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Published in:
2019 | OriginalPaper | Chapter
7. Complex Supermanifolds
Author : Enno Keßler
Published in: Supergeometry, Super Riemann Surfaces and the Superconformal Action Functional
Publisher: Springer International Publishing
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Abstract
In this chapter the theory of smooth families of complex supermanifolds is introduced. Families of complex supermanifolds are locally given by \(\mathbb {C}^{m|n}\) and patched by smooth families of holomorphic coordinate changes. Consequently, every smooth family of complex supermanifolds has an underlying (real) family of smooth supermanifolds with an almost complex structure. However, not every smooth family of supermanifolds with almost complex structure lead to a smooth family of complex supermanifolds. A “super” version of the Newlander–Nirenberg-Theorem, originally due to McHugh (J Math Phys 30(5):1039–1042, 1989), Vaintrob (Almost complex structures on supermanifolds. In: Leites D (ed) Reports of the Department of Mathematics, University of Stockholm. Seminar on supermanifolds No 24(6), pp 140–144, 1988) applies to families of supermanifolds.