Products Of Projective Spaces Research Articles

Gopakumar-Vafa invariants and the Emergent String Conjecture

The Emergent String Conjecture of Lee, Lerche, and Weigand holds that every infinite-distance limit in the moduli space of a quantum gravity represents either a decompactification limit or an emergent string limit in some duality frame. Within the context of 5d supergravities coming from M-theory compactifications on Calabi-Yau threefolds, we find evidence for this conjecture by studying (a) the gauge couplings and (b) the BPS spectrum, which is encoded in the Gopakumar-Vafa invariants of the threefold. In the process, we disuss a testable geometric consequence of the Emergent String Conjecture, and we verify that it is satisfied in all complete intersection Calabi-Yau threefolds in products of projective spaces (CICYs).

Borel (α,β)-multitransforms and quantum Leray–Hirsch: Integral representations of solutions of quantum differential equations for [formula omitted]-bundles

In this paper, we address the integration problem of the isomonodromic system of quantum differential equations (qDEs) associated with the quantum cohomology of P1-bundles on Fano varieties. It is shown that bases of solutions of the qDE associated with the total space of the P1-bundle can be reconstructed from the datum of bases of solutions of the qDE associated with the base space. This represents a quantum analog of the classical Leray–Hirsch theorem in the context of the isomonodromic approach to quantum cohomology. The reconstruction procedure of the solutions can be performed in terms of some integral transforms, introduced in [17], called Borel(α,β)-multitransforms. We emphasize the emergence, in the explicit integral formulas, of an interesting sequence of special functions (closely related to iterated partial derivatives of the Böhmer–Tricomi incomplete Gamma function) as integral kernels. Remarkably, these integral kernels have a universal feature, being independent of the specifically chosen P1-bundle. When applied to projective bundles on products of projective spaces, our results give Mellin–Barnes integral representations of solutions of qDEs. As an example, we show how to integrate the qDE of blow-up of P2 at one point via Borel multitransforms of solutions of the qDE of P1.

Syzygies of curves in products of projective spaces

Syzygies of curves in products of projective spaces

Miyaoka-Yau type inequalities of complete intersection threefolds in products of projective

Geography of projective varieties is one of the fundamental problems in algebraic geometry. There are many researches toward the characteristics of Chern number of some projective spaces, for example Noethers inequalities, the theorem of Chang-Lopez, and the Miyaoka-Yau inequality. In this paper, we compute the Chern numbers of any smooth complete intersection threefold in the product of projective spaces via the standard exact sequences of cotangent bundles. Then we obtain linear Chern number inequalities for (c_1 (X)c_2 (X))/(c_1^3 (X)) and (c_3 (X))/(c_1^3 (X)) on such threefolds under conditions of d_ij4 and d_ij6 respectively. They can be considered as a generalization of the Miyaoka-Yau inequality and an improvement of Yaus inequality for such threefolds.

Projective varieties with nef tangent bundle in positive characteristic

Let $X$ be a smooth projective variety defined over an algebraically closed field of positive characteristic $p$ whose tangent bundle is nef. We prove that $X$ admits a smooth morphism $X \to M$ such that the fibers are Fano varieties with nef tangent bundle and $T_M$ is numerically flat. We also prove that extremal contractions exist as smooth morphisms. As an application, we prove that, if the Frobenius morphism can be lifted modulo $p^2$, then $X$ admits, up to a finite étale Galois cover, a smooth morphism onto an ordinary abelian variety whose fibers are products of projective spaces.

Conjecture 𝒪 for Projective Complete Intersections

Abstract Conjecture ${{\mathcal{O}}}$ and Gamma conjectures for quantum cohomology of Fano manifolds were proposed by Galkin, Golyshev, and Iritani [ 16]. We prove Conjecture ${{\mathcal{O}}}$ for Fano complete intersections in projective spaces. The main tools to compute relevant genus-zero Gromov–Witten invariants with primitive insertions are the genus-one topological recursion relation and Zinger’s standard-versus-reduced formula. We also prove a related conjecture of Galkin [ 15] for projective Fano complete intersections.

Cohomology algebra of orbit spaces of free involutions on the product of projective space and 4-sphere

Cohomology algebra of orbit spaces of free involutions on the product of projective space and 4-sphere

Euler-symmetric complete intersections in projective space

Euler-symmetric complete intersections in projective space

Gromov–Witten theory of complete intersections via nodal invariants

AbstractWe provide an inductive algorithm computing Gromov–Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. We also prove that all Gromov–Witten classes of all smooth complete intersections in projective space belong to the tautological ring of the moduli space of stable curves. The main idea is to show that invariants with insertions of primitive cohomology classes are controlled by their monodromy and by invariants defined without primitive insertions but with imposed nodes in the domain curve. To compute these nodal Gromov–Witten invariants, we introduce the new notion of nodal relative Gromov–Witten invariants. We then prove a nodal degeneration formula and a relative splitting formula. These results for nodal relative Gromov–Witten theory are stated in complete generality and are of independent interest.

Flops for complete intersection Calabi-Yau threefolds

We study flops of Calabi-Yau threefolds realised as Kähler-favourable complete intersections in products of projective spaces (CICYs) and identify two different types. The existence and the type of the flops can be recognised from the configuration matrix of the CICY, which also allows for constructing such examples. The first type corresponds to rows containing only 1s and 0s, while the second type corresponds to rows containing a single entry of 2, followed by 1s and 0s. We give explicit descriptions for the manifolds obtained after the flop and show that the second type of flop always leads to isomorphic manifolds, while the first type in general leads to non-isomorphic flops. The singular manifolds involved in the flops are determinantal varieties in the first case and more complicated in the second case. We also discuss manifolds admitting an infinite chain of flops and show how to identify these from the configuration matrix. Finally, we point out how to construct the divisor images and Picard group isomorphisms under both types of flops.

Wonderful compactifications and rational curves with cyclic action

Abstract We prove that the moduli space of rational curves with cyclic action, constructed in our previous work, is realizable as a wonderful compactification of the complement of a hyperplane arrangement in a product of projective spaces. By proving a general result on such wonderful compactifications, we conclude that this moduli space is Chow-equivalent to an explicit toric variety (whose fan can be understood as a tropical version of the moduli space), from which a computation of its Chow ring follows.

Free quotients of favorable Calabi-Yau manifolds

Non-simply connected Calabi-Yau threefolds play a central role in the study of string compactifications. Such manifolds are usually described by quotienting a simply connected Calabi-Yau variety by a freely acting discrete symmetry. For the Calabi-Yau threefolds described as complete intersections in products of projective spaces, a classification of such symmetries descending from linear actions on the ambient spaces of the varieties has been given in [16]. However, which symmetries can be described in this manner depends upon the description that is being used to represent the manifold. In [24] new, favorable, descriptions were given of this data set of Calabi-Yau threefolds. In this paper, we perform a classification of cyclic symmetries that descend from linear actions on the ambient spaces of these new favorable descriptions. We present a list of 129 symmetries/non-simply connected Calabi-Yau threefolds. Of these, at least 33, and potentially many more, are topologically new varieties.

Stable Submanifolds in the Product of Projective Spaces

We provide a classification theorem for compact stable minimal immersions (CSMI) of codimension 1 or dimension 1 (codimension 1 and 2 or dimension 1 and 2) in the product of a complex (quaternionic) projective space with any other Riemannian manifold. We characterize the complex minimal immersions of codimension 2 or dimension 2 as the only CSMI in the product of two complex projective spaces. As an application, we characterize the CSMI of codimension 1 or dimension 1 (codimension 1 and 2 or dimension 1 and 2) in the product of a complex (quaternionic) projective space with any compact rank one symmetric space.

Algebraic reduced genus one Gromov–Witten invariants for complete intersections in projective spaces, Part 2

In Lee and Oh (Preprint, arXiv:1809.10995, 2018), we provided an algebraic proof of Zinger's comparison formula (Geom. Topol. 12 (2008), no. 2, 1203–1241; J. Differential Geom. 83 (2009), no. 2, 407–460) between genus one Gromov–Witten invariants and reduced invariants when the target space is a complete intersection of dimension 2 or 3 in a projective space. In this paper, we extend the proof in Lee and Oh (Preprint, arXiv:1809.10995, 2018) to all dimensions and to descendant invariants.

Cohomology Rings and Algebraic Torus Actions on Hypersurfaces in the Product of Projective Spaces and Bounded Flag Varieties

In this paper, for any Milnor hypersurface, we find the largest dimension of effective algebraic torus actions on it. The proof of the corresponding theorem is based on the computation of the automorphism group for any Milnor hypersurface. We find all generalized Buchstaber–Ray and Ray hypersurfaces that are toric varieties. We compute the Betti numbers of these hypersurfaces and describe their integral singular cohomology rings in terms of the cohomology of the corresponding ambient varieties.

Deep multi-task mining Calabi–Yau four-folds

We continue earlier efforts in computing the dimensions of tangent space cohomologies of Calabi–Yau manifolds using deep learning. In this paper, we consider the dataset of all Calabi–Yau four-folds constructed as complete intersections in products of projective spaces. Employing neural networks inspired by state-of-the-art computer vision architectures, we improve earlier benchmarks and demonstrate that all four non-trivial Hodge numbers can be learned at the same time using a multi-task architecture. With 30% (80%) training ratio, we reach an accuracy of 100% for and 97% for (100% for both), 81% (96%) for , and 49% (83%) for . Assuming that the Euler number is known, as it is easy to compute, and taking into account the linear constraint arising from index computations, we get 100% total accuracy.

Flops, Gromov-Witten invariants and symmetries of line bundle cohomology on Calabi-Yau three-folds

The zeroth line bundle cohomology on Calabi-Yau three-folds encodes information about the existence of flop transitions and the genus zero Gromov-Witten invariants. We illustrate this claim by studying several Picard number 2 Calabi-Yau three-folds realised as complete intersections in products of projective spaces. Many of these manifolds exhibit certain symmetries on the Picard lattice which preserve the zeroth cohomology.

Homological and combinatorial aspects of virtually Cohen–Macaulay sheaves

When studying a graded module $M$ over the Cox ring of a smooth projective toric variety $X$, there are two standard types of resolutions commonly used to glean information: free resolutions of $M$ and vector bundle resolutions of its sheafification. Each approach comes with its own challenges. There is geometric information that free resolutions fail to encode, while vector bundle resolutions can resist study using algebraic and combinatorial techniques. Recently, Berkesch, Erman, and Smith introduced virtual resolutions, which capture desirable geometric information and are also amenable to algebraic and combinatorial study. The theory of virtual resolutions includes a notion of a virtually Cohen--Macaulay property, though tools for assessing which modules are virtually Cohen--Macaulay have only recently started to be developed. In this paper, we continue this research program in two related ways. The first is that, when $X$ is a product of projective spaces, we produce a large new class of virtually Cohen--Macaulay Stanley--Reisner rings, which we show to be virtually Cohen--Macaulay via explicit constructions of appropriate virtual resolutions reflecting the underlying combinatorial structure. The second is that, for an arbitrary smooth projective toric variety $X$, we develop homological tools for assessing the virtual Cohen--Macaulay property. Some of these tools give exclusionary criteria, and others are constructive methods for producing suitably short virtual resolutions. We also use these tools to establish relationships among the arithmetically, geometrically, and virtually Cohen--Macaulay properties.

Indecomposable vector bundles via monads on P2n+1 × P2n+1

The existence of monads on products of projective spaces $P^{a_1}\times\cdots\times\ P^{a_n}$ is nontrivial. In this paper, we construct monads over the polycyclic variety $P^{2n+1}\times\ P^{2n+1}$, we prove that cohomology vector bundle associated to these monads is simple. We also construct a monad on $P^1\times P^1\times\ P^2$. We also study the vector bundles associated to monads and prove stability and simplicity.

Horrocks splitting on Segre\u2013Veronese varieties

We prove an analogue of Horrocks’ splitting theorem for Segre–Veronese varieties building upon the theory of Tate resolutions on products of projective spaces.