Yau Manifolds Research Articles

Generalized holomorphic Cartan geometries

This is largely a survey paper, dealing with Cartan geometries in the complex analytic category. We first remind some standard facts going back to the seminal works of Felix Klein, Elie Cartan and Charles Ehresmann. Then we present the concept of a branched holomorphic Cartan geometry which was introduced by Biswas and Dumitrescu (Int Math Res Not IMRN, 2017. https://doi.org/10.1093/imrn/rny003 , arxiv:1706.04407 ). It generalizes to higher dimension the notion of a branched (flat) complex projective structure on a Riemann surface introduced by Mandelbaum. This new framework is much more flexible than that of the usual holomorphic Cartan geometries (e.g. all compact complex projective manifolds admit branched holomorphic projective structures). At the same time, this new definition is rigid enough to enable us to classify branched holomorphic Cartan geometries on compact simply connected Calabi–Yau manifolds.

JKLMR conjecture and Batyrev construction

We study a mirror interpretation of the relation between the exact partition functions of gauged linear sigma-models (GLSM) on S2 and Kähler potentials on the moduli spaces of the CY manifolds proposed by Jockers et al. We use the Batyrev mirror construction for establishing the explicit relation between GLSM and the corresponding mirror family of the Calabi–Yau manifolds, defined as hypersurfaces in weighted projective spaces. We demonstrate how to do this by the explicit calculation in the case of the quintic threefold and its mirror.

Local Calabi–Yau manifolds of type [formula omitted] via SYZ mirror symmetry

We carry out the SYZ program for the local Calabi–Yau manifolds of type A˜ by developing an equivariant SYZ theory for the toric Calabi–Yau manifolds of infinite-type. Mirror geometry is shown to be expressed in terms of the Riemann theta functions and generating functions of open Gromov–Witten invariants, whose modular properties are found and studied in this article. Our work also provides a mathematical justification for a mirror symmetry assertion of the physicists Hollowood–Iqbal–Vafa (Hollowood et al., 2008).

Gluing and deformation of asymptotically cylindrical Calabi–Yau manifolds in complex dimension three

We develop some consequences of the connection between Calabi–Yau structures and torsion-free G2 structures on compact and asymptotically cylindrical six- and seven-dimensional manifolds. Firstly, we improve the known proof that matching asymptotically cylindrical Calabi–Yau threefolds can be glued. Secondly, we give an alternative proof that the moduli space of Calabi–Yau structures on a six-dimensional real manifold is smooth, and extend it to the asymptotically cylindrical case. Finally, we prove that the gluing map of Calabi–Yau threefolds, extended between these moduli spaces, is a local diffeomorphism: that is, that every deformation of a glued Calabi–Yau threefold arises from an essentially unique deformation of the asymptotically cylindrical pieces.

Type I Almost-Homogeneous Manifolds of Cohomogeneity One—IV

This paper is one of a series in which we generalize our earlier results on the equivalence of existence of Calabi extremal metrics to the geodesic stability for any type I compact complex almost homogeneous manifolds of cohomogeneity one. In this paper, we actually carry all the earlier results to the type I cases. In Part II, we obtained a substantial amount of new Kähler–Einstein manifolds as well as Fano manifolds without Kähler–Einstein metrics. In particular, by applying Theorem 15 therein, we obtained complete results in the Theorems 3 and 4 in that paper. However, we only have partial results in Theorem 5. In this note, we provide a report of recent progress on the Fano manifolds N n , m when n > 15 and N n , m ′ when n > 4 . We provide two pictures for these two classes of manifolds. See Theorems 1 and 2 in the last section. Moreover, we present two conjectures. Once we solve these two conjectures, the question for these two classes of manifolds will be completely solved. By applying our results to the canonical circle bundles, we also obtain Sasakian manifolds with or without Sasakian–Einstein metrics. These also provide open Calabi–Yau manifolds.

Local non-collapsing of volume for the Lagrangian mean curvature flow

We prove an optimal control on the time-dependent measure of a measurable set under a reparametrized Lagrangian mean curvature flow of almost calibrated submanifolds in a Calabi–Yau manifold. Moreover we give a classification of those Lagrangian translating solitons in $${{\mathbb {C}}^{m}}$$ that evolve by this reparametrized flow.

Complete intersection Calabi–Yau manifolds with respect to homogeneous vector bundles on Grassmannians

Based on the method by Kuchle (Math Z 218(4), 563–575, 1995), we give a procedure to list up all complete intersection Calabi–Yau manifolds with respect to direct sums of irreducible homogeneous vector bundles on Grassmannians for each dimension. In particular, we give a classification of such Calabi–Yau 3-folds and determine their topological invariants. We also give alternative descriptions for some of them.

Two-Sphere Partition Functions and Kähler Potentials on CY Moduli Spaces

We study the relation between exact partition functions of gauged N = (2, 2) linear sigma-models on S2 and Kahler potentials of Calabi–Yau manifolds proposed by Jockers et al. We suggest using a mirror version of this relation. For a class of manifolds given by Fermat hypersurfaces in weighted projective spaces, we check the relation by explicit calculation.

Reflective modular forms and applications

The reflective modular forms of orthogonal type are fundamental automorphic objects generalizing the classical Dedekind eta-function. This article describes two methods for constructing such modular forms in terms of Jacobi forms: automorphic products and Jacobi lifting. In particular, it is proved that the first non-zero Fourier–Jacobi coefficient of the Borcherds modular form (the generating function of the so-called Fake Monster Lie Algebra) in any of the 23 one-dimensional cusps coincides with the Kac–Weyl denominator function of the affine algebra of the root system of the corresponding Niemeier lattice. This article gives a new simple construction of the automorphic discriminant of the moduli space of Enriques surfaces as a Jacobi lifting of the product of eight theta-functions and considers three towers of reflective modular forms. One of them, the tower of , gives a solution to a problem of Yoshikawa (2009) concerning the construction of Lorentzian Kac–Moody algebras from the automorphic discriminants connected with del Pezzo surfaces and analytic torsions of Calabi–Yau manifolds. The article also formulates some conditions on sublattices, making it possible to produce families of `daughter' reflective forms from a fixed modular form. As a result, nearly 100 such functions are constructed here. Bibliography: 77 titles.

Non-Perturbative Superpotentials and Discrete Torsion

We discuss the non-perturbative superpotential in $${{E}_{8}} \times {{E}_{8}}$$ heterotic string theory on a non-simply connected Calabi–Yau manifold X, as well as on its simply connected covering space $$\tilde {X}.$$ The superpotential is induced by the string wrapping holomorphic, isolated, genus zero curves. We show, in a specific example, that the superpotential is non-zero both on $$\tilde {X}$$ and on X avoiding the no-go residue theorem of Beasley and Witten. On the non-simply connected manifold X, we explicitly compute the leading contribution to the superpotential from all holomorphic, isolated, genus zero curves with minimal area. The reason for the non-vanishing of the superpotental on X is that the second homology class contains a finite part called discrete torsion. As a result, the curves with the same area are distributed among different torsion classes and their contributions do not cancel each other.

Transversely holomorphic branched Cartan geometry

In Biswas and Dumitrescu (2018), we introduced and studied the concept of holomorphic branched Cartan geometry. We define here a foliated version of this notion; this is done in terms of Atiyah bundle. We show that any complex compact manifold of algebraic dimension d admits, away from a closed analytic subset of positive codimension, a nonsingular holomorphic foliation of complex codimension d endowed with a transversely flat branched complex projective geometry (equivalently, a ℂPd-geometry). We also prove that transversely branched holomorphic Cartan geometries on compact complex projective rationally connected varieties and on compact simply connected Calabi–Yau manifolds are always flat (consequently, they are defined by holomorphic maps into homogeneous spaces).

Calabi Yau Hypersurfaces and SU-Bordism

V. V. Batyrev constructed a family of Calabi–Yau hypersurfaces dual to the first Chern class in toric Fano varieties. Using this construction, we introduce a family of Calabi–Yau manifolds whose SU-bordism classes generate the special unitary bordism ring $${\Omega ^{SU}}[\frac{1}{2}] \cong Z[\frac{1}{2}][{y_i}:i \geqslant 2]$$ . We also describe explicit Calabi–Yau representatives for multiplicative generators of the SU-bordism ring in low dimensions.

Conway subgroup symmetric compactifications of heterotic string

We investigate special compactifications of the heterotic string for which the space of half-BPS states is, in a natural way, a representation of various subgroups of the Conway group. These compactifications provide a useful framework for analyzing the action of some of the large symmetry groups appearing in discussions of moonshine in the physics literature. We investigate toroidal compactifications of heterotic string with 16 supersymmetries as well as asymmetric toroidal orbifolds with N = 2 supersymmetry in four dimensions that arise as compactifications. The latter Conway subgroup symmetric compactifications of the heterotic string might have some interesting implications for D-brane bound states on Calabi–Yau manifolds.

A family of compact complex and symplectic Calabi–Yau manifolds that are non-Kähler

We construct a family of $6$-dimensional compact manifolds $M(A)$, which are simultaneously diffeomorphic to complex Calabi-Yau manifolds and symplectic Calabi-Yau manifolds. They have fundamental groups $\mathbb{Z} \oplus \mathbb{Z}$, their odd-dimensional Betti numbers are even, they satisfy the hard Lefschetz property, and their real homotopy types are formal. However, $M(A) \times Y$ are not homotopy equivalent to any compact K\"ahler manifold for any topological space $Y$.

Four-qubit systems and dyonic black Hole–Black branes in superstring theory

Using dyonic solutions in the type IIA superstring theory on Calabi–Yau (CY) manifolds, we reconsider the study of black objects and quantum information theory using string/string duality in six dimensions. Concretely, we relate four-qubits with a stringy quaternionic moduli space of type IIA compactification associated with a dyonic black solution formed by black holes (BHs) and black 2-branes (B2B) carrying eight electric charges and eight magnetic charges. This connection is made by associating the cohomology classes of the heterotic superstring on [Formula: see text] to four-qubit states. These states are interpreted in terms of such dyonic charges resulting from the quaternionic symmetric space [Formula: see text] corresponding to a [Formula: see text] sigma model superpotential in two dimensions. The superpotential is considered as a functional depending on four quaternionic fields mapped to a class of Clifford algebras denoted as [Formula: see text]. A link between such an algebra and the cohomology classes of [Formula: see text] in heterotic superstring theory is also given.

Lagrangian mean curvature flows and moment maps

In this paper, we construct various examples of Lagrangian mean curvature flows in Calabi–Yau manifolds, using moment maps for actions of abelian Lie groups on them. The examples include Lagrangian self-shrinkers and translating solitons in the Euclidean spaces. Moreover, our method can be applied to construct examples of Lagrangian mean curvature flows in non-flat Calabi–Yau manifolds. In particular, we describe Lagrangian mean curvature flows in 4-dimensional Ricci-flat ALE spaces in detail and investigate their singularities.

Lagrangian Constant Cycle Subvarieties in Lagrangian Fibrations

Abstract We show that the image of a dominant meromorphic map from an irreducible compact Calabi–Yau manifold X whose general fiber is of dimension strictly between 0 and $\dim X$ is rationally connected. Using this result, we construct for any hyper-Kähler manifold X admitting a Lagrangian fibration a Lagrangian constant cycle subvariety ΣH in X which depends on a divisor class H whose restriction to some smooth Lagrangian fiber is ample. If $\dim X = 4$, we also show that up to a scalar multiple, the class of a zero-cycle supported on ΣH in CH0(X) depend neither on H nor on the Lagrangian fibration (provided b2(X) ≥ 8).

Branched Holomorphic Cartan Geometries and Calabi–Yau Manifolds

Abstract We introduce the concept of a branched holomorphic Cartan geometry. It generalizes to higher dimension the definition of branched (flat) complex projective structure on a Riemann surface introduced by Mandelbaum [25]. This new framework is much more flexible than that of the usual holomorphic Cartan geometries. We show that all compact complex projective manifolds admit a branched flat holomorphic projective structure. We also give an example of a non-flat branched holomorphic normal projective structure on a compact complex surface. It is known that no compact complex surface admits such a structure with empty branching locus. We prove that non-projective compact simply connected Kähler Calabi–Yau manifolds do not admit any branched holomorphic projective structure. The key ingredient of its proof is the following result of independent interest: if E is a holomorphic vector bundle over a compact simply connected Kähler Calabi–Yau manifold and E admits a holomorphic connection, then E is a trivial holomorphic vector bundle and any holomorphic connection on E is trivial.

A new approach for computing the geometry of the moduli spaces for a Calabi–Yau manifold

It is known that moduli spaces of Calabi–Yau (CY) manifolds are special Kähler manifolds. This structure determines the corresponding low-energy effective theory which arises in superstring compactifications on CY manifolds. In the case where a CY manifold is given as a hypersurface in the weighted projective space, we propose a new procedure for computing the Kähler potential of the moduli space. Our method is based on the fact that the moduli space of CY manifolds is a marginal subspace of the Frobenius manifold which arises on the deformation space of the corresponding Landau–Ginzburg superpotential.

A criterion for the primitivity of a birational automorphism of a Calabi–Yau manifold and an application

We shall give a sufficient condition on the primitivity of a birational automrphism of a Calabi-Yau manifold in purely algebro geometric terms. As an application, we shall give an explicit construction of Calabi-Yau manifolds of Picard number $2$ of any dimension $\ge 3$, with primitive birational automrphisms of first dynamical degree $>1$.