Examples Of Calabi Research Articles

On the Curvature of the Bismut Connection: Bismut–Yamabe Problem and Calabi–Yau with Torsion Metrics

We study two natural problems concerning the scalar and the Ricci curvatures of the Bismut connection. Firstly, we study an analog of the Yamabe problem for Hermitian manifolds related to the Bismut scalar curvature, proving that, fixed a conformal Hermitian structure on a compact complex manifold, there exists a metric with constant Bismut scalar curvature in that class when the expected constant scalar curvature is non-negative. A similar result is given in the general case of Gauduchon connections. We then study an Einstein-type condition for the Bismut Ricci curvature tensor on principal bundles over Hermitian manifolds with complex tori as fibers. Thanks to this analysis, we construct explicit examples of Calabi–Yau with torsion Hermitian structures and prove a uniqueness result for them.

On the Numerical Dimension of Calabi–Yau 3-Folds of Picard Number 2

Abstract We show that for any smooth Calabi–Yau threefold $X$ of Picard number $2$ with infinite birational automorphism group, the numerical dimension $\kappa _\sigma $ of the extremal rays of the movable cone of $X$ is $\frac {3}{2}$. Furthermore, we provide new examples of Calabi–Yau threefolds of Picard number $2$ with infinite birational automorphism group.

On the Chow Ring of Cynk–Hulek Calabi–Yau Varieties and Schreieder Varieties

AbstractThis note is about certain locally complete families of Calabi–Yau varieties constructed by Cynk and Hulek, and certain varieties constructed by Schreieder. We prove that the cycle class map on the Chow ring of powers of these varieties admits a section, and that these varieties admit a multiplicative self-dual Chow–Künneth decomposition. As a consequence of both results, we prove that the subring of the Chow ring generated by divisors, Chern classes, and intersections of two cycles of positive codimension injects into cohomology via the cycle class map. We also prove that the small diagonal of Schreieder surfaces admits a decomposition similar to that of K3 surfaces. As a by-product of our main result, we verify a conjecture of Voisin concerning zero-cycles on the self-product of Cynk–Hulek Calabi–Yau varieties, and in the odd-dimensional case we verify a conjecture of Voevodsky concerning smash-equivalence. Finally, in positive characteristic, we show that the supersingular Cynk–Hulek Calabi–Yau varieties provide examples of Calabi–Yau varieties with “degenerate” motive.

A Lagrangian sphere which is not a vanishing cycle

We give examples of Calabi–Yau threefolds containing Lagrangian spheres which are not vanishing cycles of nodal degenerations, answering a question of Donaldson in the negative.

Defect formula for nodal complete intersection threefolds

In this paper, we give formulas for the Hodge numbers of a nodal complete intersection in a complex projective space. We apply these formulas to construct examples of Calabi–Yau threefolds with [Formula: see text].

On Yau’s theorem for effective orbifolds

In 1978 Yau (Yau, 1978) confirmed a conjecture due to Calabi (1954) stating the existence of Kähler metrics with prescribed Ricci forms on compact Kähler manifolds. A version of this statement for effective orbifolds can be found in the literature (Joyce, 2000; Boyer and Galicki, 2008; Demailly and Kollár, 2001). In this expository article, we provide details for a proof of this orbifold version of the statement by adapting Yau’s original continuity method to the setting of effective orbifolds in order to solve a Monge–Ampère equation. We then outline how to obtain Kähler–Einstein metrics on orbifolds with negative first Chern class by solving a slightly different Monge–Ampère equation. We conclude by listing some explicit examples of Calabi–Yau orbifolds, which consequently admit Ricci flat metrics by Yau’s theorem for effective orbifolds.

Tonoli's Calabi–Yau threefolds revisited

We find a simple construction of Tonoli's examples of Calabi–Yau threefolds in complex P6. We prove that the rank of the Picard group of elements of one of these families is at least 2.

Some Calabi–Yau fourfolds verifying Voisin’s conjecture

Motivated by the Bloch–Beilinson conjectures, Voisin has made a conjecture concerning zero-cycles on self-products of Calabi–Yau varieties. This note contains some examples of Calabi–Yau fourfolds verifying Voisin’s conjecture.

$${\mathrm {SU}}(4)$$ SU ( 4 ) -holonomy via the left-invariant hypo and Hitchin flow

The Hitchin flow constructs eight-dimensional Riemannian manifolds (M, g) with holonomy in $${\mathrm {Spin}}(7)$$ starting with a cocalibrated $$\mathrm {G}_2$$ -structure on a seven-dimensional manifold. As $${\mathrm {Sp}}(2)\subseteq {\mathrm {SU}}(4)\subseteq {\mathrm {Spin}}(7)$$ , one may also obtain Calabi–Yau fourfolds or hyperkahler manifolds via the Hitchin flow. In this paper, we show that the Hitchin flow on almost Abelian Lie algebras and on Lie algebras with one-dimensional commutator always yields Riemannian metrics g with $$Hol(g)\subseteq {\mathrm {SU}}(4)$$ but $$Hol(g)\ne {\mathrm {Sp}}(2)$$ . We investigate when we actually get $$Hol(g)={\mathrm {SU}}(4)$$ and obtain many new explicit examples of Calabi–Yau fourfolds. The results rely on the connection between cocalibrated $$\mathrm {G}_2$$ -structures and hypo $${\mathrm {SU}}(3)$$ -structures and between the Hitchin and the hypo flow and on a systematic study of hypo $${\mathrm {SU}}(3)$$ -structures and the hypo flow on Lie algebras. This study gives us many other interesting results: We obtain full classifications of hypo $${\mathrm {SU}}(3)$$ -structures with particular intrinsic torsion on Lie algebras. Moreover, we can exclude reducible or $${\mathrm {Sp}}(2)$$ -holonomy for the Riemannian manifolds obtained by the hypo flow on Lie algebras with initial values lying in certain intrinsic torsion classes and show that for initial values in other torsion classes we always get Riemannian metrics with holonomy equal to $${\mathrm {SU}}(4)$$ .

A NEW FAMILY OF POISSON ALGEBRAS AND THEIR DEFORMATIONS

Let $\Bbbk$ be a field of characteristic zero. For any positive integer $n$ and any scalar $a\in \Bbbk$, we construct a family of Artin–Schelter regular algebras $R(n,a)$, which are quantizations of Poisson structures on $\Bbbk [x_{0},\ldots ,x_{n}]$. This generalizes an example given by Pym when $n=3$. For a particular choice of the parameter $a$ we obtain new examples of Calabi–Yau algebras when $n\geqslant 4$. We also study the ring theoretic properties of the algebras $R(n,a)$. We show that the point modules of $R(n,a)$ are parameterized by a bouquet of rational normal curves in $\mathbb{P}^{n}$, and that the prime spectrum of $R(n,a)$ is homeomorphic to the Poisson spectrum of its semiclassical limit. Moreover, we explicitly describe $\operatorname{Spec}R(n,a)$ as a union of commutative strata.

Quantum periods of Calabi–Yau fourfolds

In this work we study the quantum periods together with their Picard–Fuchs differential equations of Calabi–Yau fourfolds. In contrast to Calabi–Yau threefolds, we argue that the large volume points of Calabi–Yau fourfolds generically are regular singular points of the Picard–Fuchs operators of non-maximally unipotent monodromy. We demonstrate this property in explicit examples of Calabi–Yau fourfolds with a single Kähler modulus. For these examples we construct integral quantum periods and study their global properties in the quantum Kähler moduli space with the help of numerical analytic continuation techniques. Furthermore, we determine their genus zero Gromov–Witten invariants, their Klemm–Pandharipande meeting invariants, and their genus one BPS invariants. In our computations we emphasize the features attributed to the non-maximally unipotent monodromy property. For instance, it implies the existence of integral quantum periods that at large volume are purely worldsheet instanton generated. To verify our results, we also present intersection theory techniques to enumerate lines with a marked point on complete intersection Calabi–Yau fourfolds in Grassmannian varieties.

Doubling Construction of Calabi–Yau Fourfolds from Toric Fano Fourfolds

We give a differential-geometric construction of Calabi–Yau fourfolds by the ‘doubling’ method, which was introduced in Doi and Yotsutani (N Y J Math 20:1203–1235, 2014) to construct Calabi–Yau threefolds. We also give examples of Calabi–Yau fourfolds from toric Fano fourfolds. Ingredients in our construction are admissible pairs, which were first dealt with by Kovalev (J Reine Angew Math 565:125–160, 2003). Here in this paper an admissible pair $$(\overline{X},D)$$ consists of a compact Kahler manifold $$\overline{X}$$ and a smooth anticanonical divisor D on $$\overline{X}$$ . If two admissible pairs $$(\overline{X}_1,D_1)$$ and $$(\overline{X}_2,D_2)$$ with $$\dim _{\mathbb {C}}\overline{X}_i=4$$ satisfy the gluing condition, we can glue $$\overline{X}_1\setminus D_1$$ and $$\overline{X}_2\setminus D_2$$ together to obtain a compact Riemannian 8-manifold (M, g) whose holonomy group $$\mathrm {Hol}(g)$$ is contained in $$\mathrm {Spin}(7)$$ . Furthermore, if the $$\widehat{A}$$ -genus of M equals 2, then M is a Calabi–Yau fourfold, i.e., a compact Ricci-flat Kahler fourfold with holonomy $$\mathrm {SU}(4)$$ . In particular, if $$(\overline{X}_1,D_1)$$ and $$(\overline{X}_2,D_2)$$ are identical to an admissible pair $$(\overline{X},D)$$ , then the gluing condition holds automatically, so that we obtain a compact Riemannian 8-manifold M with holonomy contained in $$\mathrm {Spin}(7)$$ . Moreover, we show that if the admissible pair is obtained from any of the toric Fano fourfolds, then the resulting manifold M is a Calabi–Yau fourfold by computing $$\widehat{A}(M)=2$$ .

NEW EXAMPLES OF CALABI–YAU 3-FOLDS AND GENUS ZERO SURFACES

We classify the subgroups of the automorphism group of the product of four projective lines admitting an invariant anticanonical smooth divisor on which the action is free. As a first application, we describe new examples of Calabi–Yau 3-folds with small Hodge numbers. In particular, the Picard number is 1 and the number of moduli is 5. Furthermore, the fundamental group is nontrivial. We also construct a new family of minimal surfaces of general type with geometric genus zero, K2 = 3 and fundamental group of order 16. We show that this family dominates an irreducible component of dimension 4 of the moduli space of the surfaces of general type.

Geometric transitions between Calabi–Yau threefolds related to Kustin–Miller unprojections

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 result 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, which are described by Pfaffian equations in weighted projective spaces.

Examples of Calabi–Yau 3-Folds of ℙ7 with ρ = 1

Abstract. We give some examples of Calabi–Yau 3-folds with $\rho =1$ and $\rho =2$, 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 Chern classes from computer based computations over some finite field ${{\mathbb{F}}_{p}}$. Three of our examples (of degree 17 and 20) are new. The two others (degree 15 and 18) are known, and we recover their well-known invariants with ourmethod. These examples are built out of Gulliksen–Neg°ard and Kustin–Miller complexes of locally free sheaves.Finally, we give two new examples of Calabi–Yau 3-folds of ${{\mathbb{P}}^{6}}$ of degree 14 and 15 (defined over $\mathbb{Q}$). We show that they are not deformation equivalent to Tonoli's examples of the same degree, despite the fact that they have the same invariants $({{H}^{3}},{{c}_{2}}.H,{{c}_{3}})$ and $\rho =1$.

Primitive contractions of Calabi-Yau threefolds II

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 $\mathbb{P}^1\times \mathbb{P}^1$.

Examples of Calabi–Yau Threefolds as Covers of Almost-Fano Threefolds

We give a method for producing examples of Calabi–Yau threefolds as covers of degree d ≤ 8 of almost-Fano threefolds, computing explicitely their Euler– Poincare characteristic. Such a method generalizes the well-knownclassical construction of Calabi–Yau threefolds as double covers of the projective space branched along octic surfaces.

New examples of Calabi–Bernstein problems for some nonlinear equations

All the entire solutions to the maximal surface equation in certain 3-dimensional Lorentzian manifolds, obeying the null energy condition, are obtained. Thus, we solve new Calabi–Bernstein problems. As a consequence, the corresponding parametric versions are also given. The behaviour of this Calabi–Bernstein property with respect to an special family of C 2 -perturbations of Lorentz–Minkowski space L 3 is investigated

Homological geometry I. Projective hypersurfaces

Introduction Consider a generic quintic hypersurfaceX inCP . It is an example of Calabi– Yau 3-folds. It follows from Riemann–Roch formula, that rational curves on a generic Calabi–Yau 3-fold should be situated in a discrete fashion. Therefore a natural question of enumerative algebraic geometry arises: find the number nd of rational curves in X of degree d for each d = 1, 2, 3, .... In 1991 Candelas, de la Ossa, Green and Parkes [1] ‘predicted’ all the numbers nd simultaneously: they conjectured that the generating function

Hosotani breaking of E6 to a subgroup of rank five

On multiply connected manifolds, it is possible to construct vacuum gauge configurations with nontrivial holonomy groups. This is the basis of the Hosotani mechanism. This naturally suggests a ‘‘Hosotani inverse problem’’: If we wish to break a gauge group G to a subgroup H, what are the possible finite holonomy groups having this effect, and what can one say about the fundamental groups of the underlying manifolds? Usually, this problem is too difficult to solve, but we show that, for G=E6 and H locally isomorphic to the rank five group SU(3)×SU(2)×U(1)×U(1), a complete solution is possible. It is hoped that the results will aid a search for examples of Calabi–Yau manifolds leading to a low-energy gauge group of rank five.