Complete intersection Calabi--Yau threefolds in Hibi toric varieties and their smoothing

We present a study of certain singular one-parameter subfamilies of Calabi-Yau threefolds realized as anticanonical hypersurfaces or complete intersections in toric varieties. Our attention to these families is motivated by the Doran-Morgan classification of variations of Hodge structure which can underlie families of Calabi-Yau threefolds with $h^{2,1} = 1$ over the thrice-punctured sphere. We explore their torically induced fibrations by $M$-polarized K3 surfaces and use th...

Birational Calabi-Yau threefolds in the same deformation family provide a `weak' counterexample to the global Torelli problem, as long as they are not isomorphic. In this paper, it is shown that deformations of certain desingularized weighted complete intersections provide examples of families containing birational varieties. The constructed examples are shown to be non-isomorphic using a specialization argument.

In this work we systematically enumerate genus one fibrations in the class of 7,890 Calabi-Yau manifolds defined as complete intersections in products of projective spaces, the so-called CICY threefolds. This survey is independent of the description of the manifolds and improves upon past approaches that probed only a particular algebraic form of the threefolds (i.e. searches for "obvious" genus one fibrations as in [1,2]). We also study K3-fibrations and nested fibration str...

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 $H^F_Z$ of infinitesimal deformations of $Z$ in $F$ to the functor of infinitesimal deformations of $Z$ is smooth. This implies the smoothness of $H^F_Z $ at the corresponding point in the Hilbert scheme. Moreover, we give some examples and include some computations on the Hodge numb...

While Calabi-Yau hypersurfaces in toric ambient spaces provide a huge number of examples, theoretical considerations as well as applications to string phenomenology often suggest a broader perspective. With even the question of finiteness of diffeomorphism types of CY 3-folds unsettled, an important idea is Reid's conjecture that the moduli spaces are connected by certain singular transitions. We summarize the results of our recent construction of a large class of new CY spac...

A primitive Calabi-Yau threefold is a non-singular Calabi-Yau threefold which cannot be written as a crepant resolution of a singular fibre of a degeneration of Calabi-Yau threefolds. These should be thought as the most basic Calabi-Yau manifolds; all others should arise through degenerations of these. This paper first continues the study of smoothability of Calabi-Yau threefolds with canonical singularities begun in the author's previous paper, ``Deforming Calabi-Yau Threefo...

In the context of string dualities, fibration structures of Calabi-Yau manifolds play a prominent role. In particular, elliptic and K3 fibered Calabi-Yau fourfolds are important for dualities between string compactifications with four flat space-time dimensions. A natural framework for studying explicit examples of such fibrations is given by Calabi-Yau hypersurfaces in toric varieties, because this class of varieties is sufficiently large to provide examples with very differ...

We consider the construction of Calabi-Yau varieties recently generalized to where the defining equations may have negative degrees over some projective space factors in the embedding space. Within such "generalized complete intersection" Calabi-Yau ("gCICY") three-folds, we find several sequences of distinct manifolds. These include both novel elliptic and K3-fibrations and involve Hirzebruch surfaces and their higher dimensional analogues. En route, we generalize the standa...

For complete intersection Calabi-Yau manifolds in toric varieties, Gross and Haase-Zharkov have given a conjectural combinatorial description of the special Lagrangian torus fibrations whose existence was predicted by Strominger, Yau and Zaslow. We present a geometric version of this construction, generalizing an earlier conjecture of the first author.

In this note, we prove combinatorial formulas for $h^{2,1}$ of prime toric divisors in an arbitrary toric hypersurface Calabi-Yau fourfold $Y_4.$ We show that it is possible to find a toric hypersurface Calabi-Yau in which there are more than $h^{1,1}(Y_4)$ non-perturbative superpotential terms with trivial intermediate Jacobian. Hodge numbers of divisors in toric CICYs are the subjects of the part two.