We describe birational models and decide the rationality/unirationality of moduli spaces AA_{d} (and AA^{lev}_{d}) of (1,d)-polarized abelian surfaces (with canonical level structure, respectively) for small values of d.
The projective lines identified in the rational/unirational moduli spaces correspond to pencils of abelian surfaces traced on nodal threefolds living naturally in the corresponding ambient projective spaces, and whose small resolutions are new Calabi-Yau threefolds with Euler characteristic zero.
We present a new construction of mirror pairs of Calabi-Yau manifolds by smoothing normal crossing varieties, consisting of two quasi-Fano manifolds. We introduce a notion of mirror pairs of quasi-Fano manifolds with anticanonical Calabi-Yau fibrations using recent conjectures about Landau-Ginzburg models. Utilizing this notion, we give pairs of normal crossing varieties and show that the pairs of smoothed Calabi-Yau manifolds satisfy the Hodge number relations of mirror symmetry. We consider quasi-Fano threefolds that are some blow-ups of Gorenstein toric Fano threefolds and build 6518 mirror…
Any Calabi-Yau threefold X with infinite fundamental group admits an étale Galois covering either by an abelian threefold or by the product of a K3 surface and an elliptic curve. We call X of type A in the former case and of type K in the latter case. In this paper, we provide the full classification of Calabi-Yau threefolds of type K, based on Oguiso and Sakurai's work. Together with a refinement of Oguiso and Sakurai's result on Calabi-Yau threefolds of type A, we finally complete the classification of Calabi-Yau threefolds with infinite fundamental group.
Most of Calabi-Yau manifolds that have been considered by physicists are complete intersection Calabi-Yau manifolds of toric varieties or some quotients of product types. Purpose of this paper is to introduce a different and rather new kind of construction method of Calabi-Yau manifolds by pasting two non-compact Calabi-Yau manifolds. We will also in some details explain a curious and mysterious similarity with construction of some $G_2$-manifolds (also called Joyce manifolds), which are base spaces for M-theory.
We carry out a systematic analysis of Calabi-Yau threefolds that are elliptically fibered with section ("EFS") and have a large Hodge number h^{2, 1}. EFS Calabi-Yau threefolds live in a single connected space, with regions of moduli space associated with different topologies connected through transitions that can be understood in terms of singular Weierstrass models. We determine the complete set of such threefolds that have h^{2, 1} >= 350 by tuning coefficients in Weierstrass models over Hirzebruch surfaces. The resulting set of Hodge numbers includes those of all known Calabi-Yau threefold…
We review the construction of families of projective varieties, in particular Calabi--Yau threefolds, as quasilinear sections in weighted flag varieties. We also describe a construction of tautological orbi-bundles on these varieties, which may be of interest in heterotic model building.
We construct stable bundle extensions on elliptically fibered Calabi-Yau threefolds. We show that these bundles can solve the topological anomaly constraint in heterotic string theory without the need of invoking background fivebranes.
This paper studies the action of the Fourier-Mukai transform on moduli spaces of vertical torsion sheaves on elliptic Calabi-Yau threefolds in Weierstrass form. Moduli stacks of semistable one dimensional sheaves on such threefolds are identified with open and closed substacksof moduli stacks of vertical semistable two dimensional sheaves on their Fourier-Mukai duals. In particular, this yields explicit conjectural results for Donaldson-Thomas invariants of vertical two dimensional sheaves on K3-fibered elliptic Calabi-Yau threefolds.
We consider the 8-supercharge 5d su(N) gauge theories from M-theory compactified on elliptic Calabi-Yau threefolds. By matching the triple intersection numbers in the elliptic Calabi-Yau with the 5d Chern-Simons levels, we determine the charged matter contents for these theories. We show that all these 5d theories can be lifted to 6d N = (1, 0) theories while satisfying the anomaly cancellation equations. This suggests that the 5d theories obtained from M-theory compactified on elliptic Calabi-Yau threefolds have a natural 12d description, which as we know is F-theory. Furthermore, we compute …
We find through a systematic analysis that all but 29,223 of the 473.8 million 4D reflexive polytopes found by Kreuzer and Skarke have a 2D reflexive subpolytope. Such a subpolytope is generally associated with the presence of an elliptic or genus one fibration in the corresponding birational equivalence class of Calabi-Yau threefolds. This extends the growing body of evidence that most Calabi-Yau threefolds have an elliptically fibered phase.
We obtain a detailed classification for a class of non-simply connected Calabi-Yau threefolds which are of potential interest for a wide range of problems in string phenomenology. These threefolds arise as quotients of Schoen's Calabi-Yau threefolds, which are fiber products over P1 of two rational elliptic surfaces. The quotient is by a freely acting finite abelian group preserving the fibrations. Our work involves a classification of restricted finite automorphism groups of rational elliptic surfaces.
We solve Bershadsky-Cecotti-Ooguri-Vafa (BCOV) holomorphic anomaly equation to determine the higher genus Gromov-Witten invariants ($g \leq 5$) of the derived equivalent Calabi-Yau threefolds, which are of the appropriate codimensions in the Grassmannian Gr(2,7) and the Pfaffian Pf(7).
Recently J.C. Rohde constructed families of Calabi-Yau threefolds parametrised by Shimura varieties. The points corresponding to threefolds with CM are dense in the Shimura variety and, moreover, the families do not have boundary points with maximal unipotent monodromy. Both aspects are of interest for Mirror Symmetry. In this paper we discuss one of Rohde's examples in detail and we explicitly give the Picard-Fuchs equation for this one dimensional family.
There are easy "polynomial" deformations of Calabi-Yau hypersurfaces in toric varieties performed by changing the coefficients of the defining polynomial of the hypersurface. In this paper, we explicitly constructed the ``non-polynomial'' deformations of Calabi-Yau hypersurfaces, which arise from deformations of the ambient toric variety.
Kreuzer and Skarke famously produced the largest known database of Calabi-Yau threefolds by providing a complete construction of all 473,800,776 reflexive polyhedra that exist in four dimensions. These polyhedra describe the singular limits of ambient toric varieties in which Calabi-Yau threefolds can exist as hypersurfaces. In this paper, we review how to extract topological and geometric information about Calabi-Yau threefolds using the toric construction, and we provide, in a companion online database (see http://nuweb1.neu.edu/cydatabase), a detailed inventory of these quantities which are…
We study the limit of Calabi-Yau modular forms, and in particular, those resulting in classical modular forms. We then study two parameter families of elliptically fibred Calabi-Yau fourfolds and describe the modular forms arising from the degeneracy loci. In the case of elliptically fibred Calabi-Yau threefolds our approach gives a mathematical proof of many observations about modularity properties of topological string amplitudes starting with the work of Candelas, Font, Katz and Morrison. In the case of Calabi-Yau fourfolds we derive new identities not computed before.
We show that "non-polynomial" deformations of semiample (minimal) nondegenerate Calabi-Yau hypersurfaces in complete simplicial toric varieties can be realized as quasismooth complete intersections in higher dimensional simplicial toric varieties.
We prove a case of the conjecture of Douglas, Reinbacher and Yau about the existence of stable vector bundles with prescribed Chern classes on a Calabi-Yau threefold. For this purpose we prove the existence of certain stable vector bundle extensions over elliptically fibered Calabi-Yau threefolds.
In this paper we present a construction of stable bundles on Calabi-Yau threefolds using the method of bundle extensions. This construction applies to any given Calabi-Yau threefold with h^{1,1}>1. We give examples of stable bundles of rank 2 and 4 constructed out of pure geometric data of the given Calabi-Yau space. As an application, we find that some of these bundles satisfy the physical constraint imposed by heterotic string anomaly cancellation.
We develop some methods to construct normal crossing varieties whose dual complexes are two-dimensional, which are smoothable to Calabi--Yau threefolds. We calculate topological invariants of smoothed Calabi--Yau threefolds and show that several of them are new examples.
We discuss Calabi-Yau and fractional Calabi-Yau semiorthogonal components of derived categories of coherent sheaves on smooth projective varieties. The main result is a general construction of a fractional Calabi-Yau category from a rectangular Lefschetz decomposition and a spherical functor. We give many examples of application of this construction and discuss some general properties of Calabi-Yau categories.
This note is a report on the observation that the Enriques-Fano threefolds with terminal cyclic quotient singularities admit Calabi-Yau threefolds as their double coverings. We calculate the invariants of those Calabi-Yau threefolds when the Picard number is one. It turns out that all of them are new examples.
We prove that a given Calabi-Yau threefold with a stable holomorphic vector bundle can be perturbed to a solution of the Strominger system provided that the second Chern class of the vector bundle is equal to the second Chern class of the tangent bundle. If the Calabi-Yau threefold has strict SU(3) holonomy then the equations of motion derived from the heterotic string effective action are also satisfied by the solutions we obtain.
We study Kustin-Miller unprojections between Calabi-Yau threefolds or more precisely the geometric transitions they induce. We use them to connect many families of Calabi-Yau threefolds with Picard number one to the web of Calabi Yau complete intersections. This enables us to find explicit description of a few known families of Calabi-Yau threefolds in terms of equations. Moreover we find two new examples of Calabi-Yau threefolds with Picard group of rank one, described by Pfaffian equations in weighted projective spaces.
We study the geometry of $3$-codimensional smooth subvarieties of the complex projective space. In particular, we classify all quasi-Buchsbaum Calabi--Yau threefolds in projective $6$-space. Moreover, we prove that this classification includes all Calabi--Yau threefolds contained in a possibly singular 5-dimensional quadric as well as all Calabi--Yau threefolds of degree at most $14$ in $\mathbb{P}^6$.
We study Calabi--Yau 3-folds with infinitely many divisorial contractions. We also suggest a method to describe Calabi--Yau 3-folds with the infinite automorphism group.
We present a list of arithmetically Gorenstein Calabi-Yau threefolds in $\mathbb{P}^7$ and give evidence that this is a complete list. In particular we construct three new families of arithmetically Gorenstein Calabi-Yau threefolds in $\mathbb{P}^7$ for which no mirror construction is known.
We study Kustin--Miller unprojections of Calabi--Yau threefolds. As an application we work out the geometric properties of Calabi--Yau threefolds defined as linear sections of determinantal varieties. We compute their Hodge numbers and describe the morphisms corresponding to the faces of their Kähler--Mori cone.
We survey the foundations for Donaldson-Thomas invariants for stable sheaves on algebraic threefolds with trivial canonical bundle, with emphasis on the case of abelian threefolds.
We establish a global Torelli theorem for the complete family of Calabi-Yau threefolds arising from cyclic triple covers of $\mathbb P^3$ branched along stable hyperplane arrangements.
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 Threefolds.'' (alg-geom 9506022). If X' is a non-singular Calabi-Yau threefold, and f:X'->X is a contraction of a divisor to a …
We consider certain elliptic threefolds over the projective plane (more generally over certain rational surfaces) with a section in Weierstrass normal form. In particular, over a del Pezzo surface of degree 8, these elliptic threefolds are Calabi-Yau threefolds. We will discuss especially the generating functions of Gromov-Witten and Gopakumar-Vafa invariants.
We prove an orbifold Riemann--Roch formula for a polarized 3--fold (X,D). As an application, we construct new families of projective Calabi--Yau threefolds.
In this note, we consider crepant resolutions of the quotient varieties of smooth quintic threefolds by Gorenstein group actions. We compute their Hodge numbers via McKay correspondence. In this way, we find some new pairs $(h^{1,1},\;h^{2,1})$ of Hodge numbers of Calabi-Yau threefolds.
We study noncompact Calabi-Yau threefolds, their moduli spaces of vector bundles and deformation theory. We present Calabi-Yau threefolds that have infinitely many distinct deformations, constructing them explicitily, and describe the effect that such deformations produce on moduli spaces of vector bundles.
We argue that the existence of genus one fibrations with multisections of high degree on certain Calabi-Yau threefolds implies the existence of pairs of such varieties that are not birational, but are derived equivalent. It also (likely) implies the existence of non-birational counterexamples to the Torelli problem for Calabi-Yau threefolds.
We study sections of a Calabi-Yau threefold fibered over a curve by K3 surfaces. We show that there exist infinitely many isolated sections on certain K3 fibered Calabi-Yau threefolds and the subgroup of the Néron-Severi group generated by these sections is not finitely generated. This also gives examples of K3 surfaces over the function field F of a complex curve with Zariski dense F-rational points, whose geometric model is Calabi-Yau.
In this article, we summarize combinatorial description of complete intersection Calabi-Yau threefolds in Hibi toric varieties. Such Calabi-Yau threefolds have at worst conifold singularities, and are often smoothable to non-singular Calabi-Yau threefolds. We focus on such non-singular Calabi-Yau threefolds of Picard number one, and illustrate the calculation of topological invariants, using new motivating examples.
We calculate the Brauer group of a certain Calabi-Yau threefold discovered by Mark Gross and Sorin Popescu, which is a small resolution of a complete intersection of type (2,2,2,2) with 64 ordinary double points. We find the Brauer group is (Z/8Z)^2, which is the largest known example of a Brauer group of a Calabi-Yau threefold.
We shall give an explicit pair of birational projective Calabi--Yau threefolds which are rigid, non-homeomorphic, but are connected by projective flat deformation over some connected base scheme.
We study the syzygetic structure of projections of del Pezzo surfaces in order to construct singular Calabi-Yau threefolds. By smoothing those threefolds, we obtain new examples of Calabi-Yau threefolds with Picard group of rank 1. We also give an example of type II primitive contraction whose exceptional divisor is the blow-up of the projective plane at a point.
We investigate the globally generated vector bundles on complete intersection Calabi-Yau threefolds with the first Chern class at most 2. We classify all the globally generated vector bundles of an arbitrary rank on quintic in $\mathbb{P}^4$ and investigate the globally generated vector bundles of rank 2 on complete intersection Calabi-Yau threefolds of codimension 2.
In this short note we construct Calabi-Yau threefolds with nonabelian fundamental groups of order 64 as quotients of the small resolutions of certain complete intersections of quadrics in $\PP^7$ that were first considered by M. Gross and S. Popescu.
In this paper we are concerned with the monodromy of Picard-Fuch differential equations associated with one-parameter families of Calabi-Yau threefolds. Our results show that in the hypergeometric cases the matrix representations of monodromy relative to the Frobenius bases can be expressed in terms of the geometric invariants of the underlying Calabi-Yau threefolds. This phenomenon is also verified numerically for other families of Calabi-Yau threefolds in the paper. Furthermore, we discover that under a suitable change of bases the monodromy groups are contained in certain congruence subgrou…
We study some conjectures about Chow groups of varieties of geometric genus one. Some examples are given of Calabi-Yau threefolds where these conjectures can be verified, using the theory of finite-dimensional motives.
We find a simple geometric construction of Tonoli's examples of Calabi--Yau threefolds of degree 17 in complex $\mathbb{P}^6$. We prove that the rank of the Picard group of elements of one of these families is at least $2$.
Kapustka and Rampazzo have exhibited pairs of Calabi-Yau threefolds $X$ and $Y$ that are L-equivalent and derived equivalent, without being birational. We complete the picture by showing that $X$ and $Y$ have isomorphic Chow motives.
In this note, an overview of Calabi-Yau varieties in positive characteristic is presented. Although Calabi-Yau varieties in characteristic zero are unobstructed, there are examples of Calabi-Yau threefolds in positive characteristic which cannot be lifted to characteristic zero, although one-dimensional and two-dimensional Calabi-Yau varieties, i.e., elliptic curves and K3 surfaces, are all liftable to characteristic zero. In this respect, Calabi-Yau threefolds in positive characteristic are interesting in view of deformation theory and they are still very mysterious.
The aim of this paper is to classify indecomposable rank 2 arithmetically Cohen-Macaulay (ACM) bundles on compete intersection Calabi-Yau (CICY) threefolds and prove the existence of some of them. New geometric properties of the curves corresponding to rank 2 ACM bundles (by Serre correspondence) are obtained. These follow from minimal free resolutions of curves in suitably chosen fourfolds (containing Calabi-Yau threefolds as hypersurfaces). Also the existence of an indecomposable vector bundle of higher rank on a CICY threefold of type (2,4) is proved.
We prove that a nef line bundle $\mathcal L$ with $c_1(\mathcal L)^2 \ne 0$ on a Calabi-Yau threefold $X$ with Picard number $2$ and with $c_3(X) \ne 0$ is semiample, that is, some multiple of $\mathcal L$ is generated by global sections.
In this paper, the numbers of rational curves on general complete intersection Calabi-Yau threefolds in complex projective spaces are computed up to degree six. The results are all in agreement with the predictions made from mirror symmetry.
This paper presents five new examples of modular rigid Calabi-Yau threefolds arising from the modular elliptic surface of level 6. Explicit correspondences to newforms of weight 4 and level 10, 17, 21, and 73 are given.
The main purpose of this article is to discuss a project relating Gopakumar-Vafa invariants to quantum K-invariants on Calabi-Yau threefolds. Results in genus zero, including recent and forthcoming works, are reported.
The concept of non-Gorenstein involutions on Calabi-Yau threefolds is a higher dimensional generalization of non-symplectic involutions on $K3$ surfaces. We present some elementary facts about Calabi-Yau threefolds with non-Gorenstein involutions. We give a classification of the Calabi-Yau threefolds of Picard rank one with non-Gorenstein involutions whose fixed locus is not zero-dimensional.
Ito-Miura-Okawa-Ueda have constructed a pair of Calabi-Yau threefolds $X$ and $Y$ that are L-equivalent and derived equivalent, but not stably birational. We complete the picture by showing that $X$ and $Y$ have isomorphic Chow motives.
We construct 16 new examples of Calabi--Yau threefolds with Picard group of rank 1. Each of these examples is obtained by smoothing the image of a primitive contraction with exceptional divisor being a del Pezzo surface of degree 6, 7 or $\mathbb{P}^1\times \mathbb{P}^1$.
