Deformations of Calabi-Yau manifolds in Fano toric varieties
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Deformations of Calabi‑Yau manifolds in Fano toric varieties Gilberto Bini1 · Donatella Iacono2 Received: 24 May 2020 / Accepted: 25 September 2020 © The Author(s) 2020
Abstract In this article, we investigate deformations of a Calabi-Yau manifold Z in a toric variety F, possibly not smooth. In particular, we prove that the forgetful morphism from the Hilbert functor HZF of infinitesimal deformations of Z in F to the functor of infinitesimal deformations of Z is smooth. This implies the smoothness of HZF at the corresponding point in the Hilbert scheme. Moreover, we give some examples and include some computations on the Hodge numbers of Calabi-Yau manifolds in Fano toric varieties. Keywords Fano toric varieties · Calabi-Yau manifolds · Deformations of subvarieties Mathematics Subject Classification 14M10 · 14M25 · 32G10
1 Introduction In this paper, we focus our attention on Calabi-Yau manifolds, i.e., projective manifolds with trivial canonical bundle and without holomophic p-forms. More precisely, if we focus on dimension greater than or equal to three, Z is a Calabi-Yau manifold of dimension n if p the canonical bundle KZ ∶= ΩnZ is trivial and H 0 (Z, ΩZ ) vanishes for p in between 0 and n. Since the canonical bundle is trivial, Z has unobstructed deformations, i.e., the moduli space of deformations of Z is smooth. This is the famous Bogomolov-Tian-Todorov Theorem [4, 5, 27, 28]. A more algebraic proof of this fact [17, 19, 24] shows that the functor DefZ of infinitesimal deformations of Z is smooth too. In particular, the dimension of the moduli space at the point corresponding to Z is the dimension of H 1 (Z, TZ ) , where TZ denotes the tangent bundle of Z. Although we know that the moduli space is smooth, we still miss a geometric understanding of it; for instance, the number of its irreducible components is unknown. A famous conjecture by M. Reid claims that the moduli space of simply connected smooth Calabi-Yau threefolds is connected via conifold transitions [25]. The * Gilberto Bini [email protected] Donatella Iacono [email protected] 1
Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, Via Archirafi n. 34, 90123 Palermo, Italy
2
Dipartimento di Matematica, Università degli Studi di Bari, Via E. Orabona 4, 70125 Bari, Italy
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G. Bini, D. Iacono
general picture is still unknown but in some cases there has been quite a lot of progress. For example, the moduli spaces of complete intersection Calabi-Yau 3-folds in products of projective spaces are connected with each other by a sequence of conifold transitions (see [29] and references therein). If Z is contained in an ambient manifold X, we can investigate the deformation functor HZX of deformations of Z in X (fixed) and the forgetful functor 𝜙 ∶ HZX → DefZ , which associates with an infinitesimal deformation of Z in X the isomorphism class of the deformation of Z. For example, if 𝜙 is smooth we can conclude that all deformations of Z lie in X and, since DefZ is smooth, the functor
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