is connected, several examples show that in general \(M_t\) and \(M_s\) have different complex structures for \(t\not =s\).
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Notes
- 1.
This is a set and not a proper class; in fact every complex structure can be interpreted as an almost complex structure and therefore as an element of the set of splittings of the complexified tangent bundle. The same fact also follows immediately from Corollary 4.3.9. We inform the reader that this footnote should be considered an exception and, for simplicity of exposition, from now on we shall frequently ignore any kind of set theoretic difficulties.
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Manetti, M. (2022). An Overview of Deformation Theory of Complex Manifolds. In: Lie Methods in Deformation Theory. Springer Monographs in Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-19-1185-9_1
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DOI: https://doi.org/10.1007/978-981-19-1185-9_1
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Online ISBN: 978-981-19-1185-9
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