Yau 3-folds Research Articles

Construction and deformations of Calabi–Yau 3-folds in codimension 4

We construct polarized Calabi–Yau 3-folds with at worst isolated canonical orbifold points in codimension 4 that can be described in terms of the equations of the Segre embedding of P2×P2 in P8. We investigate the existence of other deformation families in their Hilbert scheme by either studying Tom and Jerry degenerations or by comparing their Hilbert series with those of existing low codimension Calabi–Yau 3-folds. Among other interesting results, we find a family of Calabi–Yau 3-fold with five distinct Tom and Jerry deformation families, a phenomenon not seen for Q-Fano 3-folds. We compute the Hodge numbers of P2×P2 Calabi–Yau 3-folds and corresponding manifolds obtained by performing crepant resolutions. We obtain a manifold with a pair of Hodge numbers that does not appear in the famously known list of 30108 distinct Hodge pairs of Kruzer–Skarke, in the list of 7890 distinct Hodge pairs corresponding to complete intersections in the product of projective spaces and in Hodge paris obtained from Calabi–Yau 3-folds having low codimension embeddings in weighted projective spaces.

Weyl symmetry for curve counting invariants via spherical twists

We study the curve counting invariants of Calabi–Yau 3-folds via the Weyl reflection along a ruled divisor. We obtain a new rationality result and functional equation for the generating functions of Pandharipande–Thomas invariants. When the divisor arises as resolution of a curve of A 1 A_1 -singularities, our results match the rationality of the associated Calabi–Yau orbifold. The symmetry on generating functions descends from the action of an infinite dihedral group of derived auto-equivalences, which is generated by the derived dual and a spherical twist. Our techniques involve wall-crossing formulas and generalized Donaldson–Thomas invariants for surface-like objects.

Hilbert schemes of points on Calabi–Yau 4-folds via wall-crossing

Gross–Joyce–Tanaka [43] proposed a wall-crossing conjecture for Calabi–Yau fourfolds. Assuming it, we prove the conjecture of Cao–Kool [14] for 0-dimensional sheaf-counting invariants on projective Calabi–Yau 4-folds. From it, we extract the full topological information contained in the virtual fundamental classes of Hilbert schemes of points which turns out to be equivalent to the data of all descendent integrals. As a consequence, we can express many generating series of invariants in terms of explicit universal power series.i)On C4, Nekrasov proposed invariants with a conjectured closed form [72]. We show that an analog of his formula holds for compact Calabi–Yau 4-folds satisfying the wall-crossing conjecture.ii)We notice a relationship to corresponding generating series for Quot schemes on elliptic surfaces which are also governed by a wall-crossing formula. This leads to a Segre–Verlinde correspondence for Calabi–Yau fourfolds.

Gopakumar–Vafa Type Invariants of Holomorphic Symplectic 4-Folds

Using reduced Gromov–Witten theory, we define new invariants which capture the enumerative geometry of curves on holomorphic symplectic 4-folds. The invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi–Yau 3-folds, Klemm and Pandharipande for Calabi–Yau 4-folds, and Pandharipande and Zinger for Calabi–Yau 5-folds. We conjecture that our invariants are integers and give a sheaf-theoretic interpretation in terms of reduced 4-dimensional Donaldson–Thomas invariants of one-dimensional stable sheaves. We check our conjectures for the product of two K3 surfaces and for the cotangent bundle of P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {P}}^2$$\\end{document}. Modulo the conjectural holomorphic anomaly equation, we compute our invariants also for the Hilbert scheme of two points on a K3 surface. This yields a conjectural formula for the number of isolated genus 2 curves of minimal degree on a very general hyperkähler 4-fold of K3[2]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K3^{[2]}$$\\end{document}-type. The formula may be viewed as a 4-dimensional analogue of the classical Yau–Zaslow formula concerning counts of rational curves on K3 surfaces. In the course of our computations, we also derive a new closed formula for the Fujiki constants of the Chern classes of tangent bundles of both Hilbert schemes of points on K3 surfaces and generalized Kummer varieties.

The 4-fold Pandharipande–Thomas vertex

We give a conjectural but full and explicit description of the (K-theoretic) equivariant vertex for Pandharipande–Thomas stable pairs on toric Calabi–Yau 4-folds, by identifying torus-fixed loci as certain quiver Grassmannians and prescribing a canonical half of the tangent-obstruction theory. For any number of non-trivial legs, the DT/PT vertex correspondence can then be verified by computer in low degrees.

Bifurcations of embedded curves and toward an extension of Taubes’s Gromov invariant to Calabi–Yau 3-folds

Bifurcations of embedded curves and toward an extension of Taubes’s Gromov invariant to Calabi–Yau 3-folds

Counting sheaves on Calabi–Yau 4-folds, I

Borisov and Joyce constructed a real virtual cycle on compact moduli spaces of stable sheaves on Calabi–Yau 4-folds, using derived differential geometry. We construct an algebraic virtual cycle. A key step is a localization of Edidin and Graham’s square root Euler class for SO(2n,C)-bundles to the zero locus of an isotropic section, or to the support of an isotropic cone. We prove a torus localization formula, making the invariants computable and extending them to the noncompact case when the fixed locus is compact. We give a K-theoretic refinement by defining K-theoretic square root Euler classes and their localized versions. In a sequel, we prove that our invariants reproduce those of Borisov and Joyce.

SU(2)^2-Invariant Gauge Theory on Asymptotically Conical Calabi–Yau 3-Folds

We give a complete description of the behaviour of Calabi–Yau instantons and monopoles with an SU(2)^2-symmetry, on Calabi–Yau 3-folds with asymptotically conical geometry and SU(2)^2 acting with co-homogeneity one. We consider gauge theory on the smoothing and small resolution of the conifold, and on the canonical bundle of mathbb{C}mathbb{P}^1 times mathbb{C}mathbb{P}^1, with their known asymptotically conical co-homogeneity one Calabi–Yau metrics, and find new one-parameter families of invariant instantons. We also entirely classify the relevant moduli-spaces of instantons and monopoles satisfying a natural curvature decay condition, and show that the expected bubbling phenomena occur in these families of instantons.

Attractor mechanisms of moduli spaces of Calabi–Yau 3-folds

We investigate the complex and Kähler attractor mechanisms of moduli spaces of Calabi–Yau 3-folds. The complex attractor mechanism was previously studied by Ferrara–Kallosh–Strominger, Moore and others in string theory. It is concerned with the minimizing problems of the normalized central charges of the middle homology classes and defines a new interesting class of Calabi–Yau 3-folds called, the complex attractor varieties. In light of mirror symmetry, we introduce the Kähler attractor mechanism and define the Kähler attractor varieties. The complex and Kähler attractor varieties are expected to possess very rich structures, in particular certain complex and Kähler rigidities.

Birational geometry for d-critical loci and wall-crossing in Calaby–Yau 3-folds

Birational geometry for d-critical loci and wall-crossing in Calaby–Yau 3-folds

K-Theoretic DT/PT Correspondence for Toric Calabi–Yau 4-Folds

Recently, Nekrasov discovered a new “genus” for Hilbert schemes of points on {mathbb {C}}^4. We extend its definition to Hilbert schemes of curves and moduli spaces of stable pairs, and conjecture a K-theoretic DT/PT correspondence for toric Calabi–Yau 4-folds. We develop a K-theoretic vertex formalism, which allows us to verify our conjecture in several cases. Taking a certain limit of the equivariant parameters, we recover the cohomological DT/PT correspondence for toric Calabi–Yau 4-folds recently conjectured by the first two authors. Another limit gives a dimensional reduction to the K-theoretic DT/PT correspondence for toric 3-folds conjectured by Nekrasov–Okounkov. As an application of our techniques, we find a conjectural formula for the generating series of K-theoretic stable pair invariants of text {Tot}_{{mathbb {P}}^1}({mathcal {O}}(-1) oplus {mathcal {O}}(-1) oplus {mathcal {O}}). Upon dimensional reduction to the resolved conifold, we recover a formula which was recently proved by Kononov–Okounkov–Osinenko.

4-Manifold topology, Donaldson–Witten theory, Floer homology and higher gauge theory methods in the BV-BFV formalism

We study the behavior of Donaldson’s invariants of 4-manifolds based on the moduli space of anti-self-dual connections (instantons) in the perturbative field theory setting where the underlying source manifold has boundary. It is well-known that these invariants take values in the instanton Floer homology groups of the boundary 3-manifold. Gluing formulae for these constructions lead to a functorial topological field theory description according to a system of axioms developed by Atiyah, which can be also regarded in the setting of perturbative quantum field theory, as it was shown by Witten, using a version of supersymmetric Yang–Mills theory, known today as Donaldson–Witten theory. One can actually formulate an AKSZ model which recovers this theory for a certain gauge-fixing. We consider these constructions in a perturbative quantum gauge formalism for manifolds with boundary that is compatible with cutting and gluing, called the BV-BFV formalism, which was recently developed by Cattaneo, Mnev and Reshetikhin. We prove that this theory satisfies a modified Quantum Master Equation and extend the result to a global picture when perturbing around constant background fields. These methods are expected to extend to higher codimensions and thus might help getting a better understanding for fully extendable [Formula: see text]-dimensional field theories (in the sense of Baez–Dolan and Lurie) in the perturbative setting, especially when [Formula: see text]. Additionally, we relate these constructions to Nekrasov’s partition function by treating an equivariant version of Donaldson–Witten theory in the BV formalism. Moreover, we discuss the extension, as well as the relation, to higher gauge theory and enumerative geometry methods, such as Gromov–Witten and Donaldson–Thomas theory and recall their correspondence conjecture for general Calabi–Yau 3-folds. In particular, we discuss the corresponding (relative) partition functions, defined as the generating function for the given invariants, and gluing phenomena.

Calabi–Yau 3-folds of Picard number 2 with hypersurface Cox rings

We classify all smooth Calabi–Yau threefolds of Picard number two that have a general hypersurface Cox ring.

Geometric Transitions with Spin(7) Holonomy via a Dynamical System

We clarify the global geometry of two 1-parameter families of cohomogeneity one Spin(7) holonomy metrics with generic orbit the Aloff–Wallach space \(N(1,-1) \cong \text {SU}(3)/\text {U}(1)\) and singular orbits \(S^5\) and \({\mathbb {C}}P^{2}\), which at short distance were shown to exist by Reidegeld. The two families fit into the geography of previously known families of cohomogeneity one metrics with exceptional holonomy and provide a Spin(7) analogue of the well-known conifold transition in the setting of Calabi–Yau 3-folds. Furthermore, we discover that there is another transition to families of Spin(7) holonomy metrics which have a similar asymptotic behaviour on one end, but are singular on the other end. We obtain our results by relating the Spin(7)-equations to a simple dynamical system on a 3-dimensional cube.

Calabi–Yau 3-folds from projective joins of del Pezzo manifolds

In this paper, we will construct new examples of derived equivalent Calabi–Yau 3-folds with Picard number greater than one. We also study their mirror Calabi–Yau manifolds and find that they are given by Schoen's fiber products of suitable rational elliptic modular surfaces.

Singularities of $1/2$ Calabi–Yau 4-folds and classification scheme for gauge groups in four-dimensional F-theory

Singularities of $1/2$ Calabi–Yau 4-folds and classification scheme for gauge groups in four-dimensional F-theory

Complete noncompact G2-manifolds from asymptotically conical Calabi–Yau 3-folds

We develop a powerful new analytic method to construct complete noncompact Ricci-flat 7-manifolds, more specifically G2-manifolds, that is, Riemannian 7-manifolds (M,g) whose holonomy group is the compact exceptional Lie group G2. Our construction gives the first general analytic construction of complete noncompact Ricci-flat metrics in any odd dimension and establishes a link with the Cheeger–Fukaya–Gromov theory of collapse with bounded curvature. The construction starts with a complete noncompact asymptotically conical Calabi–Yau 3-fold B and a circle bundle M→B satisfying a necessary topological condition. Our method then produces a 1-parameter family of circle-invariant complete G2-metrics gϵ on M that collapses with bounded curvature as ϵ→0 to the original Calabi–Yau metric on the base B. The G2-metrics we construct have controlled asymptotic geometry at infinity, so-called asymptotically locally conical (ALC) metrics; these are the natural higher-dimensional analogues of the asymptotically locally flat (ALF) metrics that are well known in 4-dimensional hyper-Kähler geometry. We give two illustrations of the strength of our method. First, we use it to construct infinitely many diffeomorphism types of complete noncompact simply connected G2-manifolds; previously only a handful of such diffeomorphism types was known. Second, we use it to prove the existence of continuous families of complete noncompact G2-metrics of arbitrarily high dimension; previously only rigid or 1-parameter families of complete noncompact G2-metrics were known.

Stable Pair Invariants of Local Calabi–Yau 4-folds

Abstract In 2008, Klemm–Pandharipande defined Gopakumar–Vafa type invariants of a Calabi–Yau 4-folds $X$ using Gromov–Witten theory. Recently, Cao–Maulik–Toda proposed a conjectural description of these invariants in terms of stable pair theory. When $X$ is the total space of the sum of two line bundles over a surface $S$, and all stable pairs are scheme theoretically supported on the zero section, we express stable pair invariants in terms of intersection numbers on Hilbert schemes of points on $S$. As an application, we obtain new verifications of the Cao–Maulik–Toda conjectures for low-degree curve classes and find connections to Carlsson–Okounkov numbers. Some of our verifications involve genus zero Gopakumar–Vafa type invariants recently determined in the context of the log-local principle by Bousseau–Brini–van Garrel. Finally, using the vertex formalism, we provide a few more verifications of the Cao–Maulik–Toda conjectures when thickened curves contribute and also for the case of local $\mathbb{P}^3$.

Derived Equivalences for the Flops of Type C2 and {A}_{4}^{G} via Mutation of Semiorthogonal Decomposition

We give a new proof of the derived equivalence of a pair of varieties connected by the flop of type C2 in the list of Kanemitsu (2018), which is originally due to Segal (Bull. Lond. Math. Soc., 48 (3) 533–538, 2016). We also prove the derived equivalence of a pair of varieties connected by the flop of type {A}_{4}^{G} in the same list. The latter proof follows that of the derived equivalence of Calabi–Yau 3-folds in Grassmannians Gr(2,5) and Gr(3,5) by Kapustka and Rampazzo (Commun. Num. Theor. Phys., 13 (4) 725–761 2019) closely.

Inception neural network for complete intersection Calabi–Yau 3-folds

We introduce a neural network inspired by Google’s Inception model to compute the Hodge number h 1,1 of complete intersection Calabi–Yau (CICY) 3-folds. This architecture improves largely the accuracy of the predictions over existing results, giving already 97% of accuracy with just 30% of the data for training. Accuracy climbs to 99% when using 80% of the data for training. This proves that neural networks are a valuable resource to study geometric aspects in both pure mathematics and string theory.