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\(\mathbb Z_2\) and \(\mathbb Z\)-deformation theory for holomorphic and symplectic manifolds. (English) Zbl 1076.32004

Kowalski, Oldřich (ed.) et al., Complex, contact and symmetric manifolds. In honor of L. Vanhecke. Selected lectures from the international conference “Curvature in Geometry” held in Lecce, Italy, June 11–14, 2003. Boston, MA: Birkhäuser (ISBN 0-8176-3850-4/hbk). Progress in Mathematics 234, 75-103 (2005).
Introduction: Deformation theories are one of the keystone settings of contemporary geometry, appearing in very different areas and providing, through moduli space constructions, a highly powerful tool to produce new invariants.
This paper, in the first part, describes a tuning up of a general machine for deformation theory, enhancing the relationships between \(\mathbb{Z}\) and \(\mathbb{Z}_2\)-theories. Then, after presenting equivalence classes of \(A^\infty\)-algebras as an example of deformation space, to show how vast the range covered by deformation theories is, it deals with complex/holomorphic deformations and symplectic deformation. In the latter case, a totally new non-naïf theory is constructed.
By means of the results established in the first part, both in the complex/holomorphic case and the symplectic case, we define and discuss the corresponding \(\mathbb{Z}_2\)-theories (complex/holomorphic and supersymplectic structures).
For the entire collection see [Zbl 1062.53001].

MSC:

32G05 Deformations of complex structures
53D05 Symplectic manifolds (general theory)
58H15 Deformations of general structures on manifolds
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