Abstract
We extend the notion of small essential deformations of Calabi–Yau complex structures from the case of the Iwasawa manifold, for which they were introduced recently by the first-named author, to the general case of page-1-partial {{bar{partial }}}-manifolds that were jointly introduced very recently by all three authors. We go on to obtain an analogue of the unobstructedness theorem of Bogomolov, Tian and Todorov for Calabi–Yau page-1-partial {{bar{partial }}}-manifolds. As applications of this discussion, we study the small deformations of certain Nakamura solvmanifolds and reinterpret the cases of the Iwasawa manifold and its 5-dimensional analogue from this standpoint.
Highlights
In this paper, we begin to investigate the role that the new class of page-1-∂∂ ̄ -manifolds introduced in [11] plays in the theory of deformations of complex structures and in the new approach to Mirror Symmetry, extended to the possibly non-Kähler context, proposed in [10].(I) On the one hand, we introduce in Sect. 3 the notion of small essential deformations of an arbitrary compact Calabi-Yau page-1-∂∂ ̄ -manifold.The special case of the 3-dimensional Iwasawa manifold I (3) was treated in [10]
As a consequence of this discussion, we propose the following definition of small essential deformations of the complex structure of X induced by a given metric ω on X
(II) On the other hand, we study in Sect. 4 the possible unobstructedness of the small deformations, both essential and ordinary, of Calabi–Yau page-1-∂∂ ̄ -complex structures
Summary
We begin to investigate the role that the new class of page-1-∂∂ ̄ -manifolds introduced in [11] plays in the theory of deformations of complex structures and in the new approach to Mirror Symmetry, extended to the possibly non-Kähler context, proposed in [10]. (I) On the one hand, we introduce in Sect. 3 the notion of small essential deformations of an arbitrary compact Calabi-Yau page-1-∂∂ ̄ -manifold. The special case of the 3-dimensional Iwasawa manifold I (3) (a complex parallelisable nilmanifold that is a Calabi–Yau page-1-∂∂ ̄ -manifold) was treated in [10]. Recall that a compact complex manifold X is said to be complex parallelisable if its holomorphic tangent
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