Toric Manifolds Research Articles

The fundamental group and Betti numbers of toric origami manifolds

Toric origami manifolds are characterized by origami templates, which are combinatorial models built by gluing polytopes together along facets. In this paper, we examine the topology of orientable toric oigami manifolds with coorientable folding hypersurface. We determine the fundamental group. In our previous paper [HP], we studied the ordinary and equivariant cohomology rings of simply connected toric origami manifolds. We conclude this paper by computing some Betti numbers in the non-simply connected case.

A bound for the splitting of smooth Fano polytopes with many vertices

The classification of toric Fano manifolds with large Picard number corresponds to the classification of smooth Fano polytopes with large number of vertices. A smooth Fano polytope is a polytope that contains the origin in its interior and is such that the vertex set of each facet forms a lattice basis. Casagrande showed that any smooth d-dimensional Fano polytope has at most 3d vertices. Smooth Fano polytopes in dimension d with at least $$3d-2$$3d-2 vertices are completely known. The main result of this paper deals with the case of $$3d-k$$3d-k vertices for k fixed and d large. It implies that there is only a finite number of isomorphism classes of toric Fano d-folds X (for arbitrary d) with Picard number $$2d-k$$2d-k such that X is not a product of a lower-dimensional toric Fano manifold and the projective plane blown up in three torus-invariant points. This verifies the qualitative part of a conjecture in a recent paper by the first author, Joswig, and Paffenholz.

A reconstruction theorem for Connes–Landi deformations of commutative spectral triples

We formulate and prove an extension of Connes’s reconstruction theorem for commutative spectral triples to so-called Connes–Landi or isospectral deformations of commutative spectral triples along the action of a compact Abelian Lie group G , also known as toric noncommutative manifolds. In particular, we propose an abstract definition for such spectral triples, where noncommutativity is entirely governed by a deformation parameter sitting in the second group cohomology of the Pontryagin dual of G , and then show that such spectral triples are well-behaved under further Connes–Landi deformation, thereby allowing for both quantisation from and dequantisation to G -equivariant abstract commutative spectral triples. We then use a refinement of the Connes–Dubois-Violette splitting homomorphism to conclude that suitable Connes–Landi deformations of commutative spectral triples by a rational deformation parameter are almost-commutative in the general, topologically non-trivial sense.

Non-Compact Symplectic Toric Manifolds

A key result in equivariant symplectic geometry is Delzant's classification of compact connected symplectic toric manifolds. The moment map induces an embedding of the quotient of the manifold by the torus action into the dual of the Lie algebra of the torus; its image is a unimodular ("Delzant") polytope; this gives a bijection between unimodular polytopes and isomorphism classes of compact connected symplectic toric manifolds. In this paper we extend Delzant's classification to non-compact symplectic toric manifolds. For a non-compact symplectic toric manifold the image of the moment map need not be convex and the induced map on the quotient need not be an embedding. Moreover, even when the map on the quotient is an embedding, its image no longer determines the symplectic toric manifold; a degree two characteristic class on the quotient makes an appearance. Nevertheless, the quotient is a manifold with corners, and the induced map from the quotient to the dual of the Lie algebra is what we call a unimodular local embedding. We classify non-compact symplectic toric manifolds in terms of manifolds with corners equipped with degree two cohomology classes and unimodular local embeddings into the dual of the Lie algebra of the corresponding torus. The main new ingredient is the construction of a symplectic toric manifold from such data. The proof passes through an equivalence of categories between symplectic toric manifolds and symplectic toric bundles over a fixed unimodular local embedding. This equivalence also gives a geometric interpretation of the degree two cohomology class.

Some remarks on the symplectic and Kähler geometry of toric varieties

Let M be a projective toric manifold. We prove two results concerning, respectively, Kähler–Einstein submanifolds of M and symplectic embeddings of the standard Euclidean ball in M. Both results use the well-known fact that M contains an open dense subset biholomorphic to $$\mathbb {{C}}^n$$ .

Hidden symmetries on toric Sasaki-Einstein spaces

We describe the construction of Killing-Yano tensors on toric Sasaki-Einstein manifolds. We use the fact that the metric cones of these spaces are Calabi-Yau manifolds. The description of the Calabi-Yau manifolds in terms of toric data, using the Delzant approach to toric geometries, allows us to find explicitly the complex coordinates and write down the Killing-Yano tensors. As a concrete example we present the complete set of special Killing forms on the five-dimensional homogeneous Sasaki-Einstein manifold T1,1.

A volume stability theorem on toric manifolds with positive Ricci curvature

In this short note, we will prove a volume stability theorem which says that if an $n$-dimensional toric manifold $M$ admits a $\mathbb {T}^n$ invariant Kähler metric $\omega$ with Ricci curvature no less than $1$ and its volume is close to the volume of $\mathbb {CP}^n$, $M$ is bi-holomorphic to $\mathbb {CP}^n$.

Torus manifolds and non-negative curvature

A torus manifold $M$ is a $2n$-dimensional orientable manifold with an effective action of an $n$-dimensional torus such that $M^T\neq \emptyset$. In this paper we discuss the classification of torus manifolds which admit an invariant metric of non-negative curvature. If $M$ is a simply connected torus manifold which admits such a metric, then $M$ is diffeomorphic to a quotient of a free linear torus action on a product of spheres. We also classify rationally elliptic torus manifolds $M$ with $H^{\text{odd}}(M;\mathbb{Z})=0$ up homeomorphism.

Young Diagrams and Intersection Numbers for Toric Manifolds associated with Weyl Chambers

We study intersection numbers of invariant divisors in the toric manifold associated with the fan determined by the collection of Weyl chambers for each root system of classical type and of exceptional type $G_2$. We give a combinatorial formula for intersection numbers of certain subvarieties which are naturally indexed by elements of the Weyl group. These numbers describe the ring structure of the cohomology of the toric manifold.

Torus manifolds in equivariant complex bordism

We restrict geometric tangential equivariant complex Tn-bordism to torus manifolds and provide a complete combinatorial description of the appropriate non-commutative ring. We discover, using equivariant K-theory characteristic numbers, that the information encoded in the oriented torus graph associated to a stably complex torus manifold completely describes its equivariant bordism class. We also consider the role of omnioriented quasitoric manifolds in this description.

Uniqueness of the direct decomposition of toric manifolds

In this paper, we study the uniqueness of the direct decomposition of a toric manifold. We first observe that the direct decomposition of a toric manifold as algebraic varieties is unique up to order of the factors. An algebraically indecomposable toric manifold happens to decompose as smooth manifold and no criterion is known for two toric manifolds to be diffeomorphic, so the unique decomposition problem for toric manifolds as smooth manifolds is highly nontrivial and nothing seems known for the problem so far. We prove that this problem is affirmative if the complex dimension of each factor in the decomposition is less than or equal to two. A similar argument shows that the direct decomposition of a smooth manifold into copies of $\mathbb{C}P^{1}$ and simply connected closed smooth 4-manifolds with smooth actions of $(S^{1})^{2}$ is unique up to order of the factors.

On the examples of Nill and Paffenholz

In 1999, Batyrev and Selivanova introduced the notion of "symmetric toric Fano manifolds" and proved that every symmetric toric Fano manifold admits an Einstein–Kähler metric ([1]). For n ≦ 4, every Einstein–Kähler toric Fano n-fold is symmetric. Hence, it was conjectured that every Einstein–Kähler toric Fano manifold is symmetric. However, in [6], Nill and Paffenholz constructed the counter-examples of this conjecture. In this paper, we shall construct higher-dimensional generalizations of their examples and also discuss the Chow unstability of these generalizations.

Complex symplectomorphisms and pseudo-Kähler islands in the quantization of toric manifolds

Let $$P$$ be a Delzant polytope. We show that the quantization of the corresponding toric manifold $$X_{P}$$ in toric Kähler polarizations and in the toric real polarization are related by analytic continuation of Hamiltonian flows evaluated at time $$t = {\sqrt{-1}}s$$ . We relate the quantization of $$X_{P}$$ in two different toric Kähler polarizations by taking the time- $${\sqrt{-1}}s$$ Hamiltonian “flow” of strongly convex functions on the moment polytope $$P$$ . By taking $$s$$ to infinity, we obtain the quantization of $$X_{P}$$ in the (singular) real toric polarization. Recall that $$X_{P}$$ has an open dense subset which is biholomorphic to $$({\mathbb {C}}^{*})^{n}$$ . The quantization of $$X_{P}$$ in a toric Kähler polarization can also be described by applying the complexified Hamiltonian flow of the Abreu–Guillemin symplectic potential $$g$$ , at time $$t={\sqrt{-1}}$$ , to an appropriate finite-dimensional subspace of quantum states in the quantization of $$T^{*}{\mathbb {T}}^{n}$$ in the vertical polarization. By taking other imaginary times, $$t= k {\sqrt{-1}}, k\in {\mathbb {R}}$$ , we describe toric Kähler metrics with cone singularities along the toric divisors in $$X_{P}$$ . For convex Hamiltonian functions and sufficiently negative imaginary part of the complex time, we obtain degenerate Kähler structures which are negative definite in some regions of $$X_{P}$$ . We show that the pointwise and $$L^2$$ -norms of quantum states are asymptotically vanishing on negative-definite regions.

SIGNED A-POLYNOMIALS OF GRAPHS AND POINCARÉ POLYNOMIALS OF REAL TORIC MANIFOLDS

Recently, Choi and Park introduced an invariant of a finite simple graph, called signed a-number, arising from computing certain topological invariants of some specific kinds of real toric manifolds. They also found the signed a-numbers of path graphs, cycle graphs, complete graphs, and star graphs. We introduce a signed a-polynomial which is a generalization of the signed a-number and gives a-, b-, and c-numbers. The signed a-polynomial of a graph $G$ is related to the Poincar\'e polynomial $P_{M(G)}(z)$, which is the generating function for the Betti numbers of the real toric manifold $M(G)$. We give the generating functions for the signed a-polynomials of not only path graphs, cycle graphs, complete graphs, and star graphs, but also complete bipartite graphs and complete multipartite graphs. As a consequence, we find the Euler characteristic number and the Betti numbers of the real toric manifold $M(G)$ for complete multipartite graphs $G$.

Four-dimensional Fano toric complete intersections.

We find at least 527 new four-dimensional Fano manifolds, each of which is a complete intersection in a smooth toric Fano manifold.

Quasimorphisms on contactomorphism groups and contact rigidity

We build homogeneous quasi-morphisms on the universal cover of the contactomorphism group for certain prequantizations of monotone symplectic toric manifolds. This is done using Givental's nonlinear Maslov index and a contact reduction technique for quasi-morphisms. We show how these quasi-morphisms lead to a hierarchy of rigid subsets of contact manifolds. We also show that the nonlinear Maslov index has a vanishing property, which plays a key role in our proofs. Finally we present applications to orderability of contact manifolds and Sandon-type metrics on contactomorphism groups.

Quantum geometry of del Pezzo surfaces in the Nekrasov-Shatashvili limit

We use mirror symmetry, quantum geometry and modularity properties of elliptic curves to calculate the refined free energies in the Nekrasov-Shatashvili limit on non-compact toric Calabi-Yau manifolds, based on del Pezzo surfaces. Quantum geometry here is to be understood as a quantum deformed version of rigid special geometry, which has its origin in the quantum mechanical behaviour of branes in the topological string B-model. We will argue that, in the Seiberg-Witten picture, only the Coulomb parameters lead to quantum corrections, while the mass parameters remain uncorrected. In certain cases we will also compute the expansion of the free energies at the orbifold point and the conifold locus. We will compute the quantum corrections order by order on $\hbar$, by deriving second order differential operators, which act on the classical periods.

Pseudotoric structures on a hyperplane section of a toric manifold

We continue to investigate pseudotoric geometry. Namely, we present a type of nontoric manifolds with pseudotoric structures. As an application, we construct a new pseudotoric structure on two-dimensional complex quadrics.

Toric Kähler–Einstein Metrics and Convex Compact Polytopes

We show that any compact convex simple lattice polytope is the moment polytope of a Kahler–Einstein orbifold, unique up to orbifold covering and homothety. We extend the Wang–Zhu Theorem (Wang and Zhu in Adv Math 188:47–103, 2004) giving the existence of a Kahler–Ricci soliton on any toric monotone manifold on any compact convex simple labeled polytope satisfying the combinatoric condition corresponding to monotonicity. We obtain that any compact convex simple polytope Open image in new window admits a set of inward normals, unique up to dilatation, such that there exists a symplectic potential satisfying the Guillemin boundary condition (with respect to these normals) and the Kahler–Einstein equation on Open image in new window. We interpret our result in terms of existence of singular Kahler–Einstein metrics on toric manifolds.

The partial quotients of Moment-angle manifolds over a polygon

In this paper, we generalize the conception of characteristic function in toric topology and construct many new smooth manifolds by using it. As an application, we classify the Moment-Angle manifolds and the partial quotients manifolds of them over a polygon. In the appendix we give a simple new proof for Orlik-Raymond’s theorem in terms of characteristic function which gives the classification for quasitoric manifolds of dimension 4.