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We investigate topological properties of Calabi-Yau fourfolds and consider a wide class of explicit constructions in weighted projective spaces and, more generally, toric varieties. Divisors which lead to a non-perturbative superpotential…
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…
In this work we study genus one fibrations in Calabi-Yau three-folds with a non-trivial first fundamental group. The manifolds under consideration are constructed as smooth quotients of complete intersection Calabi-Yau three-folds (CICYs)…
In a previous study, we constructed a family of elliptic Calabi-Yau 4-folds possessing a geometric structure that allowed them to be split into a pair of rational elliptic 4-folds. In the present study, we introduce a method of classifying…
We construct examples of primitive contractions of Calabi--Yau threefolds with exceptional locus being $ \mathbb{P}^1 \times \mathbb{P}^1$, $\mathbb{P}^2$, and smooth del Pezzo surfaces of degrees $\leq 5$. We describe the images of these…
Given a smooth 3-fold $Y$, a line bundle $L \to Y$, and a section $s$ of $L$ such that the vanishing locus of $s$ is a normal crossings surface $X$ with graph-like singular locus, we present a way to reconstruct the singularity category of…
This is the last in a series of five papers math.DG/0211294, math.DG/0211295, math.DG/0302355, math.DG/0302356 studying compact special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities x_1,...,x_n…
It is a fundamental problem in geometry to decide which moduli spaces of polarized algebraic varieties are embedded by their period maps as Zariski open subsets of locally Hermitian symmetric domains. In the present work we prove that the…
We present a complete classification of all arrangements of eight planes in projective threespace that give rise to double octic Calabi-Yau threefolds. Building on earlier work, we determine all 455 combinatorial types and describe the…
We give some examples of Calabi-Yau 3-folds with $\rho=1$, defined over $\mathbb{Q}$ and constructed as 4-codimensional subvarieties of $\mathbb{P}^7$ via commutative algebra methods. We explain how to deduce their Hodge diamond and top…
We give a simple construction, starting with any elliptic curve E, of an n-dimensional Calabi-Yau variety of Kummer type (for any n>1), by considering the quotient Y of the n-fold self-product of E by a natural action of the alternating…
Torus-fibered Calabi-Yau threefolds Z, with base dP_9 and fundamental group pi_1(Z)=Z_2 X Z_2, are reviewed. It is shown that Z=X/(Z_2 X Z_2), where X=B X_{P_1} B' are elliptically fibered Calabi-Yau threefolds that admit a freely acting…
We prove that a generic complete intersection Calabi-Yau 3-fold defined by sections of ample line bundles on a product of projective spaces admits a conifold transition to a connected sum of S^{3} \times S^{3}. In this manner, we obtain…
We describe a new kind of transition between topologically distinct $N=2$ type II Calabi--Yau vacua through points with enhanced non-abelian gauge symmetries together with fundamental charged matter hypermultiplets. We connect the…
We analyze freely-acting discrete symmetries of Calabi-Yau three-folds defined as hypersurfaces in ambient toric four-folds. An algorithm which allows the systematic classification of such symmetries which are linearly realised on the toric…
Calabi-Yau threefolds with infinitely many flops to isomorphic manifolds have an extended Kahler cone made up from an infinite number of individual Kahler cones. These cones are related by reflection symmetries across flop walls. We study…
Calabi-Yau algebras are particularly symmetric differential graded algebras. There is a construction called `Calabi-Yau completion' which produces a canonical Calabi-Yau algebra from any homologically smooth dg algebra. Homologically smooth…
We find new examples of compact Spin(7)-manifolds using a construction of Joyce. The essential ingredient in Joyce's construction is a Calabi-Yau 4-orbifold with particular singularities admitting an antiholomorphic involution, which fixes…
We study graded and ungraded singularity categories of some commutative Gorenstein toric singularities, namely, Veronese subrings of polynomial rings, and Segre products of some copies of polynomial rings. We show that the graded…
The SYZ Conjecture explains Mirror Symmetry between mirror Calabi-Yau 3-folds M,M' in terms of special Lagrangian fibrations f : M --> B and f' : M' --> B over the same base B, whose fibres are dual 3-tori, except for singular fibres. One…